The Mollification Method and the Numerical Solution of Ill-Posed Problems

o^t
NUMERICAL IMPLEMENTATIONS

a piecewise interpolant of f (t)

at the grid points f,. H e r e ,

m

Ψ ο Κ

39

fl,

if 0 < ί < A / 2 ,

\θ,

otherwise,

9 ] K

(1,

if


\o,

otherwise,

and

\ 0,

otherwise,

Once the data function has been extended and the radius of mollification selected—as indicated in Sec. 2.2—we evaluate (Ps*P )(ti)

= E/ ,*(Pa*
m

m

k k = /3 ,,., ι « 0 , 1 , . . . , AT, with the weights a , computed exactly. Recalling the Lipschitz property of f and the fact that p constant interpolant of f at the grid points, we have

(2.36)

m>a

s

ky

m

m

is a piecewise

m

H/m - J » J k / -

max | / ( / ) -p (t) m

m

\ < Lh

or f (t)=p (t)+0(h). m

m

Hence, (Pa * / J ( 0

= (P*

+ 0(A).

0 <; t < 1,

(2.37)

and the smoothness of ( p * p X f ) , repeating the previous argument, implies that 6

m

Ν

Σ /3 , .,
( =0

m

+ O(h),

m

5

O n the other hand, the smoothness of ( p * f y s

(Pa*U(0=

L(Pa*U(ti)

i=0

m

0 <; ί < 1.

ensures 0 < t < 1,

(2.38)

40

ABEL'S INTEGRAL EQUATION

and also

(PA*/J'C) = ^ [ ( P A * / J ( ' , , ) -

(Ps*L)(t -i)]+0(h ), 2

+

i

i=

1,2,...,

j - 1, (2.39)

and

(PA*/J'('Y)

^[(PA*/J(0)

=

~

(P«

* / * . ) ( > , - . ) ]

+

/=

1,2,...,

Ν.

From (2.38), we have

(PS*/J'C)

=

Σ(Ρ**/

)'(Ί>/(0

Μ

ο

<

+ 0 ( A ) ,

r <

=

i , 2 , . . . , i v .

Using expressions (2.37) and (2.39) in the last equality,

(PA*/J'C) =

l[(p *U(t )

-

0

s

( P W J C J W O

ι +

Σ Z

[(p */»J(i i) e

i+

(PE*PM)('/-L)]
" z=0

+ ^[(PS*/J(>,) - (PE*/J(I>-i)]^(0 + 0 ( A ) , o
o,i,...,7v.

Inserting this expression into (2.11) and integrating exactly the factors

w'i =

/ V(0('y - 0 "

1 / 2

o

i,) = 0,1,...,

N,

(2.40)

we get *!(',•)=**.*(';)

+

0

( ^ ) .

y= 0,l,...,JV,

(2.41)

NUMERICAL IMPLEMENTATIONS

41

where

g»M

= o, ι

>-

(

1

+ [/8«, ,/-/ «,«.;-i] /}' 3

7 = 0,1,...,Λ,,

w

e

(2.42)

taking into account (2.36). T h e approximate inverse Abel transform is computed with formula (2.42) after evaluating the factors defined in (2.36) and (2.40). Combining estimate (2.41) with the restriction to the grid points of t h e error estimate (2.12) and using the triangle inequality, we have the following theorem. Theorem 2.4 (Discrete Error Estimate) U n d e r the conditions of T h e o r e m 2.2, if f (t) is uniformly Lipschitz on / = [0,1] with Lipschitz constant L and if the approximate solution g£ (fy) is computed using (2.42), then m

)A

\8 ~ SlhlUi

^

2

Μ

3δ

2

+

ε

+ C(L,5,e)y77,

\TT ο

where

C(L,S,e)

=

CIL,

d

i = 1,2,3. oo,/

oo,/

T h e error estimate for the discrete case is obtained by adding t h e global truncation error to the error estimate of the nondiscrete case. Method II T h e convolution ρ * k requires an extension of the singular kernel k for values of t less than or equal to 0. In our numerical implementation we use the following symmetric extension: μ

* ( 0 ) = -JL,

k(t)=k(-t),t<0.

T h e discrete approximation is now straightforward.

42

ABEL'S INTEGRAL EQUATION

Writing s, = (ρ * kXtj), j = 0,l,...,N, t h e discrete convolution formulas corresponding t o (2.13) a r e as follows with £*,/,(*,) indicating t h e approximate inverse Abel transform at t h e grid points. μ

*:.*(o) = o,

ti.hi'j)

1 = T I

> Σ Sj-k(fm,k

+i

J = 1,2,...,

Ν - 1,

and 1

ΞμΜ) =glhO -l)

+

N

Soifm.N-fm.N-l)7Γ

Method III In this case, given t

jt

j = 0 , 1 , . . . , N, we define

PmiO = Lfm,i4>M>

0
i=0

a piecewise linear interpolant of f (t) m

ΦοΟ) Φ,(')

at t h e grid points f . H e r e , ;

1 - t/h,

0 ^ t < h,

0,

otherwise,

1 + ( f - tj)/h,

t_

0,

otherwise,

j

< t < tj,

1

and (1 + (t - t,)/h, *,·(')=

ί,_,<ί<ί„

1 - ( / - O A

f, < f <

(θ,

+

otherwise, i = 1 , 2 , . . . , j - 1, 1 <j


T h e q u a d r a t u r e formula is obtained by directly convolving t h e kernel function H against p as follows: y

m

sl {t ) h

j

= =

HH * ){t ) y

ττ 1 rt,

o 1 L J

TT

Pm

j

-f H {t -s)p {s)ds i

y

j

m

NUMERICAL IMPLEMENTATIONS

43

where

is evaluated exactly for i = 0 , 1 , . . . , ; ' . T h u s , we have

*;,*(o) = o, J

1

(2.43)

g'yM'-LLAM* 1 7

ι,2,...,λγ.

; =

«=o

F r o m (2.18) and (2.43), it follows that

k U O

~g'y{t,)\-

^|/ '^r('y 0

*

-Pjk// V (iy - ')|*· Q

r

(2-44)

F r o m the definition (2.18) of the kernel function H , we immediately obtain y

{\HAtj •Ό

- s)\ds

= 2γ-'/

2

- ί" / 1

< 2y"

2

1 / 2

Recalling the Lipschitz property of f and the fact that p interpolant of f at the grid points, we have m

m

.

(2.45) is a piecewise

m

max \f {t)-p (t)\ m

m

=

max

{

max

\f {t)-p (f)|> m

< Lh.

m

(2.46) Using the estimates (2.45) and (2.46) in inequality (2.44), we get

k ( 0 ) - s U 0 ) M ^ ^ >

; = 0,1,...,W.

Combining this u p p e r b o u n d with the restriction to the grid points of t h e error estimate of T h e o r e m 2.3, we have the following proposition. Theorem 2.5 (Discrete Error Estimate) U n d e r the conditions of T h e o r e m 2.3, if f (t) is uniformly Lipschitz on / = [0,1] with Lipschitz constant L and if the approximate solution of Abel's inverse problem £ γ , ( ί , ) is computed m

Λ

44

ABEL'S INTEGRAL EQUATION

using (2.43), then

m n

a

x

„ k ( ' / )

0<,j^N

-8Wj)\* j z ^ LIT

+ Ιη=(* TT y y

M

+

)

Lh

+ °(y

3 / 2

)-

T h e error estimate for the discrete case is obtained by adding the global truncation error to the error estimate of the nondiscrete case. Method IV Iterative Algorithm Discretization leads to a finite-dimensional version of the combined regularization-adjoint-conjugate gradient algorithm of Sec. 2.2. T h e operators A and A* are now represented by an Ν Χ Ν matrix A and its transpose A , respectively. T h e approximate discrete solution g "? , obtained after η iterations, the gradient VS (g ), d , r , g ® , and the residual av are now Af-dimensional real vectors. From (2.2), a simple discretization gives the lower triangular system of linear equations T

{

m

( n)

a

(

a m

n

n

( n)

m

a m

j hY

a

.

e

= f

{n)

*• L-t **j + 1 — to a, m,i J m,i' where 1

/ = 1,2

TV,

indicates the (j - 1) subdiagonal of the Ν Χ Ν matrix A. T h e discrete algorithm for the conjugate gradient method follows exactly the steps described previously in Sec. 2.2, and we only have to add the necessary stopping criteria, given by ||„(n)

l|S ,m

_

„(n-l)||

So,m

a

<

1 1 2 , / —

TOLlI *-ΉΙ 1

»o,

mII \\2,1'

where T O L is a small positive tolerance p a r a m e t e r entered by the user and

'-\NTl£o l

Uh

[fl]

is the discrete—weighted—I

{ 2 A 1 )

norm on I = [0,1],

2

Direct Algorithm T h e finite-dimensional version of the canonical equations (2.24) gives the system of Ν linear equations with Ν unknowns (A A T

+al)y

=A b, T

a

(2.48)

NUMERICAL IMPLEMENTATIONS

45

with the Ν Χ Ν matrix A and its transpose A as described previously, I the NXN identity matrix, y = (yf, y j , . . . , y £ ) , and b = ( / „ f , . . . , We assume that the vector y is the unique solution of the least squares problem T

r

2

a

A Ay=A b. T

(2.49)

T

In other words, we assume that the symmetric matrix A A is positive definite and, consequently, invertible. Moreover, the eigenvalues {A,}/Ij of A A satisfy T

0 < λ[ < λ

2

< ··· < λ

However, the matrices A A associated with discretized models of ill-posed problems are, most of the time, nearly numerically singular; the smallest eigenvalue, λ,, is " t o o close to 0." Equivalently, we say that the system (2.49) is highly ill conditioned for inversion or that small perturbations in the data vector b might produce gigantic changes in the solution vector y, or that the condition n u m b e r of the matrix A A is very large. T h e condition n u m b e r of a matrix depends on the norm being used and for symmetric matrices it is common to consider the / - n o r m condition n u m b e r given by T

T

2

W e would like to investigate how sensitive the solution of system (2.48) is to changes in the data vector b. F o r instance, if lib — b | | » , / < ε, what can we say about ||y - y J L , / ? [Notice that we always have | | y - ya<mh,i < \\y - y „ , l U , / when using the weighted I norm (2.47).] Let us begin by looking at the partitioned 2 Ν X 2 Ν symmetric eigenvalue problem m

a

a >

a

2

a

m

It follows immediately that A\

= Au

(2.50)

λν.

(2.51)

and

Hence. Α Αν τ

= λν 2

and

AA u T

= A u. 2

(2.52)

46

ABEL'S INTEGRAL EQUATION

Consider the set of orthonormal eigenvectors { u } / I ! and {v )/l [ associated with the symmetric matrices AA and A A, respectively. The vector y solution of (2.48) can be uniquely represented as a linear combination of the eigenvectors v . For example, we can write (1)

T

(/)

T

a

(i)

Ν y=

Σ

a

y, v .

y" = ( y « ) \ ) ,

a

( 0

i = 1,2,...,

n.

Thus, Ν

A Ay

Ν

= Σ y?A Av

T

= Σ

T

a

0)

i=l

ν?λ ν , 2

i=l

( 0

Ν 1= 1

and (A A T

= £ (λ + «)y<%,,

+ al)y

(2-53)

2

a

1= 1

using (2.52). T h e data vector b can also b e uniquely represented as a linear combination—this time—of the eigenvectors u , that is, (l)

Ν b =

Σ M

i =

b

i=l

b 7 u



i = 1 , 2 , . . . , iV.

Then Ab T

=

Σ Μ

Γ

ι <

1=1

0

=

(2-54)

- > M o 1=1

by equality (2.51). F r o m (2.53) and (2.54), using (2.48), it follows that (λ + α)ν« = λ Α , 2

ι = 1 , 2 , . . . , AT,

or

*•

* ^

* τττ$ ">bTa

"

h

2

"·

<"

5

)

Notice that if in (2.55) we consider a = 0, we get a characterization for the

NUMERICAL IMPLEMENTATIONS

47

solution vector y of the system (2.49). Indeed, we obtain y, = — b u r

( 1 )

,

/'= 1,2,.,.,/V.

(2.56)

T h e effect of the "regularization" can be seen very clearly now by comparing (2.55) and (2.56). If the data vector b in (2.48) is replaced by the noisy data vector b , we obtain, in the same manner, m

=-^(bjSo.

{y )i a

m

i - 1,2,...,/V.

(2.57)

(Χ ~Γ Λ:

O n c e again, we are facing a familiar situation. From (2.55) and (2.56), we have

ι

ly,-y,"

b u,,N r

a + k\

λ,

a < ^-Ι^ΙΙ ./||η || 2

(<)

=

bu T

—-,

λ,(α + λ ) 2

(n

2>/

TjllDlh,/,

which implies ly - yJL,/

<

(2.58)

-jllblU,/,

and we have consistency. In the absence of noise in the data vector, as a -* 0,

Hy-yJk/-»o.

•

F o r stability, we compare (2.55) and (2.57). W e get

\y° - (y )i\ m

= -a ^+^AX ) 2

<

^Af... —lib -

a

λ

Ν

< —ε, a

- (·>™) »(οΙ Γ

Moo, /

48

ABEL'S INTEGRAL EQUATION

or (2.59) Thus, the method is stable. For fixed a > 0, as ε -» 0, ||y - y As usual, from inequalities (2.58) and (2.59), we obtain e

a

m

I U , / -* 0.

(2.60)

a and if a = 0 ( \ / ε ) , as ε -» 0, we have convergence—in norm—with rate 0(·/ε ) . •

the

uniform

Remarks 1. If a lower uniform b o u n d — i n d e p e n d e n t of TV—is available for λ and if λ is known as a function of the dimension TV, it is possible to estimate the error between the assumed unique solution function g of the operator equation A*Ag = A*f and the TV-dimensional solution vector y given by (2.48) with the vector b replaced by the vector b . Setting a = 0(1/N) in (2.60) and selecting TV as a function of ε—minimizing the u p p e r b o u n d for the error t e r m — w e can obtain convergence as ε -» 0. [The uniform lower bound for λ = A (TV) is generally provided by the knowledge—when available—of the first eigenvalue of the operator A*A] ϊ

Ν

m

{

t

2. T h e analysis in the finite-dimensional setting mimics exactly the situation in the infinite-dimensional case. This is always the case when we discretize the continuous—nondiscrete—model. However, we should k e e p in mind that there are situations where the physical model is inherently discrete in nature and better adapted—directly—to computer analysis than its continuous approximation. 3. T h e symmetric positive definite matrix Q a family of new norms in R as follows:

= A A + al, a > 0, induces T

a

N

iiviiL = ( e « v ) v . r

It is relative to this norm that the error estimates for the iterations in the conjugate gradient method are presented. It is possible to show—see the references at the end of this c h a p t e r — t h a t

<«)

8,a,m

_

8a,

m

Q, a

I —

^ 8, a

m

8 ,m a

[*2(C)]

1 / 2

-l

Q ,l a

[*2(fi.)]

1 / 2

+

l

NUMERICAL RESULTS AND COMPARISONS

2.4

49

N U M E R I C A L RESULTS AND C O M P A R I S O N S

In this section we describe some tests that we have implemented in o r d e r to compare the performance of the methods introduced in the previous sections. W e tested the methods on three examples. In all of them, the exact data function is denoted / ( r ) and the noisy data function f U ) is obtained by adding an ε random error to / ( / ) , that is, f U ) = / ( / , ) + ε σ , where r - = jh, j = 0 , 1 , . . . , N, Nh = 1, and σ is a uniform random variable with values in [ - 1 , 1 ] such that m

m

}

;

y

;

max | / ( i , . ) - / ( i . ) | < e .

0<.j
m

;

T h e exact inverse Abel transform is denoted g(t) and its approximation—given by any of the methods—is denoted gp Oj), where ρ repre
TABLE 2.1. Error Norms as Functions of ε (Example 1 with Ν = 500)

Method I II III IV

Parameter δ μ γ α

= = = =

0.0080 0.0080 0.0040 0.0064

ε = 0.000

ε = 0.005

0.0000 0.0005 0.0302 0.0279

0.0048 0.0137 0.0315 0.0294

e = 0.010 0.0096 0.0274 0.0349 0.0359

TABLE 2.2. Error Norms as Functions of ε (Example 2 with Ν = 500)

Method I II III IV

Parameter δ μ γ α

= = = =

0.0080 0.0080 0.0010 0.0064

ε = 0.000

ε = 0.005

0.0001 0.0005 0.0174 0.0275

0.0048 0.0136 0.0263 0.0293

ε = 0.010 0.0096 0.0273 0.0431 0.0365

TABLE 2.3. Error Norms as Functions of ε (Example 3 with Ν = 500)

Method I II III IV

Parameter 5 μ γ α

= = = =

0.0080 0.0080 0.0010 0.0064

Β = 0.000

ε = 0.005

0.0052 0.0295 0.0648 0.0615

0.0052 0.0330 0.0678 0.0618

ε = 0.010 0.0053 0.0411 0.0760 0.0641

50

ABEL'S INTEGRAL EQUATION

sents the regularization parameter of the particular method; that is, ρ = δ for M e t h o d I, ρ = μ for Method II, ρ = γ for M e t h o d III, and ρ = a for Method IV. E X A M P L E 1 As a first example we consider the data function fit) = t with exact inverse Abel transform g(t) = 2/τη/ί". This data function satisfies all the necessary hypotheses for the convergence estimates of Sees. 2.2 and 2.3. EXAMPLE 2

T h e data function \2t ,

0 < t < 1/2,

2

Λ

° "

1-2(1-0'.

1/2 < t < 1,

1.20-1

-0.20

t-vaIues

Fig. 2.2 Reconstructed inverse Abel transform. Example 3, Method I: ε = 0.01, δ = 0.0080, Ν = 500.

NUMERICAL RESULTS AND COMPARISONS

51

is only once continuously differentiable on I = [0,1], partially violating the required conditions for the theoretical analysis of Sees. 2.2 and 2.3. In this example, the exact inverse Abel transform is given by '(16/3ΤΓ)/

g(t)

3 / 2

= { (16/3π)[ί / 3

, 2

0 < t <

+ (t-

l/2)

3 / 2

] - (8/ττ)(ί - 1/2)

1 / 2

1/2,

( 2 ί - 1),

1 / 2 < ί < 1. EXAMPLE 3

T h e data function is defined as follows 0 < t < 0.2,

ίο, / ( , ) = J 2 ( i - 0.2)

1/2

, 2(i - 0 . 2 )

1 / 2

0.2 < t < 0.6, - 2(f - 0 . 6 )

1 / 2

0.6 < t < 1.

,

1.20-

c

ο

Ο c

0.Β0Η

3

1~U

CD

*J 0 D L· *J (0 C

0.60 -

0.40-

Ο 0 φ

0.20H

-0.00-

-0.20

τ—ι—ι—ι—ι—!—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι 0.00

0.20

0.40 t-vo

0.60

0.Θ0

I Lies

Fig. 2.3 Reconstructed inverse Abel transform. Example 3, Method II: ε = 0.01, μ = 0.0080, Ν = 500.

52

ABEL'S INTEGRAL EQUATION

The first derivative is not continuous on / = [0,1], strongly violating the necessary hypotheses for the convergence estimates of Sees. 2.2 and 2.3. However, this reconstruction constitutes an important challenging test for the practical utilization of the methods. T h e exact inverse Abel transform is given by

8{t) =

{k

0.2 < t < 0.6, otherwise.

The four methods were t e s t e d — M e t h o d IV in its iterative version—for three different values of JV, namely Ν = 200, 500, and 1000, three different values of ε, ε = 0.000, 0.005, and 0.010, and several values of the correspond-

1.20-,

1.00 Η

0.Θ0D

1Ό

0.60-

0 3

C «-> 0 . 4 0 CO c Ο 0 Ο 0.20H

-0.00

-0.20

r—ι—ι ι ι ι ι—ι—ι—ι—ι—ι—r—ι—|—ι—ι—r—ι—| 0.00

0.20

0.40

0.60

0.Θ0

t-volues

Fig. 2.4 Reconstructed inverse Abel transform. Example 3, Method III: ε = 0.01, γ = 0.0010, Ν = 500.

NUMERICAL RESULTS AND COMPARISONS

53

1.20-1

0.Θ0Η 3

H0.60-

Ό

Φ <-> Ο 3

i>

0.40-1

- -20-|—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—Γ 0.00 0.20 0.40 0.60 0.80 t- ν β Iue s 0

Fig. 2.5 Reconstructed inverse Abel transform. Example 3, Method IV: ε = 0.01, a = 0.0064, Ν = 500.

ing regularization p a r a m e t e r s that were determined numerically—after several trials—except for Method I where the radius of mollification was selected automatically. These quasi-optimal p a r a m e t e r values are used in the actual tables and figures. Different values of ε provide a crucial test for stability. Tables 2.1 to 2.3 illustrate this point. T h e error norms in the tables are computed according to U s £Ρ,ΛΙΙ2,/ defined in (2.47). In the tables, each row corresponds to one of t h e methods with a fixed regularization p a r a m e t e r value, and shows t h e change in the error n o r m due to changes in the level of noise in the data. T h e numerical results indicate stability. T h e columns in the tables allow us to compare the performance of the different methods under similar conditions. Figures 2.2 to 2.5 show the reconstructions of the step function of Example 3 provided by the four methods for t h e same number of sample data _

a s

54

ABEL'S INTEGRAL EQUATION 0.75

-0.16

τ—ι—ι—ι—ι—ι—ι—ι t-νβIues

0.76

1.00

Fig. 2.6 Reconstructed inverse Abel transform. Example 1, Method I: ε = 0.001, δ = 0.0080, Ν = 500. points, Ν = 500, the same level of noise, ε = 0.01, and quasi-optimal regularization parameters. T h e qualitative behavior is quite good taking into consideration the high amount of noise in the data. Figures 2.6 and 2.7 show the plots of the computed solutions for Examples 1 and 2, respectively, obtained with M e t h o d I for TV = 500, ε = 0.001, and 8 = 0.008. T h e consistency and stability of the four methods are clearly confirmed throughout experimentation and very weak dependency on the p a r a m e t e r TV is observed. Method I provides an automatic mechanism to select δ = δ ( ε ) b u t — a s a consequence of the stability of the four m e t h o d s — i t is easy to find, by numerical experimentation, sharp lower and upper bounds for quasi-optimal values of the regularization parameters for Methods II, III, and IV. All the results are very competitive. However, mollification solutions are slightly better in terms of accuracy and M e t h o d I I I — t h e easiest to implement—seems to be a bit more sensitive to perturbations in the data.

EXERCISES

55

0.75-Τ

- 0 · 16 -j—ι—|—ι—ι—j—ι—ι—ι—ι—ι—ι—ι—ι—ι—|—ι—ι—|—ι—| 0.00 0.25 0.50 0.75 1.00

t- Vβ IUθ S

Fig. 2.7 Reconstructed inverse Abel transform. Example 2, Method I: e = 0.005, δ = 0.0080, Ν = 500.

2.5

EXERCISES

2.1. Show that if kit) = 1 / v7, / > 0, and kw) = (1 - i ) / 2 y w , i = / ^ T .

kit)

= k{-t),

t < 0,

then

2.2. Consider the set of canonical equations (A*A + al)g = A*f, where g and / are smooth functions in L ( 7 ) , / = [0,1], and α e R, a > 0. Show that if A*f Φ 0, then Hgjb,/ is a decreasing function of a. a

a

2

2.3. U n d e r the same conditions of Exercise 2.2, show that \\Ag increasing function of a.

a

2.4. (Choice of the regularization parameter, \\Ag

3

- f\\2,i

= ε

a, in Tikhonov's

- / | | , / is an 2

method).

(Discrepancy principle)

If (2.61)

56

ABEL'S INTEGRAL EQUATION

for a > 0, show that g is the unique solution of the problem MinllgJI .,/ over all possible functions g subject to the constraint \\Ag — /111, / ^ ε , where g is the solution of (Λ*Λ + al)g = A*f with a = a. This is a characterization of the solution g obtained using t h e parameter criterion given by (2.61). 2

s

a

a

2

3

a

5

2.5. Given the Ν Χ Ν system of linear equations Ax = b, with A = A and x Ax > 0 for χ Φ 0, the vectors { p } ^ form a conjugate basis for if they are a basis for U and ( p ) ^ 4 p = 0, V / Φ j , 1 < i,j < N. Any conjugate direction method to solve Ax = b has the form T

T

( t )

N

= 1

7

( i )

χ<* > = < * > - 0 ρ +

ifc = 0 , 1 , . . . , / V - 1,

1

χ

Λ

0 )

α ) >

x

( 0 )

= 0.

Show that _

( p

) ( ^ x ^ - b ) r

w

(Ρα)) ^ρ Γ

( λ )

2.6. (a) From the fact that an Ν Χ Ν matrix is singular if and only if its determinant is 0, one might expect that ill-conditioned matrices must have small determinants. Consider A = A and find A such that T

(1)

d e t ( ^ ) = 1, k (A)

(2)

2

= 10 . 60

(b) If 0.2161 \l.2969

0.1441 \ 0.8648]

, a

n

d

D

0.1480 \ ~ \0.8642)'

an " a p p r o x i m a t e " solution for Ax = b seems to be -

=

/ 0.9911 \ 1-0.4870Γ

In fact, the residual Ab = Ax - b = (0.00000001, - 0 . 0 0 0 0 0 0 0 1 ) However, the exact solution is

7

Explain. 2.7. Solve the canonical equations corresponding to Example 3 of Sec. 2.4 using Choleski's method to take advantage of the symmetry and positive

REFERENCES AND COMMENTS

definiteness of the Ν Χ Ν matrix A A + al. 1 0 " ' , 1 0 ~ , and Ι Ο " , and ε = 1 0 " . T

4

6

57

Consider Ν = 100, a =

3

2.8. For Ν = 10, 50, and 100, look at the eigenvalues of the matrix A in Exercise 2.7 (no need to compute anything since A is lower triangular) and compare them with the eigenvalues of A A. To find the eigenvalues of A A, use your favorite package; if you want to have some fun, program the classical method of Jacobi. Pay particular attention to the variation of the / -condition number of A A as a function of N. T

7

2

T

2.9. Program Methods I, II, III, and IV to solve Example 1 of Sec. 2.4. Try some of the parameter values shown in Table 2.1.

2.6

R E F E R E N C E S AND C O M M E N T S

T h e following references and comments serve to expand the basic material covered in the corresponding sections. 2.1. T h e solution of many practical problems is reduced to the inversion of Abel's equation by means of suitable computations. Orthogonal function approaches have b e e n used by E. L. Kosarev, The numerical solution of Abel's integral equation, U.S.S.R. Computational Math, and Math. Phys. 13 (1973), 271-277. G. N. Minerbo and Μ. E. Levy, Inversion of Abel's integral equation by means of orthogonal polynomials, SLAM J. Numer. Anal. 6 (1969), 598-616. Product integration techniques were utilized by Κ. E. Atkinson, The numerical solution of Abel integral equation by a product trapezoidal method, SLAMJ. Numer. Anal. 11 (1974), 97-101. R. F. Cameron and S. McKee, High-accuracy product integration methods for the Abel equation, and their application to a problem in scattering theory, Lntemat. J. Numer. Methods in Engrg. 19 (1983), 1527-1536. Regularization procedures have been analyzed by R. S. Anderssen, Stable procedures for the inversion of Abel's equation, J. Lnst. Math. Appl. 17 (1976), 329-342. and more recently by Κ. M. Hanson, A Bayesian approach to nonlinear inversion: Abel inversion from x-ray attenuation data, Transport Theory, Invariant Imbedding, and Integral Equations, Lecture Notes in Pure and Applied Mathematics 115, P. Nelson et al. (eds.), Marcel Dekker, New York, 1989, pp. 363-378. For a more general point of view, a systematic treatment of numerical methods for Abel's equations with an extensive and complete bibliography is

58

ABEL'S INTEGRAL EQUATION

available in R. Gorenflo and S. Vesella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics, No 1461 Springer-Verlag, Berlin; New York, 1991. See also chapter 1, sections 1.3-1.6. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985. A general review and a comprehensive treatment of R a d o n ' s transform and tomography can be found in F. Natterer, The Mathematics of Computerized Tomography, Wiley, Chichester, 1986. T h e three-dimensional image reconstruction procedure from cone-beam projections is a field of presently stimulating extensive research. See B. D. Smith, Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imaging MI-4 (1985), 14-25. 2.2-2.3. For Method I we follow closely the article D. A. Murio, Stable numerical inversion of Abel's integral equation by discrete mollification, Theoretical Aspects of Industrial Design, Proceedings in Applied Mathematics 58, D. A. Field and V. Komkov (eds.), SIAM, Philadelphia, 1992, pp. 92-104. Method II was originally introduced by K. Miller, An optimal method for the x-ray reconstruction problem, preliminary report, Amer. Math. Soc. Notices (January 1978), A-161. K. Miller, New results on reconstruction methods from x-ray projections, Lecture Notes (unpublished), Department of Mathematics, University of Firenze, December 1978, available from the author. T h e particular way of dealing with the singularity in the convolution kernel in Method HI was first reported by Β. K. P. Horn, Density reconstruction using, arbitrary ray-sampling schemes, Proc. IEEE 66 (1978), 551-562. See also D. A. Murio, D. Hinestroza, and C. E. Mejia, New stable numerical inversion of Abel's integral equation, Comput. Math. Appl. 23 (1992), 3-11. T h e generalization to three-dimensional reconstructions is available in a comprehensive review article by B. D. Smith, Cone-beam tomography: recent advances and a tutorial review, Opt. Engrg. J. 29 (1990), 524-534.

REFERENCES AND COMMENTS

59

T h e iterative form of Method IV can b e found in the paper by D. A. Murio and C. E. Mejia, Comparison of four stable methods for Abel's integral equation, Inverse Design Concepts and Optimization, G. S. Dulikravich (ed.), Washington, DC, 1991, pp. 239-252. A complete exposition of the conjugate gradient method in the infinitedimensional setting is presented in W. M. Patterson, Iterative Methods for the Solution of Linear Operator Equations in Hilbert Spaces, Lecture Notes in Mathematics 394, Springer-Verlag, Berlin; New York, 1974. Its finite-dimensional counterpart is clearly discussed in P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989. For the general theory of the Tikhonov method, see chapter 1, sections 1.3-1.6. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, London, 1984. T h e error estimate for the conjugate gradient method in terms of the induced Q norm is taken from a

J. T. King, A minimal error conjugate gradient method for ill-posed problems, /. Optim. Theory Appl. 60 (1989), 297-304. 2.4. H e r e we follow in part the paper by D . A. Murio and C. E. Mejia mentioned previously. 2.5. For an in-depth discussion of the difficulties associated with the accurate computation of matrix eigenvalues, the reader may consult—and enjoy—the book by B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

3 INVERSE HEAT CONDUCTION PROBLEM

T h e classical heat conduction problem—where the heat flux or temperature histories at the surface of the body are known as functions of time and the interior t e m p e r a t u r e distribution is then determined—is termed a direct problem. However, in many dynamic heat transfer situations, it is necessary to estimate the surface heat flux a n d / o r surface temperature histories from transient, measured temperatures inside a thermally conducting solid. This inverse heat conduction problem (IHCP) is frequently encountered, for example, in the determination of thermal constants in some quenching processes, the estimation of surface heat transfer measurements taken within the skin of a reentry vehicle, the determination of aerodynamic heating in wind tunnels and rocket nozzles, the design and development of calorimetertype instrumentation, and infrared computerized tomography. We begin with the investigation of the I H C P in its simplest geometry— associated with a one-dimensional infinite conductor—and t h e n continue with the analysis of the one-dimensional I H C P corresponding to the finite slab case. T h e much more general and challenging situations related with the two-dimensional I H C P are studied next, in C h a p . 4. In what follows, we assume all the functions involved to be L ( - 0 0 , 0 0 ) after extending them—if necessary—to the whole t axis by defining t h e functions to be 0 for t < 0. 2

3.1

ONE-DIMENSIONAL IHCP IN A SEMI-INFINITE BODY

Consider a semi-infinite slab with one-dimensional symmetry. Linear heat conduction with constant thermal properties is assumed and, after appropri60

61

ONE-DIMENSIONAL IHCP IN A SEMI-INFINITE BODY

ate changes in the space and time scales, without loss of generality, the problem is normalized by using dimensionless quantities. T h e problem can be described mathematically as follows. T h e unknown t e m p e r a t u r e u(x, t) satisfies xx

U

=

M

o

0

u(\,t)=F(t),

i

>

0

(3.1b)

> F (t),

with corresponding approximate data function u(x,0)

=0,

w(0, r ) = / ( i ) ,

(3.1a)

ί > 0,

< JC,

m

0 < x,

(3.1c)

t > 0, unknown,

(3.1d)

u(x, t) bounded as χ -* °°, t > 0,

(3.1e)

where t is time and χ is the distance measured from the heated surface. W e notice that the unknown boundary condition might be replaced by u (0, x

t) = q(t),

(3.1d')

t > 0, unknown.

T h e objective is to estimate the surface t e m p e r a t u r e fit) a n d / o r the surface heat flux q(t) given the interior measurements at χ = 1, denoted F (t). W e assume that the measured data function F (t) satisfies the L ( - ° ° , OO) error bound m

2

m

(3.2)

\F-Fj<e,

where ε is a known positive upper bound. Taking the Fourier transform of (3.1a) with respect to time, we obtain the differential equation u (x,w)

= iwu(x, w),

xx

0 < Λ : < Ο Ο , - Ο Ο < Η ; < Ο Ο ,

which has the general solution

u(x,w)

= A(w)

exp

— (1 + 2 v

ia)x '

+ B(w)

exp

νT

(

1

+

ί σ ) χ

ν - 1 , σ = sgn w. 7

After imposing the boundary conditions (3.Id) and (3.1e), the general solution reduces to

u(x, w) = B(w)

exp

(1 +

ίσ)χ

62

INVERSE HEAT CONDUCTION PROBLEM

and, in particular, for χ = 1,

F(w)

=f(w)

exp

- V

T

( i +

*V)

and

/ O )

=

F(w)

ia)

(1 +

EXP

(3.3)

It is clear from (3.3) that the I H C P — a t t e m p t i n g to go from F to /—magnifies the high-frequency component by the factor explYlvvl/2 ]. Since we have assumed that / belongs to L ( - ° ° , °°), this implies that the exact data function F is such that its high-frequency components die out faster than exp[y/\w\/2] for the product Fexp[^\w\/2] = f, and consequently / , to be in L ( - o o , oo). However, when we actually have to solve the inverse problem, instead of the ideal data function F(t) we are given a noisy, measured data function F (t) which is an L ( - o o , o o ) function with its high-frequency components not subject to such a rapidly decreasing behavior, and we cannot, in general, guarantee that the product F explY|iv|/2] will be in L ( - oo, oo). Thus, we conclude that the I H C P is a highly ill-posed problem in the high-frequency Fourier components. Notice that for the I H C P the high-frequency components of the error grow exponentially, whereas for the differentiation problem they grow linearly; that is, the I H C P is much more ill posed—from this point of view—than the differentiation process. 2

2

2

m

m

2

Remark. In this chapter we always assume zero initial temperature for the I H C P . In fact, if z(x, t) is the t e m p e r a t u r e distribution that satisfies the conditions stated in (3.1), with z(x,0) = h(x) instead of z(x, 0) = 0, we can first solve a direct problem with t e m p e r a t u r e function u(x, t) that satisfies v(0, t) = 0 and v(x,0) = h(x). Considering the t e m p e r a t u r e function u(x, t) = z(x, t) — v(x, t), it follows—by superposition—that u(x, t) satisfies all the conditions (3.1). In particular, u(x,0) = 0.

3.2

STABILIZED P R O B L E M S

T h e r e are many ways to stabilize the I H C P . In this section we discuss and compare three different approaches to the approximate solution of the I H C P that successfully restore a certain type of continuous d e p e n d e n c e on the data. All these procedures share the important property that they can be numerically implemented by means of suitable explicit finite-difference schemes marching in space and, consequently, can be applied to problems with nonconstant coefficients and to nonlinear problems. In all cases, we

STABILIZED PROBLEMS

63

study the formal stability of the method and analyze the relationship between this property a n d the behavior of the high-frequency components of the noisy data. O t h e r methods, based on the reformulation of the I H C P as a Volterra integral equation of the first kind, will be reviewed later in Sec. 3.5 when we present the numerical algorithms. O u r first method is the familiar mollification method (MM), introduced in C h a p . 1 for the regularization of the numerical differentiation process. Instead of attempting to find the point values of the t e m p e r a t u r e function / ( f ) , we lower our goal and attempt, instead, to reconstruct the δ-mollification of the function / at time t, given by J f(t) = ( p * / X f ) . T h e stability analysis for this m e t h o d follows n o w — a s usual—from the definition of J f and the mollification of (3.3). Indeed, we have s

s

s

{JJ)M

1

=

2s2

W 8

|W|

-j^

— ( 1 + ia) 2 '

exp

4

v

F(w).

-

(3.4)

If C / / X w ) denotes the Fourier transform of the mollified surface temperature function / ( f ) when the Fourier transform of the ideal data t e m p e r a t u r e Fit) is replaced in (3.3) by the Fourier transform of the noisy data function F U), we obtain s

m

m

w\

1

W e observe that the function w\ sup

exp

JJ

m

e

(1 +

L (-°°,°°) 2

(1 + ia)

-

w 2 8s 2

ΐσ)

F (w). m

(3.5)

and

w 2 8s 2

< exp

0<>v
3

'

[48 ^\ 2

(3-6)

Subtracting (3.5) from (3.4), squaring, integrating with respect to w, and using the estimate (3.6), we get 2

Jsf) '

{hfn

< ^

1

exp

26 / 2

3

\F -

F

m

(3.7)

Thus, from Parseval's equality and assumption (3.2), it follows that 1 4 δ

2/3

(3.8)

that is, the mollification method is formally stable. W e have Lipschitz

64

INVERSE HEAT CONDUCTION PROBLEM

continuous dependence on the data as ε tends to 0, provided we keep the radius of mollification, δ, fixed. • For the estimation of the error | | / - J f \\, we need to study the quantity \\f-J f\\—related with the consistency of the m e t h o d — b e c a u s e in the absence of noise in the data, we require the mollified surface temperature to be "close" to the exact t e m p e r a t u r e solution. W e know—recall T h e o r e m A.9 — t h a t 11/ - / / | | -» 0 as δ -» 0, but in order to estimate the rate of convergence, it is convenient to assume some extra smoothness on the ideal temperature solution function / . For instance, if / ' e L ( - o o , a > ) and | | / ' | | < M, we immediately have s

m

s

s

2

(3.9)

11/ - V I I

νν2δ» 2 2

/ ( H > ) | I

-

exp

1 -

iwf(w)

εχρ[-νν δ /4] 2

IW

1 -

<\\f

sup — oo<W
< Μ

1 -

sup —oo<

2

βχρ[-νν δ /4] 2

2

w εχρ[-νν δ /4] 2

2

w

w < oo

using Parseval's equality and recalling that iwf(w) = f'(w). Evaluating the right-hand side of expression (3.9) at the point w = 2.2418δ where the maximum is achieved, we get the consistency estimate \\f-JJ\\<\M8.

(3.10)

Combining this result with inequality (3.8), we obtain the following proposition. Theorem 3.1 (Error Estimate)

Suppose that | | F - F \\ < ε. Then m

(i) T h e problem 0 < x, t > 0, J u(l,t) s

J u(x,0) s

/ , « ( ( ) , r) J u(x, s

=J F„(t), t

t > 0, 0

=0, =JJ {t), m

<x,

(3.11)

t > 0, unknown,

t) b o u n d e d as χ - » oo, t > 0,

is a formally stable problem with respect to perturbations in the data.

STABILIZED PROBLEMS

(ii) If the exact boundary t e m p e r a t u r e / ( r ) is such that / ' e with II/'|| < M, then JJJt) verifies 1 I I / - /«/«II ^ -M8

1 e-j=exp

+

4 δ

65

L (-00,00) 2

2/3

(3.12)

F r o m a strictly theoretical point of view, with the p a r a m e t e r choice δ = [ 1 η ( 1 / ε ) ] ~ , for example, we get convergence. Indeed, this particular c h o i c e — n o " o p t i m a l " c h o i c e of t h e p a r a m e t e r is attempted—gives the error estimate, for small ε > 0, 3/2

2[1η(1/ε)] T h e second m e t h o d in this section—the hyperbolic m e t h o d ( H M ) — i s based on a completely different approach. T h e fundamental idea consists now on replacing the heat diffusion operator by a closely related hyperbolic one. F r o m this point of view, the procedure can be classified as a singular perturbation method. In an attempt to stabilize the I H C P , we might solve instead, the approxim a t e hyperbolic problem 1 -τυ„ c

υ= χχ

+v.,

v(l,t)=F(t),

0<*,i>0,

(3.13a)

t>0,

(3.13b)

with corresponding approximate data function

F (t), m

v(x,0)=0,

0<x,

(3.13c)

v (x,0)=0, t

0<:;c,

(3.13d)

y ( 0 , f ) =f"(t),

t > 0, unknown,

(3.13e)

v(x,

t) b o u n d e d as χ - » oo, t > 0.

(3.13f)

T h e p a r a m e t e r c, c » 1, can be interpreted as a finite thermal propagation speed. For most conduction problems its effect is negligible, although the traditional Fourier equation for heat flow implicitly assumes t h e thermal propagation speed to be infinite. Taking the Fourier transform of (3.13a) with respect to time, we obtain the differential equation v (x,w) xx

= (iw - w /c )v(x, 2

2

w),

0 < x, - 0 0 < w < 0 0 ,

66

INVERSE HEAT CONDUCTION PROBLEM

which has the general solution

' [M

A(w)

exp y - y

+ B(w) exp

I(w,c)x

I(w,c)x

where " 1/2

I(w,c)

=

1 +

\w\

w

7Γ

1/2

/ w \w\ + ia 1/ c c 4

2

Applying t h e boundary conditions (3.13e) a n d (3.13f), t h e general solution is reduced to 0(x,w)

= (// )(w)exp /

w

—

I{w,c)x

and, in particular, for χ = 1,

Π")

= ( / ? ) ( " ) exp

I(w,c)

and

(fc )(»)

=/ (w)exp

H

?

/(w,c)

For c fixed, | e x p [ V | / 2 / ( M ' , c)]| is an increasing function of M v

(3.14)

in (0,oo),

bounded above. In fact,

sup

exp

—

I(w,c)x

< exp

(3.15)

-oo<w>
Notice that for low-frequency components, I(w, c) = 1 + ia a n d the hyperbolic model approximates t h e parabolic o n e very closely.

(.f" Xw)

If denotes the Fourier transform of t h e hyperbolic surface m t e m p e r a t u r e function when t h e Fourier transform of t h e ideal data temperature F(t) is replaced in (3.14) by t h e Fourier transform of t h e noisy data function F (t), we obtain m

(f )(»>)=F ( )cxp H

c m

m W

(3.16)

STABILIZED PROBLEMS

67

O n c e again, using inequality (3.15), we see that the function f" e L ( - oo, oo) since F e L ( - oo, oo). Subtracting (3.16) from (3.14), taking norms, and using inequality (3.15) and Parseval's equality, it follows that 2

m

2

m

ll/c"-/^ll£e«p(!).

•

(3-17)

This shows that t h e hyperbolic m e t h o d is also formally stable. W e have restored continuous d e p e n d e n c e on the data as ε tends to 0, provided that we k e e p the finite thermal propagation speed, c, fixed. If / ' e L ( - o o , o o ) and M = max(||/||, | | / ' | | ) , it is possible to show—see t h e references at the end of this c h a p t e r — t h a t 2

Μ

2

"-ττ^·

(3

·

18)

This consistency estimate together with inequality (3.17) gives the following error estimate. Suppose that \\F - F \\ < ε. T h e n

Theorem 3.2 (Error Estimate)

m

(i) Problem (3.13) is a formally stable problem with respect to perturbations in the data. (ii) If t h e exact boundary t e m p e r a t u r e / ( r ) is such that / ' e L ( - o o , oo) with max(||/||, | | / ' | | ) < M, then f Jt) verifies 2

H

c

||/-/

w c

m

||< exp(^) + ^ e

^

.

(3.19)

A s with t h e mollification method, theoretically, with the choice c = 2 I n ( l / ve"), for example, we obtain convergence as e -» 0. For small e, evaluating the u p p e r bound (3.19) at c, we have

\\f-f "J<2^M s

ye

+ νε".

Remark A closer look at the formal stability analysis of both the mollification and t h e hyperbolic methods reveals that, for the former, according to (3.5),

68

INVERSE HEAT CONDUCTION PROBLEM

showing that the high-frequency components of F the kernel function

are effectively filtered;

m

belongs to L ( - ° ° , oo) and the stability bound (3.8) is not sharp at all for high-frequency components. O n the other hand, for the hyperbolic method, from (3.16), we have 2

and

T h e kernel function now allows all the frequencies to enter. T h e stability estimate (3.17) is now sharp for high-frequency components; they are amplified by the factor = e x p ( c / 2 ) . T h e third method in this section is motivated from the realization that it is possible to design a procedure "in b e t w e e n " the two previous methods. This method—appropriately named the intermediate method (IM)—should be stable, barely filter (not necessarily eliminate), and certainly not amplify the high-frequency components of the measured data function. Exploiting the powerful idea of singular perturbation, we propose to replace the heat diffusion equation by the approximating partial differential equation ΖXX

=

ζ,

-

0 <x,

tlx'

z(l,t)=F(t),

t > 0,

with corresponding approximate data function z(x,0)

t > 0,

=0,

z ( 0 , t) = fy{t),

0

(3.20a) (3.20b)

F (t), m

<x,

t > 0, unknown

(3.20c) (3.20d)

z(x,

t) bounded as χ - > oo, t > 0,

(3.20e)

z(x,

t) bounded as t - > oo, χ > 0.

(3.20f)

Condition (3.20f) is imposed to ensure the well-posedness of system (3.20)

STABILIZED PROBLEMS

69

considered as a Cauchy problem in the time variable with z(0, i ) known and condition (3.20b) deleted. In dimensionless quantities, if q(x,t) and τ(χ, t) represent, respectively, the heat flux per unit area and the t e m p e r a t u r e in a solid, the first law of thermodynamics for the one-dimensional heat flow is given by (3.21)

q + r, = 0, x

and the Fourier equation of thermal conductivity for heat flow states m a t h e matically that (3.22)

q + τ = 0.

T h e contribution of these two equations forms the governing parabolic partial differential equation for transient heat conduction (3.1a). Physically, the model equation (3.20a) can be obtained by combining (3.21) (with τ = z ) and the hypothetical—experimentally not established—Fourier law y z„ + q + z

(3.23)

= 0,

2

x

where y , y - « 1, is a nonnegative constant and γ can b e interpreted as a relaxation time parameter. Taking t h e Fourier transform of (3.20a) with respect to time, we obtain t h e differential equation 2

z (x,w) xx

- w y z (x,w) 2

- iwz(x,w)

2

x

0 < x, - o o < w < oo,

= 0,

whose general solution is given by

z(x,

w) = A(w)

— γ ν ν + K(w,y)

exp

+ B(w)

2

exp

)x

2

-y w 2

+

2

K{w,y)\x

where K{w,y)

= 2- /2J[( 8 ,s + 16* ) 3

2

r

w

+ ia[(y w*+ s

1/2

1 / 2

+ yV ] ' 4

16»v ) 2

1 / 2

1

-y w ] 4

4

1 / 2

}.

Applying the boundary conditions (3.20d) and (3.20e), t h e general solution

70

INVERSE HEAT CONDUCTION PROBLEM

becomes 1 Z ( * , H O

=

γ w

- K(w,

z

exp

γ)\x

and, in particular, for χ = 1,

Η»)

( / ^ ( n O e x p -y w 2

=

-

2

K(w,y)

and (f!){ ) W

-y H> +

=F(.v)exp

For each y, the function G (w) m u m at the point

2

= |exp[- | γ ι ν 2

y

tf(w,y)

2

2

+ K(w,y)]\

(3.24) attains its maxi-

1/6

9yT7 - 37 \

s

0.54768γ~

4 / 3

(3.25)

and G (w)

lim

= 1.

7

(3.26)

|W|—»oo

Furthermore, exp - - y w

sup

z

+

2

K(w,y)

-oo<W
1 exp

yw 2

<exp|--y

2

2

/

+

K(w ,y) 0

(3.27)

3

Notice that for low-frequency components, t h e intermediate m e t h o d behaves like the original parabolic problem since

exp[K(w,y)]

s exp

y

(l + «r)

If (f', )(w) denotes the Fourier transform of the unknown surface temperature for the intermediate m e t h o d when the Fourier transform of the ideal m

STABILIZED PROBLEMS

71

data t e m p e r a t u r e F(t) is replaced in (3.24) by the Fourier transform of t h e noisy data function F (t), we have m

1

+

yw 2

2

K(w,y)

(3.28)

It follows immediately, using inequality (3.27), that t h e function f L ( - o o , oo). F r o m (3.28) and (3.24) it follows, in the usual manner, that

ym

2

| | /

Y

' - /

y

' , J | <

E

e x p ( p -

2

/

3

(3.29)

) .

This shows that t h e new singular perturbation method is formally stable. W e have restored continuous dependence on t h e data as ε tends to 0 provided we k e e p t h e time relaxation parameter, γ, fixed. W e observe that t h e stability bound (3.29) for t h e intermediate m e t h o d is t h e exact counterpart of t h e one corresponding to t h e mollification m e t h o d after interchanging t h e parameters δ and γ [see (3.8)]. Also, we notice that for t h e new method, t h e kernel function

exp

-]p w 2

2

+K{w,y)

as

\w\

and t h e stability bound (3.29) is not sharp for the high-frequency components. In fact, these frequencies are barely amplified, according to (3.26). In this respect, t h e intermediate method is " i n s e r t e d " between t h e mollification method—which eliminates t h e high-frequency components of t h e noisy data function—and t h e hyperbolic method—which amplifies those frequencies by a factor = e x p ( c / 2 ) W e shall see in Sec. 3.6 that t h e particular control of the high-frequency components introduced by t h e intermediate m e t h o d is sufficient for t h e successful implementation of a conditionally stable, fully explicit, finite-difference space marching scheme. If / ' e L ( - o o , o o ) and Μ = maxfll/H, l l / ' l l ) , it is possible to s h o w — s e e t h e 2

references at t h e end of this c h a p t e r — t h a t | | /

/ y

- / | | < ( y + 2

/ v

r")M =

0( /r"). v

(3.30)

Theoretically, t h e choice y = [ l n ( l / ε ) ] , for example, implies convergence as ε -» 0. From (3.29) and (3.30) plus t h e triangle inequality, we have 3 / 2

M

l l / - # J I < 3

LNL-

ε

l

- 3 / 4

Μ + νε".

For completeness, we state t h e following proposition.

(3.31)

72

INVERSE HEAT CONDUCTION PROBLEM

Theorem 3.3 (Error Estimate)

Suppose that \\F - F \\ < ε. T h e n m

(i) Problem (3.20) is a formally stable problem with respect to perturbations in t h e data. (ii) If the exact boundary t e m p e r a t u r e / ( F ) is such that / ' e with maxfll/H, l l / ' l l ) < M, then f verifies inequality (3.31).

L

2

( - « > ,

oo)

l

y

3.3

ONE-DIMENSIONAL I H C P W I T H F I N I T E SLAB SYMMETRY

T h e most striking and challenging difference in going from the I H C P in the semi-infinite slab to the I H C P in the finite slab is that now we have to deal with a system of two equations with two unknowns instead of just one equation with one unknown. W e consider a one-dimensional I H C P in which t h e t e m p e r a t u r e and the heat flux histories / ( F ) and qit) on the left surface χ = 0 are desired and unknown, and the t e m p e r a t u r e and heat flux histories Fit) and Q(t), respectively, are approximately measured at the right surface χ = 1. In dimensionless quantities, the normalized linear I H C P can b e described mathematically as follows: The unknown t e m p e r a t u r e uix, t) satisfies u

0 < JC < 1, f > 0,

(3.32a)

F>0,

(3.32b)

= u,,

xx

U(L,F)

=

F(t),

F (t),

with corresponding approximate data function u (l,t) x

=Q(t),

m

F>0, Q it),

with corresponding approximate data function U(0,F)=/(0,

(3.32c)

m

F>0,

(3.32d)

t h e desired but unknown t e m p e r a t u r e function, «,(0,0=9(0.

i

>

0

>

( -32e) 3

the desired but unknown heat flux function, M ( * , 0 ) = 0,

0 < X < 1 .

(3.32f)

It is now quite natural to assume that the exact extended data functions Fit) and Qit) and the extended measured data functions F (t) and Q it) satisfy m

m

73

ONE-DIMENSIONAL IHCP WITH FINITE SLAB SYMMETRY

the L error bounds 2

\\F

-

F J I

<

| | β - QJ

and

ε

< ε.

(3.33)

T h e classical—well-posed—direct problems associated with the finite slab geometry are of course obtained by specifying one of the functions F or Q at the surface χ = 1 and one of the functions / or q at the surface χ = 0. Applying a Fourier transformation with respect to time to (3.32a), we obtain u (x,w)

= iwu(x, w),

xx

0 < χ < 1, - o o < w <

(3.34)

Expression (3.34) is a second-order differential equation which has the general solution

u(x, w) = A(w)

— (1 + i a ) x

exp

+ B(w)

exp - V y

(1 + i
where σ = sgn w. With a = y V l / 2 and j8 = 1 + ί'σ, differentiation of (3.35) with respect to χ gives u {x,w)

= αβΑε

αβχ

x

-

αβΒε-" . βχ

(3.36)

W h e n JC = 0, it follows from expressions (3.35) and (3.36), using conditions (3.32d) and (3.32e), that f(w)

=A(w)

+

B(w)

and q(w)

= αβ(Α(\ν)

-

B(w)).

Solving this system for the functions A(w) and B(w), we get
1 2αβ\αβ

B(w)

{

1

αβ

/(*)

- 1 \4(w)

(3.37)

O n the other hand, according to (3.35) and (3.36), for χ = 1, using conditions (3.32b) and (3.32c), we have ,-αβ

F(w) Q( ) w

J

,αβε αβ

_ αβε-

αβ

\A{w) ]\B(w)

74

INVERSE HEAT CONDUCTION PROBLEM

Combining this last equality with (3.37), it follows that F{w) I

Q{w)

-αβ

1

_

2αβ\ βε

/(")

αβ

-αββ'^ \αβ

αβ

α

Thus, ι F(w)

cosh αβ

\ I

— sinh αβ αβ

[ αβ sinh αβ

cosh αβ

\

(3.38) \q(w)j

J

and \

ι

/(")

1

cosh α β

\q(w)

\ /

sinh α β

αβ cosh αβ

[ -αβ sinh αβ

F(w)

(3.39)

\QM)

The transformations (3.38) and (3.39) clearly illustrate the symmetry of the I H C P in the finite slab. Moreover, since |sinhaj8| = |sinh(yVl/2(l + ϊσ))\ - » OO as Μ -» OO and |cosh αβ\ = Icoshiyriv/^(1 -I- ι'σ))| -> OO as \w\ - » OO, we see that solving the inverse problem (3.32)—obtaining /(f) and q(t) from Fit) and Q(t)—amplifies the error in a high-frequency component by the factor e x p ( y V l / 2 ) , showing that the I H C P is highly ill posed in the highfrequency components. The mollification method stabilizes this problem by attempting to reconstruct the δ-mollifications J f and J q of the functions / and q, respectively, at time t. Mollifying system (3.32) with respect to f, we obtain the following associated problem. Attempt to find J f (t) = J u(0, t) and J q U) = J u (0, t) at some time / of interest and for some radius 8 > 0, given that J u(x,t) satisfies s

s m

s

s

s

s

m

x

s

0 < χ < 1, t > 0, / u(l,i) e

=J F (t),

t > 0,

=J Q (t),

f > 0,

s

J u {\,t) 8

x

J u(x,0) s

m

s

= 0,

m

0 <x < 1.

This problem and its solutions satisfy the following proposition.

(3.40)

75

ONE-DIMENSIONAL IHCP WITH FINITE SLAB SYMMETRY

Assume that \\F - F \\ < ε and \\Q - Q \\ <

Theorem 3.4 (Error Estimate) ε. T h e n

m

m

(i) Problem (3.40) is a formally stable problem with respect to perturbations in t h e data. (ii) If t h e exact boundary t e m p e r a t u r e f(t) a n d heat flux q(t) have uniformly bounded first derivatives on some finite interval / = [0, T], that is, maxfll/'H/, Wq'Wi) < M, then J f U) a n d J q (0 verify s m

s

Wf-J fJ,<

1 -MS + e e x p

H<7 -Jalmh

1 ^ - M S + 2eexp

s

(25)

m

2/3

and

(2δ)

2/3

Proof (i) Repeating t h e same procedure as t h e o n e used to obtain (3.39) a n d taking into account t h e definitions of J f and J q, we have s

(hf)-(hf )

*

m

s

e-^^cosh

2π

« -

*

2

/2ττ

2

/ 4

oj8(/-/ ) m

J - s i n h a / 3v ( ! 2 - ( 2 ; ) m

αβ

(3.41)

and (J q) 3

-

(J q ) s

α

Ι2π

+

β(Ρ-Ρ )

β -» δ /4

sinh

α

e-» /*

cosh

αβ(ύ-ύ„,) .

2

m

2

ε

2s2

/2ττ

η

W e notice that /

, Icosh αβ\ < — e x p y — 1 + exp M

1

\ 1

r—

| a / 3 s i n h a / 3 | < —\\w\ exp

[~\ Γ / \w\ ~ V

+ 1 .

τ

\

(3.42)

76

INVERSE HEAT CONDUCTION PROBLEM

and 1/2

αβ

sinh α β < < ,

(cosh yj2\w\ - cos v 2 M ) /

\ n\w\

'

x

Replacing these estimates in (3.41), we get w\ hf)

~ (hfm) \ ^ -;==

max (exp

νπ

2

2

\F-F„

2 w\

max (exp

- p = -

+

w8

w8 2

2

W\Q-Q„

2

M > O

ι re exp-

Ι2π '

4(2δ)

2 / 3

<

6

(2δ)

2 / 3

and by Parseval's equality, we obtain 1

(3.

(28) Similarly, using the previous estimates in (3.42), we have w\

h i ) ~ (hln) \ Ζ ^ = max j-yitvlexp /2ir

ε

| |AO W

I2

I

/ 2 T T IwlsO max (exp

w\

w8

2

4

2

2

w8 2

max < exp 1 + y/2\w\ !2π \w\>a

+

w\ /2TT

max <exp

exp 1 +

+

Ι2π

< 2ε exp

2

M>0

exp

(2δ)

4(2S)

2/3

3 4(25)

2/3 •

2/3

2

w8 2

2

w8 2

2

FINITE-DIFFERENCE APPROXIMATIONS

77

and by Parseval's equality, 1

\\J q-J qJ<2eexp—-jyj. s

a

(3.44)

(26)

Inequalities (3.43) and (3.44) indicate that for a fixed δ > 0, the errors tend to 0 as ε -» 0. This shows that the mollified inverse problem (3.40) is formally stable with respect to perturbations in the data, (ii) Write /(')

-hMt)

°(<)

-J Qm(t)

= /(')

-JafO)

+ hf{t)

-JJM)

-JBQO)

+J q(0

-J»Q {t).

and

S

=Q(')

s

m

Taking norms, using the estimates (3.10), (3.43), (3.44), and the triangle inequality, we have

W

- hfJi

<\Μδ

+ ε e x p - ^ j , ()

L

2δ

ι

lUaQ -JaQmWi £ τΜ8

+ 2ε exp

z

ι

3

·

2 / 3

( °) 2

< · »

•

45

T h e space marching finite-difference implementation of the methods introduced in the last two sections is described next.

3.4

FINITE-DIFFERENCE APPROXIMATIONS

Finite Slab Geometry Mollification Method In this section we consider the approximate solution of the mollified problem (3.40) by means of finite differences in detail. Without loss of generality, we seek to reconstruct the unknown boundary temperature J f and heat flux J q in the time interval / = [0,1]. W e consider a uniform grid in the (x, t) plane: {(x = sh, tj = jk), s = 0 , 1 , . . . , N; Nh = 1; j = 0 , 1 , . . . , M ; Mk = L}, where L depends on h and A: in a way to be specified later, L > 1. W e first notice that the differential equation J u = (J u) is being treated as a second-order ordinary differential equation when marching— a

m

s

m

s

s

xx

s

t

78

INVERSE HEAT CONDUCTION PROBLEM

backward—in space. This initial value problem—technically a Cauchy problem in the setting of partial differential e q u a t i o n s — n e e d s two initial conditions at χ = 1, given by t h e approximate knowledge of t h e mollified temperature data function J ii(l, t) = J F (t) and the mollified heat flux J u (l, t) a

a

m

a

x

Thus, we consider the mollified problem 0 < x < 1,

J&xx J u{\,t)

= W ) .

t > 0,

J u {\,t)

=

t > 0,

J u(x,0)

= o,

a

a

x

s

J Q (0, s

m

0 <x

J u(0,t)

0,

(3.46)

< 1,

unknown, t > 0,

a

J u (0,t) a

ί>

unknown, t > 0.

= 4>

x

Introducing the new variables ν = J u equivalent to

and w = dv/dx,

s

dw

— dx

dv = — dt

0 <

'

dv W

= — dx

v(x,0)

0

'

X <

< x

<

0 <

t,

= hQm>

0 <

t,

= 0,

0

< x

1, 0 <

t

1,0 <

t

system (3.46) is

(3.47)

<

1,

0 < t, unknown,

v(0,t)

0 < t, unknown.

" ( 1 , 0 = S J

With the grid functions V and W respectively defined by Vj = v(x ,

tj)

s

WJ = w(x ,

and

t ),

5

}

0 < s < N,0

< j < M,

we notice that Κ = Wtj),

= J Q (tj), a

m

0 <j

< M,

and V ° = 0, s

0 < ί < Ν.

Approximating the partial differential equations in (3.46) with the consistent

FINITE-DIFFERENCE APPROXIMATIONS

79

finite-difference schemes

wj-wL, h

vj -vj+ l

2k

and

yi - yi_ W*_ =

,

s

x

,

Ν >s > 1, 1 < ; <M-

1,

we obtain a local truncation error that behaves like 0(h + k ) for t h e discrete mollified t e m p e r a t u r e and 0(h + k ) for t h e discrete mollified heat flux as h, k -> 0. T h e discrete system associated with (3.47) is now given by 2

2

2

v>_ = v > - h w u x

s = Ν, Ν — 1,... ,1, j = 1,2,..., V}< = JsF (ik),

j=l,2,...,M,

Wa = J Q (jk),

; = 1,2,..., M,

^

5 = 0,1,...,JV.

m

s

m

= 0,

Μ — Ν + s — 1, (3.48)

Notice that, as we march backward in space, at each step we must drop the estimation of t h e interior t e m p e r a t u r e from the highest previous point in time. Since we want to evaluate {V^} and {W$ at t h e grid points of / = [0,1] after N iterations, t h e minimum initial length of t h e data sample interval, L = Mk, has to be decided. T h a t is, L needs to satisfy t h e condition Mk-Nk = 1. H e n c e , L = Mk = 1 + k/h. Next, in order to analyze t h e numerical stability of system (3.48), assuming that t h e equations hold in the entire discrete plane (x, t), after applying t h e discrete Fourier transform—recall Definition A.16—system (3.48) becomes

^ - i ( w ) = W {w) s

- i1

urv(k )V ( ), W

s W

(3.49) M")

= {Js~Fm)(w),

W (w)

=

N

(JslQM,

given,

given, s = N,N-

1,...,1,0 < Μ < ^ . k

INVERSE HEAT CONDUCTION PROBLEM

80

Equivalently, ι 1 + i — sin(»vA:) Κ

K-l

—i—

\

\

—h

sm(wk) s =N,N-

1,0 s; \w\ <

1

π

Hence,

\V _ \ < | l + h + y S

X

s i n ( > | * ) J max(|Fj,\W \) S

and \W _ \ < | l + - s i n ( M * ) J max(|K,|, | ^ | ) . S

X

Thus, < ( l + h + £ s i n ( M * ) ) max(|Kj, | ^ | )

maxfll/.,!, and

max(|K |, | ^ | ) < J l + h + £ s i n ( M * ) ) 0

0

< (l + h + - sin(M*)) h

Consequently, we can write

K\ ^ 2

1 + Λ + - sin(M*)) h

and

|H> | < | ( l + h + - s i n ( M * ) ) 2

H

0

p

s

j J

max(|K„|, \W \) N

\p Ujj.

max(|p /j,

B

e

I p / f l / J + IOJ ) 2

2

|p | (|/j 2

e

2

+

I g J

2

) .

(3.50a)

(3.50b)

Recalling that the relationship between the continuous Fourier transform and the discrete Fourier transform p is given by Poisson's summation 8

FINITE-DIFFERENCE APPROXIMATIONS

81

formula Ps(w)

Σ

=

/ p \w

2TT\

TT

+ j—J,

0 < k | < - ,

(w + Ijtr/k)

8

s

we have

|p (w)| ^

Σ

]/2lr

8

exp wS 2

4ΪΤΤ

exp

2

Σ

exp

w8 2

4

ί

w8 {ϊττ~

2

4 νν δ 2

2

< 4 exp

1),π δ } 2

2

P 1

1 -

εχρ(-π δ /* ) 2

2

2

1 1 + 21 - exp(-ττ )

2

exp * δ

{(> e x

J"=I

1 + 2

2

2

Σ

1 + 2

2

A:

2

2

exp

(wAyir + / π ) δ

2

4

exp

2

2

after utilizing t h e inequality 8 >k. Using this last estimate in (3.50a) and (3.50b), integrating with respect to w, and taking square roots, we get max(||K || , ||tf || ) < 4 0

2

0

2

sup

j(l

OSMUIR/*

\ \

+ h + \

sinflH*)] K

w 82x2 z

Xexp

< 4

sup

I exp hN

»(ll/JI + 2

hN + —

sin(Mfc)

K>

OsM £ir/k \

_ νν δ 2

Xexp

sup

<4exp[l +

< exp ι

+ M

w8

+

WQJh)

2

F j i + \\Q h) n

e- ](||F || + | | ( 2 J | ) , 2

m 2

2

(\\FJz 2

< 4

IIGJl2)

2

82

INVERSE HEAT CONDUCTION PROBLEM

provided that δ > yjlk/ir.

Thus

K II|| < 44e x p [ l + 6 - ] ( | | F | | I|K 2

00

22

m

2

+ ||QJ| )

(3.51)

2

and ||lK || <4exp[l + 5 - ] ( | | F j | + 2

0

2

by Parseval's equality.

2

(3.52)

|iej| ) 2

•

The estimates (3.51) and (3.52) show that, for a given δ > 0, the finitedifference scheme (3.48)—consistent with the differential equation J u = (J u) —is unconditionally stable and, consequently, the finite-difference solution converges to the solution of the mollified problem (3.46) by the usual arguments. s

s

xx

t

Semi-infinite Geometry W e still regard the heat equation u = u as a second-order differential equation when attempting to march in space. Consequently, we also need two initial conditions at χ = 1 but, this time, we only have one, given by the approximate knowledge of the temperature w(l, f) = F (t). However, the second data function can be easily obtained after solving the well-posed direct problem associated with the heat equation in the quarter plane χ > 1, t > 0, with initial condition u(x,0) = 0 and boundary condition « ( 1 , / ) » —hF U) f° mollification method. T h e temperature solution for the direct problem needs to be computed only at the points (1 + h, f •), j = 1,2, M , in order to estimate the discrete data function G (tj) « u ( l + h, tj)—J G Uj) for the mollification method. Alternatively, it may be possible sometimes to physically implant another thermocouple at the χ location 1 + h and directly record the temperature history values G (tj). xx

t

m

r

t n e

m

m

s

m

m

Mollification Method There are many ways to implement a space marching finite-difference scheme when the initial data are given by two temperature history measurements. H e r e we describe a simple variation of the algorithm introduced in the previous section, obtained by replacing the heat fluxes Wj_, and Wj by their respective definitions in terms of the corresponding temperatures at the nodes. The discrete system now reads 2Vj-—(Vr -Vj- )-Vj l

l

+l

v

N

η

N+l

s = N,NJ F (jk), J G (jk), 0, s

m

s

m

l,...,l,y = l,2,...,M-/V + s - 1 (3.53) j = 1,2,..., M, j= 1,2,..., M, s = 0,1,..., N.

FINITE-DIFFERENCE APPROXIMATIONS

83

T h e value of V^, j = 0 , 1 , . . . , Μ - Ν, is the calculated approximation for the exact t e m p e r a t u r e f(t) at the grid points of the active surface JC = 0. Hyperbolic Method T h e numerical implementation of the hyperbolic method is straightforward. Assuming that the t e m p e r a t u r e function v(x, t) is sufficiently smooth, using the finite-difference quotients (Vj - 2VJ + V>_ )/h , (Vj - 2VJ + V>~ )/k , and (V - VJ~ )/2k to approximate v , u,,, and v , respectively, the local truncation error behaves like 0(h + k ) as h, k -» 0. With h, k and c > 1 set to satisfy the hyperbolic stability condition h < c k , substituting the finite-difference approximations in the differential equation (5.13a)—after dropping the truncation terms—we obtain the following space marching scheme: 2

+l

x

2

J+l

+l

x

x

s

xx

2

t

2

2

h

h

2

J-i

~ 2VJ + VJ~ )

+γ

l

2

s = N,NV

N

Hi +

2

2

= -ΪΤϊ{Κ' ck

V

2

=

F (jk),

=

G (jk),

+ xr(K 2k

; + 1 s

- Vt )

+ 2VJ -

1

Vj , +l

1 , . . . , 1 , ; = 1 , 2 , . . . , M - Λ / + s - 1,

m

; = 1,2,....M,

m

V ° = o,

s = 0, ! , . . . , # + 1.

s

T h e value of j = 0 , 1 , . . . , Μ - Ν, is the computed approximation for the unknown boundary temperature. Applying the discrete Fourier transform to system (3.54), imagining the equations to hold in the entire discrete plane, we get K-i(")=2

1

+ p > p ( c o s ( w * ) - 1) + i— uu{wk)V (w)

-

s

Κ

Ί

+

1

Ί » ,

V (w)=F (w), N

m

V M

= G (w),

N+

s =

m

N,N -

1,.

ι,ο<Μ<

-,

k

or, equivalently, in matrix form, /

= A,

h

Ν 1

2

2 1 +

ck 2

2

(cos(w&) - 1)

+ i— sin(wA:)

-1

1

ο < Μ < - . k

T h e discrete stability analysis—which depends on the behavior of the eigen-

84

INVERSE HEAT CONDUCTION PROBLEM

values of the iteration matrix—is very similar to the o n e already done for the mollification method and it is left as an exercise. Intermediate Method W e approximate t h e partial differential equation (5.20a) with the consistent finite-difference scheme Zj+\ ~

2Zj +

Ζ' _ $

γ

h

2

Z /

=

+

- Z R '

1

_

y {ZJX\

-

2

2ZJ

+ Zj-\ - Zj

+l

+l

2k

2Z{

+

-

Ζ Γ

1

)

kh 2

(3.55)

s =N,N-

= 1,2,..., M,

obtaining a local truncation error that behaves as Oih + k ) as h, k -» 0, assuming that t h e partial derivatives z , z„„ a n d z are uniformly bounded in the quarter plane χ > 0, t > 0. F o r computational purposes, (3.55) should b e rewritten as 2

xxxx

h

+ 2Zi-Zi-i)-Zi

+

+l

+ 2Zi,

s = N,N - l,...,l,j Z' = F (Jk),

; = 1,2,...,M,

= G (jk),

j=l,2,...,M,

Ν+Ϊ

t

2

-Zi >

Ζ'

m

yh

2

N

2

m

m

Z° =0,

s = 0,1,...,

s

= 1,2,...,M

- Ν + s - 1,

+ 1.

(3.56)

After marching backward in space with this finite-difference scheme, t h e solution Z' , j = 0 , 1 , . . . , Μ - Ν, is then taken as t h e accepted value for t h e approximate boundary temperature history. T h e discrete Fourier transform of system (3.56) gives, in the usual manner, Q

Z (w) 0

0 < \w\ <

k'

(3.57)

INTEGRAL EQUATION APPROXIMATIONS

85

where

_ / 2/3(cos(H>)t) - 1) + 2 ( 1 - ia sin(wA:))

2/3(1 - cos(wfc)) - 1

with a = h /2k and β = y h/k . T h e details of the general discussion of the conditional stability of t h e m e t h o d are left as a challenging exercise. (See Sec. 3.7 for some special cases.) Notice t h a t — a s was t h e case for the hyperbolic m e t h o d — w e n e e d to know two interior t e m p e r a t u r e s (at χ = 1 and χ = 1 + h) in o r d e r to b e able to initiate the space marching scheme. Also notice that at each s t e p — d u e to the structure of t h e computational molecule—we have to drop the estimation of t h e t e m p e r a t u r e from t h e last highest previous point in time. 2

3.5

2

2

INTEGRAL EQUATION APPROXIMATIONS

T h e simple version of the semi-infinite I H C P presented before in this chapter —system (3.1)—is equivalent to t h e Volterra integral equation of the first kind

F( ) = //ω—4; ds, L

(3.58)

t

where φ(χ,ί)

=

e

r

f

c

( ^

T h e corresponding Volterra integral equation for the boundary heat flux history is given by 3φ(1,ΐ-5)

[ t

(3.59)

w h e r e ψ(1, t)—the t e m p e r a t u r e response at χ = 1 for a unit step rise of t h e surface heat flux at t = 0—is defined by

ψ(χ,ί)

=

\ft

(

x

fir

\

4t

2-p^exp

[0,

2

χerfci

— p - 1 ,

t > 0, t ^ 0.

86

INVERSE HEAT CONDUCTION PROBLEM

In the spirit of Chap. 2, (3.58) and (3.59) are both of the general abstract form (2.20) and, as such, can be solved by t h e regularization procedure of Tikhonov (Method IV in Chap. 2). T h e iterative algorithm (2.32) and the direct solution of the canonical equations (2.48) apply without changes, except that in this case the (j — 1) subdiagonal of the Ν Χ Ν matrix A associated with the integral equation (3.45) is given by U j

= Αφ = φ([]

+ l]k)

ί

- φ(]^,

j = \,2,...,N.

(3.60)

These entries are obtained, for example, when using t h e midpoint quadrature formula to approximate (3.58) with Nk = 1 and t = jk, j = 1 , 2 , . . . , N. O n e of the first algorithm originally designed for the numerical solution of the I H C P — t h e future temperatures method (FTM)—will be discussed now. It is based on a discretization of the integral equation (3.58) and a least squares fitting of future t e m p e r a t u r e data points to effectively stabilize the ill-posed problem. From (3.58), the exact data t e m p e r a t u r e at time t is approximately given by j;

M

Μ-i

ΣΛΔΦ*- -Λ#Δφο>

FOM)=

= 0,

/o

Β

(3.61)

n= 0

where t = Mk, Δ $ = φ([ΐ + l]k) - $(/fc), / „ = / ( ( « - \)k), and the caret on /„ denotes a previously estimated value of a t e m p e r a t u r e component. T h e only unknown in (3.61) is f , which is to b e estimated in a sequential manner. T h e t e m p e r a t u r e at time (M + j)k, j = 0 , 1 , 2 , . . . , is given by M

(

M

M-l

=

F{t ) M+j

Σ

fn *4>M-n

+j

ΔΦ; + · · '

+ f

U

+f

M + J

Δ . 0

(3.62)

n= 0

T h e temperature at time ( M + i - \)k, i = 1 , 2 , . . . , may be expressed as f

M+i

= A

0

+A,i

+ Ai

+ ••• +Α ϊ*

2

2

(3.63)

β

and the coefficient A , which is equal to f , is to be found. F o r instance, if β = 0, the t e m p e r a t u r e is constant, and if β = 1, there is a linear variation of t h e surface t e m p e r a t u r e . Assuming that the maximum value of ;' in (3.62) is equal to r - 1, then it is required that 0

S = r

M

Σ [F((M 7=0

+ j)k)

- F ((M m

+ j)k)}

2

(3.64)

INTEGRAL EQUATION APPROXIMATIONS

87

be m a d e a minimum with respect to A , A ..., Α . T o obtain a solution, r must b e at least as large as β + 1 and, in order to introduce some least squares smoothing, it is necessary that r > β + 2. Equation (3.64) uses r future temperatures to obtain the single temperature component f . Differentiation of (3.62) with respect to A gives the set of linear equations 0

v

β

M

p

r-1

Σ [F((M

+ j)k)

-F((M

+ j)k)]C , pj

ρ = 0 , 1 , . . . , β,

j-o

where C=

Σ* ΔΦί-„

Ρ = 0 , 1 , . . . , β.

ρ

pj

This set of equations is equivalent to β

Σ « * ρ ^ = 7ρ.

Ρ = 0 , 1 , . . . , β,

(3.65)

Ι-Ο

where r-1 a

i

P

=

a P

i

= Σ

i = 0 , 1 , . . . , β, ρ = 0 , 1 , . . . , β,

CijCpj,

j-o

r-1

M-l

y = LF ((M P

m

Σ

+ J)k)c pj

I

r-1

\

/ „ Σ ΔΦΜ-η+jcJ,

^o = 0.

After k, β, and r are chosen ( r > β + 2), the set of β + 1 linear equations given by (3.65) is solved for A . This solution is then taken as the accepted value for fit) over the M t h single time step only. Next, the right-hand side of (3.65) is updated, the analysis interval is shifted one time step, and the entire process is repeated. For β = 0—piecewise constant solution—there is just one linear equation with one unknown, and if β = 1—piecewise linear solution—a system of two linear equations with two unknowns is obtained. 0

Remarte 1. T h e implementation of fully explicit space marching finite-difference methods for the numerical solution of the I H C P gives maximum computational flexibility. These methods are extremely simple to program and, more importantly, they very naturally provide all the intermediate information related to the interior temperature and heat flux function values as part of the marching process itself. T h e same general information—using methods

88

INVERSE HEAT CONDUCTION PROBLEM

based on the integral representation of the I H C P — c a n be obtained only by solving the direct heat conduction problem after the unknown boundary history temperature has been recovered. However, there are several important situations—related to free boundary problems, for example—where the location of the active surface is not known! This distinct feature of space marching schemes will play an important role in some of the applications to be studied later in Chap. 5. 2. All the methods mentioned in this chapter—with the exception of the regularization procedure of Tikhonov—are sequential, in the sense that only one component of the unknown boundary t e m p e r a t u r e a n d / o r heat flux vector is produced at each step. In contrast, the iterative or direct implementation of Tikhonov's method — M e t h o d IV in Chap. 2 — a r e both, essentially, "whole d o m a i n " type of implementations. They utilize all of the data to estimate simultaneously all the components of the approximate solution vectors. Thus, if the solutions are needed over a long period of time or if the sample data frequency is very small, the dimension of the matrices and the n u m b e r of computations involved increase accordingly. From this point of view, sequential methods are much more computationally efficient than whole-domain regularization methods. 3. T h e automatic p a r a m e t e r selection criterion discussed in Sec. 3.5—which determines δ in a m a n n e r consistent with the amount of noise in the data—should be used in the numerical implementation of the mollification method if a reliable u p p e r bound, ε, for the I data error is known. 2

4. It is always possible to combine the data filtering aspect of the mollification method with any of the other sequential methods described previously in this section. This is particularly t r u e in the presence of very high noise, where the combined algorithms are, in general, substantially more stable and accurate when using small time sample steps. See some of the examples presented in Sec. 3.6.

3.6

NUMERICAL RESULTS

Semi-infinite Case In order to test the actual stability and accuracy of the methods presented in the previous sections, the approximate reconstruction of a surface temperature f(t) of value 1 between t = 0.2 and t = 0.6 and 0 at all other times is investigated. Such a curve has the difficult characteristics of an abrupt rise and an equally abrupt drop and constitutes a severe test because the algorithms anticipate changes in the solution temperature and also give delayed values. W e consider the data function F —the approximate temperature at the interior point χ — 1—to be a discrete function measured at equally spaced m

NUMERICAL RESULTS

89

sample points t , = jk, j = 0 , 1 , . . . , M, Mk = L in the interval I = [0,L], where L = 1 + k/h. W h e n needed, the data functions G and J G —the approximate t e m p e r a t u r e at the interior point χ = 1 + h and the mollified approximate t e m p e r a t u r e at χ = 1 + h—are introduced and treated in exactly the same way as F (t) after numerically solving (just one step forward in space is n e e d e d ) the corresponding well-posed direct initial value problem in the quarter plane χ > 1, t > 0, with zero initial condition and boundary condition F it) and J F U), respectively. T h e exact data t e m p e r a t u r e function is denoted Fit) and the noisy data function F {t) is obtained by adding random errors to Fit). Hence, for every grid point tj in I , )

L

m

s

m

m

m

s

m

m

L

FM=ntj)

+ *j>

where the e-'s are Gaussian random variables with variance σ T h e exact data t e m p e r a t u r e is given by Fit)

= φ(ί - 0.2) - φ(ί

2

= ε . 2

- 0.6),

where (r) = erfc T o test the numerical stability of the algorithms, we use different average perturbations for ε = 0.000, 0.003, and 0.005. If the exact true component of the surface t e m p e r a t u r e solution restricted to the grid points is denoted by fj = fit), we use the weighted I n o r m — o r sample root mean square norm —defined by (2.47), to measure the errors in the discretized interval / = [0,1]. Tables 3.1 and 3.2 show the results associated with the space marching finite-difference methods and the integral methods, respectively. In order, they are the mollification method (3.35) with h/2 = k = 0.01 and the automatic selection of 8; the hyperbolic method (3.40) with h/2 = k = 0.01 and quasi-optimal choices of c; the intermediate method with h/2 = k = 0.01 and quasi-optimal choices of γ ; the whole-domain regularization technique of Tikhonov, implemented by solving the system of canonical equations (2.48) 2

2

TABLE 3.1. Space Marching Algorithms (Error Norms as Functions of ε; k = At = 0.01, h = Ax = 0.02)

Method

Param.

Mollification

δ = k

Hyperbolic

ε = 0.000 0.0856

c = 43

0.0855

Intermediate y = 10~ 2

6

0.0869

Param.

ε = 0.003

Param.

ε = 0.005

δ = 6k

0.1673

δ = 7k

0.1903

c = 7

0.2053

c = 6

0.2230

y = 0.002 2

0.1650

γ = 0.002 2

0.1904

90

INVERSE HEAT CONDUCTION PROBLEM

TABLE 3.2. Integral Equation Algorithms (Error Norms as Functions of ε; k = At = 0.01) Method Canonical

Param. a = 5

X

1(T

= 4

Χ

10"

5

Iterative Fut.-Temp.

Param.

ε = 0.000 5

r= 5

0.1275 0.0798

a = 6 VA"

0.0698

Χ

10

ε = 0.003 - 5

= 0.004

r = 10

0.1302

Param. a = 7

X

10~

ε - 0.005 5

0.1397

0.1030

•Ja = 0.005

0.1109

0.1230

r = 10

0.1364

1.10-1

1.00 φ

L

0.90

D ^ 0.Θ0 L Φ

α 0.70 ε Φ

0.60 "Ό Φ

+j 0.50 ϋ L 0.40 (0 c

0.30

Ο Ο Φ

(γ

0.20 0.10 -0.00

-β-10 -|—ι—ι—ι—ι—|—ι—ι—ι—ι—ι—ι—ι—ι—ι—|—ι—ι—ι—ι—| 0.00

0.20

0.40

0.60

0.Θ0

t I ma Fig. 3.1 Reconstructed temperature function. Mollification method: e = 0.005, δ = 0.07, h/2 = k = 0.01.

NUMERICAL RESULTS

91

1.10-1

-0.10

t I me Fig. 3.2 Reconstructed temperature function. Hyperbolic method: e = 0.005, c = 6, h/2 = k = 0.01.

with Ν = 100 using the midpoint quadrature formula to generate the subdiagonal entries of the matrix A [given by (3.60)]; the iterative implementation with the adjoint-conjugate gradient algorithm (2.32)—see chap. 2—with k = 0.01, T O L = 1.6 Χ 1 0 " , and initial guess = 0; and the future temperatures method (3.65) with k = 0.01, constant t e m p e r a t u r e assumption (β = 0), and quasi-optimal choice of the number of future temperatures r. Convergence for the iterative scheme is achieved after 52, 22, and 22 iterations for ε = 0.000, 0.003, and 0.005, respectively. Figures 3.1 to 3.6 illustrate the qualitative behavior of the reconstructed boundary t e m p e r a t u r e function with the methods mentioned previously, in that order, for ε = 0.005 and the other parameters as shown in the tables. Figures 3.7 and 3.8 show the solution obtained with the combined mollification-hyperbolic method procedure with h/2 = k = 0.01, δ = 4k, c = 6, 5

92

INVERSE HEAT CONDUCTION PROBLEM 1.20-1

o

1.00-

L 3

*J O L Φ

0.80-

a

ε a

0.60-

a *-> o zs 0 . 4 0 L *J (0 c

o o

Œ

0.20-

-0.00

-0.20-)—r-r 0.00

t tm© Fig. 3 J Reconstructed temperature function. Intermediate method: e = 0.005, y2 = 0.002, h/2 = k = 0.01.

and ε = 0.005 and the combined mollification-future temperatures method with k = 0.01, 8 = 4k, β = 0, r = 10, and ε = 0.01, respectively. Figure 3.9 illustrates the "solution" obtained with the mollification scheme without filtering the initial data for ε = 0.003, A/2 = k = 0.01, and 5 = 0. Remarks 1. The horizontal inspection of Tables 3.1 and 3.2 confirms the numerical stability of all the methods tested in these trials with the indicated selection of the regularization parameters. A vertical reading of the tables allows for a comparison of the accuracy of the different methods. As expected for the semi-infinite case, the integral-based methods do hold a slight edge over the space marching schemes.

NUMERICAL RESULTS

93

1.20

t Ime Fig. 3.4 Reconstructed temperature function. Tikhonov's method (canonical equations): e = 0.005, a = 7 X IO"5, N = 100. 2. The I2 error norm of the combined mollification-hyperbolic method with A/2 = k = 0.01, δ = 4k, c = 6, and ε = 0.005 shown in Fig. 3.7 is approximately 0.2073. This represents a somewhat modest improvement of about 10% with respect to the original value shown in Table 3.1 for the hyperbolic method. However, it is important to notice the influence that the filtering of the initial data has on the computed solution. The incipient oscillations that we observed in Fig. 3.2—indicative of some slight numerical instability—are not present at all when using the combined algorithm. Finite Slab Case In this section we apply the mollification method—with the finite-difference implementation given by (3.48)—to two problems in the finite slab. First, in Problem 1, the approximate reconstruction of a surface temperature /(f) at

94

INVERSE HEAT CONDUCTION PROBLEM 1.20-

Φ L

1.00

Η

D

*J Ο

L· 0.80 Φ

Ο. ε Φ

0.60 -Η Φ

<-> ϋ D £_ 0 . 4 0 Η -Ι->

CO c Ο ϋ 0.20 Φ

α:

-0.00

-β.2β—(—ι—ι—I—I | ι 0.00 0.20

ι—I I | I 1 I I | I I I I | 0.40 0.60 0.80 t

ΙΓΠΘ

Fig. 3.5 Reconstructed temperature function. Tikhonov's method (adjoint conjugate gradient): ε = 0.005, να" = 0.005, Ν = 100.

χ = 0 of value 1 between t = 0.2 and t = 0.6 a n d of value 0 at other times is sought. T h e exact data for the I H C P are obtained by solving the direct problem 0 < χ < 1, t > 0, « ( 0 , i ) = 1,

t > 0,

«(1,0 = 0 ,

t > 0,

M(JC,0)

0 <x < 1.

=0,

T h e t e m p e r a t u r e solution for this problem is given by

u(x,t)

= (l-x)

+

2 7 7

" (-1) - Σ η «=i

and u(x, t) = 0 if / < 0.

exp[ - ( R T 7 R ) r ] sin ηπ(1 2

- χ),

t > 0,

NUMERICAL RESULTS

95

1.20-,

-0-2a

|

0.00

i

i

i

i

|

8.20

i

i—i—i—i

0.40

i

i

t Ime

i

i

i

0.60

i

i i

0.80

Fig. 3.6 Reconstructed temperature function. Future temperatures method: ε = 0.005, r = 10, β = 0, k = 0.01. The corresponding heat flux solution is given by ux(x,t)

= - < 1 + 2 Σ ( _ 1 ) " e x p [ - ( m r ) 2 i ] cosrc7r(l -x)\, «=i

and w x (*, i) = 0 if t < 0. Thus, the exact data functions for the IHCP are F(t)

=u(l,t)

=0

and G(f) = " * ( ! . ' - 0 . 2 )

-Μ,(1,ί-0.6).

t > 0,

96

INVERSE HEAT CONDUCTION PROBLEM

1.00-

-0.00-

-0.10 I ι ι—ι ι ι ι—ι—ι—ι ι—ι—ι—r—ι—|—ι—ι—r~ι—| 0.00

0.20

0.40

0.60

0.80

t Ime Fig. 3.7 Reconstructed temperature function. Combined mollification-hyperbolic method: ε = 0.005, δ = 4k, c = 6, h/2 = k = 0.01.

In Problem 2 we investigate the approximate reconstruction of a surface heat flux q(t) at χ = 0 for a finite slab exposed to a nonlinear heat flux of value 0 for t < 0.2 with a unit jump at t = 0.2 and satisfying the relationship q( ) = 1/(1 + 0.1/(f)) for t > 0.2. T h e exact data for the I H C P are obtained by solving the direct nonlinear problem t

( 1 + 0 . 1 u > , = ( ( 1 + 0.1u)u ) ,

0 < χ < 1, t > 0,

x x

«,(1,0=0,

i>0, 1

",(0,0 u(x,0)

(1 + 0.1M(0,0) = 0,

,

t > 0, 0 <x

< 1.

NUMERICAL RESULTS

97

-0·10-|—ι—ι—ι—ι—|—ι—ι—ι—ι—|—ι—ι—ι—ι—|—ι—ι—ι—ι—| β.ββ

0.20

0.40

0.60

0.80

t Ime Fig. 3.8 Reconstructed temperature function. Combined mollification-future temperatures method: e = 0.01, δ = 4k, r = 10, β = 0, k = 0.01.

U n d e r the transformation

v(x,t)=

f ( 1 + 0.1s) ds = u + - O . l u , 2

t h e problem in the new variable ν becomes =

v , KX

0 < χ < 1, t > 0,

« x ( < U ) = 1,

t > 0,

= 0,

t > 0,

= 0,

0 < χ < 1,

v(x,0)

(3.66)

98

INVERSE HEAT CONDUCTION PROBLEM 12.60-i

10.00O

c 3 «-> Ο

7.E

\

t-

Φ CL

II

5.00-

ε

Φ

2.50-

Ό Φ +J ϋ 3 C *J CO c ο ϋ CD

'

\

Ί ,ιΊ.'

1"l HI I,

I'l I'

0.001

-2.50H

»

"

Ί 'I!'!'''!',

IΊ ϋ 1 ι, ί

4n -5.00H

11 -7.60-H

-10.00-(—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—r—1—ι—ι 0.00

0.20

0.40

0.60

0.90

t I me Fig. 3.9 Reconstructed temperature function. Mollification method: ε = 0.003, δ = 0, h/2 = it = 0.01.

with exact temperature and heat flux solutions given respectively by v(x,t)

= -t+

v(x,t)

2 Ζ

Ϊ

7Γ

=0

^[3(1 - x )

1

- l]

" _cos ηπ(1 Σ ( - 1 ) 1

- χ) , , -exp[-(«T7) i], r

2

*>0,

n = l

(3.67)

t < 0,

if

and v {x,t) x

v (x, x

2 " „ sin η π ( 1 - χ) ι = 1 - J + [ ( - 1 ) e x p [ - ( « i r ) t\, r

:

t) = 0

if

17

n=l

/ < 0.

2

t > 0,

NUMERICAL RESULTS

99

F r o m (3.66), we readily obtain u(x, 0 = o7 [ V L + 0 . 2 ϋ ( χ , / ) - L ] .

(3.68)

In particular, the exact data functions for the nonlinear I H C P are given by = W ( L , t - 0.2) = 10(71 + 0 . 2 u ( L , f - 0.2)

F(t)

- L)

and

= «,(1,/-0.2) =0.

Q(t)

Equations (3.67) and (3.68) are also used to obtain the exact solution for t h e surface heat flux q(t) in t h e I H C P which must satisfy Β(Ί)-«,(0,Ί-0.2)=

1

1 + 0.1w(0,I -

0.2)

1 Γ Ί + 0 . 2 u ( 0 , F - 0.2)

t > 0.2,

'

and q(t)

=0

t < 0.2.

if

This function is n e e d e d for comparison purposes against our computed solution. T o numerically solve Problem 2, we rewrite the equation (1 + 0 . 1 « Κ

= ((1

+

0.1Ι0Μ,)

Χ

as

o.i(",)

u. ' and then apply t h e h

, (V

- ψι

yj-i

= yj — hWJ_

x

s

y ° = o, s

, J

2 k

[

Μ

Η

"

finite-difference

ψι ~ s

=

s

l<

+

l

• ,x - V ~) + J

s

l

}

2

1.0.1M scheme (3.48) modified as follows: 0.1/i(W/) ' 1 + 0.1K/' 2

v

s = TV, Ν - 1 , . . . , 1, j = 1 , 2 , . . . , Μ - Ν + s - 1,

s = o,i,...,/v,

with h = k = 0.01 and automatic selection of δ.

100

INVERSE HEAT CONDUCTION PROBLEM TABLE 3.3. Temperature Error Norms, Problem 1

ε

δ

Error Norm

0.000 0.003 0.005

0.01 0.03 0.04

0.0884 0.1066 0.1189

TABLE 3.4. Heat Flux Error Norms, Problem 2

ε

δ

Error Norm

0.000 0.003 0.005

0.01 0.06 0.07

0.0584 0.1208 0.1337

1.10—1

-0-10 I ι ι—ι—ι ι ι ι—ι ι ι ι ι—r—ι—| ι ι ι ι—| 0.00

0.20

0.40

0.60

0.80

t I me Fig. 3.10 Reconstructed temperature function. Mollification method—Problem 1: ε = 0.005, δ = Ak, h = k = 0.01.

EXERCISES

101

1.20

1.00

X 3

0.80 O Θ JC

0.60-

"O Φ •P Ü 3 C*J (0 e O Ü

0.40-

0.20-

a

■0.00

-0.20

-0.40

t Ime Fig. 3.11 Reconstructed heat flux function. Mollification method—Problem 2: e = 0.005, ô = 7k,h = k = 0.01.

Tables 3.3 and 3.4 show the discrete error norms—computed according to formula (2.47)—as functions of the amount of noise in the data, ε, for the computed surface temperature and surface heat flux in Problems 1 and 2, respectively. The qualitative behavior of the reconstructed surface functions are shown in Figs. 3.10 and 3.11 for the linear and nonlinear models, respectively. The computations do verify the theoretical stability properties of the mollification method and—in the case of the nonlinear example—show the versatility and good accuracy of the space marching explicit scheme even for small time sample intervals and a relative high level of noise in the data. 3.7 EXERCISES 3.1. Show the numerical stability of the hyperbolic method using formula (3.54) under the conditions h2 < c2k2 and h = k.

102

INVERSE HEAT CONDUCTION PROBLEM

3.2. Show that the hyperbolic system (3.54) is equivalent to the Volterra equation of the second kind (formally stable)

v(x,t)

Γ

X \

/

c (t-s) 2

- /c

CX

CX]

x

= /[r - -Jexpl - y j +

/(i)exp

2 21 ! / \ 2

X

[(/ - s) -

(x/cff

2

2

7

· ( τ [

(

,

-

)

2

- ( 7 )

2

]

)

d

s

'

t >

and v(x, f ) = 0,

t < -, c

where v(0, t) = f(t) and 2*

2 ^

(z/2)

2^ο

*K*+1)!

denotes the modified Bessel function of the first kind of order 1. 3.3. Show that the intermediate method is numerically stable under the special implementation h = k and γ = Λ / 2 . 2

3.4. Replace condition (2.22) by the more general statement

1 1 * * 1 1 ! , / = (Bg,Bg)

=

j\Bg{t)] dt<E . 2

2

(B represents, in general, a "smoothing" operator. For instance, we can have Bg = g", etc.) Now choose your approximation for the function g as t h e one minimizing t h e functional S (g) a

= j(\\Ag

- /111,/ + aWBgWlj),

Find the canonical equations associated with

a > 0.

S (g). a

3.5. (Combined future temperatures-regularization procedure) Describe the combined algorithm obtained by requiring to minimize—with respect t o

REFERENCES AND COMMENTS A, 0

A ,...,

Α —the

x

β

103

functional

S,,a=L[n(M+m

-F {{M+j)k)}

2

m

,' = 0 r-1 + «

Σ

(QM+J) ,

«

2

>

0.

7= 0

instead of (3.64). 3.6. Numerically recover the surface temperature function f(t) = fi in the discrete time interval / = [0,1] using the mollification method with h = k = 0.01 and ε = 0.001. T h e exact data temperature at χ = 1 is given by

F(t)

1

= V7 j e x p

νπ

/ rjteric

2VF

4f

1 τ -

.. ,

\2VS"//'

/>0,

and F(/)=0

if f < 0.

3.7. Same as in Exercise 3.6 but solving the canonical equations from Exercise 3.4 if Bg = g + g'. Use k = 0.01 and ε = 0.005. D e t e r m i n e a by the "discrepancy" criterion (2.61). 3.8. Solve Exercise 3.6 by the combined future temperatures-regularization method of Exercise 3.5 with h = k = 0.01, β = 1, r = 10, and ε = 0.005. Determine a by the discrepancy principle (2.61). 3.9. Solve Exercise 3.6 by the iterative algorithm (2.32) with k = 0.01 and ε = 0.005. Use a = 1 0 , Κ Γ , and 10 " Λ _ 2

3.8

4

REFERENCES AND COMMENTS

T h e following references and comments serve to expand the basic material covered in the corresponding sections. In the last two decades more than 300 papers have been written about the I H C P . See Appendix Β for an extensive bibliography. 3.1. A n introduction and systematic overview of the I H C P with emphasis on the future temperatures method can be found in the pioneer textbook J. V. Beck, B. F. Blackwell, and C. R. St. Clair, Inverse Heat Conduction—III Posed Problems, Wiley, New York, 1985.

104

INVERSE HEAT CONDUCTION PROBLEM

Another excellent general treatise of inverse heat conduction problems and iterative techniques is available in Ο. M. Alifanov, Inverse Heat Transfer Problems, Springer-Verlag, to appear. For a good description of several practical aspects of the I H C P , see E. C. Hensel, Inverse Theory and Applications in Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1990. 3.2. H e r e we follow in part the work of D. A. Murio and L. Guo, A stable marching finite difference algorithm for the inverse heat conduction problem with no initial filtering procedure, Comput. Math. Appl. 19 1990, 35-50. T h e mathematical concept of formal stability for the mollification method was originally introduced in P. Manselli and K. Miller, Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side, Ann. Mat. Pura Appl. 123 (1980), 161-183. The first extensive numerical experiments involving the integral representation of the mollified I H C P , introducing the approximated "inverse kernels" of the discrete operators, were performed by D. A. Murio, The mollification method and the numerical solution of an inverse heat conduction problem, SIAMJ. Sci. Statist. Comput. 2 (1981), 17-34. For a recent generalization of the mollification method as a filtering procedure in abstract Banach spaces with stability estimates of Holder type, the reader may consult Dinh Nho Hao and R. Gorenflo, A non-characteristic Cauchy problem for the heat equation, Acta Appl. Math. 24 (1991), 1-27. Dinh Nho Hao, A mollification method for ill-posed problems, Preprint A-91-35, Freie Universitat Berlin, 1991. T h e interesting suggestion of replacing the I H C P by a stable approximate hyperbolic problem is d u e to C. Weber, Analysis and solution of the ill-posed inverse heat conduction problem, Internal. J. Heat Mass Transfer 24 (1981), 1783-1792. A theoretical analysis of Weber's procedure can be found in L. Elden, Hyperbolic approximations for a Cauchy problem for the heat equation, Inverse Problems 4 (1988), 59-70.

REFERENCES AND COMMENTS

105

For an integral solution of the I H C P based on the hyperbolic approximation method, see D. A. Murio and C. Roth, An integral solution for the inverse heat conduction problem after the method of Weber, Comput. Math. Appl. 15 (1988), 39-51. C. Roth, An integral solution approach for the inverse heat conduction problem, Ph.D. Dissertation, University of Cincinnati, 1989. 3.3-3.4. T h e numerical implementation of the mollification m e t h o d in the finite slab case—as described in Sec. 3.4—is taken from L. Guo, D. A. Murio, and C. Roth, A mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry, Comput. Math. Appl. 19 (1990), 75-89. The first space marching finite-difference procedure based on a new interpretation of the mollification m e t h o d by initially replacing the noisy data function F with the filtered data function J F was introduced in m

s

m

E. C. Hensel and R. G. Hills, A space marching finite difference algorithm for the one dimensional inverse heat conduction transfer problem, ASME Paper 84-HT-48, 1984. T h e questions of numerical stability and choice of the radius of mollification for a related fully explicit space marching finite-difference method were addressed in D. A. Murio, The mollification method and the numerical solution of the inverse heat conduction problem by finite differences, Comput. Math. Appl. 17 (1989), 1385-1396. A combined space marching finite-element scheme and initial data filtering by means of Tikhonov's regularization was originally analyzed and implemented in A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math. 42 (1982), 558-574. An efficient two-step mixed procedure consisting of a weighted average of a space marching step and a time marching step was presented in M. Raynaud and J. Bransier, A new finite difference method for the nonlinear heat conduction problem, Numer. Heat Transfer 9 (1986), 27-42. Several time marching finite-difference algorithms combining the numerical solution of a direct heat diffusion problem at each time step with the future temperatures method are described in the textbook by Beck, Blackwell, and St. Clair mentioned previously.

106

INVERSE HEAT CONDUCTION PROBLEM

3.5. T h e first formulation of the future temperatures method was described in J. V. Beck, Nonlinear estimation applied to the nonlinear inverse heat conduction problem, Internal. J. Heat Mass Transfer 13 (1970), 703-716. T h e first application of the regularization technique of Tikhonov to the I H C P appeared in A. N. Tikhonov and V. B. Glasko, Methods of determining the surface temperature in a body, U.S.S.R. Computational Math, and Math. Phys. 7 (1967), 267-273. F o r a discussion of t h e interesting alternatives mollification-future temperatures procedure, see

of

the

combined

D. A. Murio and J. R. Paloschi, Combined mollification-future temperatures procedure for solution of inverse heat conduction problem, / . Comput. Appl. Math. 23 (1988), 235-244. A n exhaustive numerical testing of the combined future t e m p e r a t u r e s - r e g u larization procedure was presented in J. V. Beck and D. A. Murio, Combined function specification-regularization procedure for solution of inverse heat conduction problem, AIAA J. 24 (1986), 180-185.

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

Analytical and computational methods for treating the I H C P have b e e n basically restricted to one-dimensional models. T h e difficulties of the twodimensional I H C P are more pronounced and very few results are available in this case. In this chapter we introduce a natural generalization of the mollification method for the two-dimensional inverse heat conduction problem in a slab and t h e n concentrate our attention on the actual application of the algorithm to the more realistic situation related to a bounded rectangular domain in the U , y) plane. A s usual, we assume all the functions involved to be L functions in I R suitably extended—when necessary for the analysis—as being 0 everywhere in {(y, f), t < 0, y < 0). 2

4.1

2

TWO-DIMENSIONAL I H C P IN A SEMI-INFINITE SLAB

W e consider a two-dimensional I H C P in a semi-infinite slab, in which the t e m p e r a t u r e and heat flux histories / ( y , t) and q(y, t) on the right-hand side (x = 1/2) are desired and unknown, and the t e m p e r a t u r e and heat flux on the left-hand surface (x = 0) are approximately measurable. T h e normalized linear problem can be described mathematically as follows. T h e unknown t e m p e r a t u r e u(x, y, t) satisfies u, — uXX + U w(0,y,0 = F ( y , i ) ,

0

<JC

<

1 / 2 , y > 0, t > 0,

y > 0, / > 0,

(4.1a) (4.1b) 107

108

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

F (y,t),

with corresponding approximate data function u (0,

y,t)

x

= Q(y,t),

m

y>0,

t > 0,

with corresponding approximate data function Q (y,

t),

m

w ( * , y , 0 ) = 0, «(l/2,y,i) =/(y,i).

(4.1c)

0 < J C < 1 / 2 , y > 0,

(4.Id)

y>0,/>0,

(4.1e)

t h e desired but unknown t e m p e r a t u r e function, y > 0, t > 0,

u (l/2,y,t)=q(y,t), x

(4.1f)

the desired but unknown heat flux function, u(x,Q,t)=0,

0<χ<1/2,

ί > 0.

(4.1g)

W e hypothesize that t h e exact data functions F(y, t) and Q(y, t) and t h e measured data functions F (y, t) and Q (y, t) satisfy the L data error bounds 2

m

m

\\F-FJ<e

and

\\Q - Q \\ <, ε.

(4.2)

m

Fourier transforming (4.1a) with respect to y and t, we obtain u (x, xx

s,w)

= (s

+ iw)u(x,

2

s,w),

0 < χ < 1 / 2 , - « < s, w < <». (4.3)

Expression (4.3) is a second-order differential equation that has the general solution u(x,s,w)

=A(s,w)e

+ B(s,w)e-" ,

ax

(4.4)

x

where a = Vs + iw . Differentiation of (4.4) with respect to χ gives 2

u (x,s,w) x

= A(s,w)ae

ax

- B(s,w)ae' .

(4.5)

ax

Solving t h e system of (4.4) and (4.5) using conditions (4.1b) and (4.1c), we get

(A(s,w)\j_ia \B( ,w)J S

l j p ( ^ ) ' 2 a \

a

-IJ\Q(S,W)

'

O n t h e other hand, according to the same equations (4.4) and (4.5), using

STABILIZED PROBLEM

109

conditions (4.1e) and (4.1f) and the preceding system, we have 1 1 — sinh —α α 2 1 cosh—α 2

cosh—α 2 q{s,w)

α sinh—α 2

(4.6) Q(s,w)

Given that a

=

V !s 7

+ iw =

(vV + w

+ ΙΣ-(ϊΜ + w

+ s)

2

2 l/1

2

-

s)

1/2

2

where σ = sgn(w), it is clear that 1 sinh—α 2

as

s + \w\

cosh—α 2

as

s + w

and

Following this observation, we see that attempting to solve problem (4.1) —obtaining f(y,t) and q(y,t) from F(y,t) and Q(y, t)—amplifies the error in a high-frequency component by the factor e x p t i v s + w + s ) ], showing that the inverse problem is highly ill posed in the high-frequency components. 4

2

2 l/1

Remark For the one-dimensional I H C P , solving for fit) and q(t) from F ( r ) and QU)—see Sec. 3.2—the error is amplified in a high-frequency component by the factor e x p [ y | H ' | / 2 ] . Comparing this result with the previous analysis, we conclude that t h e two-dimensional I H C P is much m o r e ill posed, in the high-frequency components, than the one-dimensional I H C P . /

4.2

STABILIZED PROBLEM

T h e two-dimensional Gaussian kernel is defined by 1 p ( y , i , 5 ! , S ) = —r-^-exp πο ο 2

ι

2

W h e n 5j = δ = δ, we denote p ( y , t, 3 δ ) by p (y, t). It is also true that Pa(v> t) falls to nearly 0 outside the circle centered at the origin with radius 2

lt

2

s

110

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

3δ and that }°ί„{°ί„ρ ()>, is given by

t)dydt

8

= 1. Moreover, the Fourier transform of p

s

1 P(> ) s

=

w

s

δ

2

(s

2

2~^

exp

+

w) 2

4

and the two-dimensional convolution of any locally integrable function g ( y , r) with t h e Gaussian kernel p (y, t) is now written as s

J g{y,t)

= Γ

={ *g){y,t)

s

Ps

f

p {y',t')g(y-y',t-t')dy'dt'. s

Mollifying system (4.1), we obtain t h e following associated problem. A t t e m p t to find J f (y, t) = 7 « ( l / 2 , y, t) and J q (y, t) = J u (l/2, y, t) at some point (y,t) of interest and for some radius δ > 0, given that J u(x, y, t) satisfies s m

s

s

m

s

x

s

(J u), s

J u{0,y,t) s

= (J u) s

+ (J u) ,

xx

a

0<x<

yy

=J F (y,t), s

y > 0, t > 0,

m

. W 0 , y , / ) =/ C? (y,0, s

J u(x,y,0)

1 / 2 , y > 0, t > 0,

y>0,r>0,

m

(4.7)

= 0,

0 < * < l / 2 , y > 0,

/ κ ( χ , 0 , r ) = 0,

0 < χ < 1 / 2 , ί > 0.

s

δ

This problem and its solutions satisfy the following theorem. Theorem 4.1

Suppose that \\F - F \\ < ε and HQ - QJ\ < ε. T h e n m

(i) Problem (4.7) is a formally stable problem with respect to perturbations in t h e data. (ii) If the exact boundary t e m p e r a t u r e f(y,t) and heat flux q(y, t) have uniformly b o u n d e d first-order partial derivatives on t h e b o u n d e d domain D = [0, Υ] X [0, T], then J f and J q verify s

m

s

m

| | / - / / J | < 0 ( S ) + 2εεχρ[7δ- ] 2

a

D

and \\q - J qJ s

D

< Ο(δ)

+2eexp[75- ]. 2

STABILIZED PROBLEM

111

Proof (i) Repeating the same procedure as the one used to obtain (4.6), we have

(hf)

~

(hf )

(Js~q) -

(J q )

n

s

m

1 cosh—α 2 1 a sinh—α

1 1 — sinh—a a 2 exp 1 cosh—a

F~F

s + w 2

2

m

Q - Q n

In particular,

pi?) - K l k «

s + w 2

cosh—α exp 2

2

WF ~

F

m

(4.8) s + w

1 1 + max —sinh—α exp a 2

2

2

lie -

Qn

and s + w z

\\(j q) s

-

p ^

m

) | | <

max a sinh—α exp 2 \m, \s\

2

WF ~

s + w 2

+ max cosh—α exp 2

2

WQ-QJ.

W e notice that cosh—α < e "! < exp(|s| + VM ), 2 1

1 1 — sinh—α < e a 2

| a |

< exp(|s| +

]/\w\),

and a cosh —a < \a\e 2

M

FJ

< exp(2|s| + 2 ^ ) .

(4.9)

112

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

Thus, s

+ w

2

max α sinh—α exp 2

2

4 s

sS 2

< max < exp 21*1 — 4 MM < max exp 2 U I -

2

exp

sd 2

+ w

2

2

72δ

2

max exp 2 M -

2

Μ

< exp(45" )exp(352

2 / 3

).

This is a uniform u p p e r b o u n d for all the related factors in expressions (4.8) and (4.9). Consequently,

\\{JJ) ~ {7ZA *

e x p ( 4 S " ) e x p ( 3 5 - / ) ( | | / - FJ 2

2

3

and ||(^) -

< e x p ( 4 5 - ) e x p ( 3 5 - / ) ( | | F - FJ 2

2

+ \\Q -

3

Qj).

Using Parseval's equality and the error bound for the data, we obtain IIV -

JJJD

< Usf

-

J J J

<

2ε e x p ( 4 5 - ) e x p ( 3 5 " / ) 2

2

3

< 2εεχρ(7δ~ )

(4.10

2

and \U Q

~ JSQJD

S

<

\U q s

< 2ε cxp^S' )

~ J qJ

2

s

exp(30~ / ) 2

3

< 2εεχρ(7δ" ).

(4.11

2

This shows that the mollified problem (4.7) is a formally stable problem with resp8ct to perturbations in the data. By the preceding inequalities, for a fixed δ > 0, the error is guaranteed to tend to 0 as ε -» 0. (ii) Consider f{y,t)-JJ {y,t)

=f(y,t)

-J f(y,t)

+J f(y,t)

q{y,t)

=g(y,t)

-J q(y>0

+ J»
m

s

8

-J f (y,t) s m

and -J q {y,t) s

m

s

-4j«m(y.O-

+

\\Q

NUMERICAL PROCEDURE AND ERROR ANALYSIS

113

Since, by assumption, all the partial derivatives of / and q are uniformly b o u n d e d in D, both differences 11/ - J f\\o and \]q — JgqWo are b o u n d e d by a constant times δ [see Sec. 3.2, (3.10)]. Combining this with the estimates (4.10) and (4.11) and using the triangle inequality, we immediately have s

\\/-Ι Ο\ <0(δ) $

+ 2εεχρ(7δ- ) 2

0

and

J q J < 0 ( δ ) + 2ε

\\q -

4.3

a

D

exp(7S" .) 2

•

NUMERICAL PROCEDURE AND ERROR ANALYSIS

With ν = J u and w = dv/dx,

system (4.7) is equivalent to

s

dw

dv ~dt w

dv 2

+

dx

+

7 7 7 ,

dp

0 < χ < 1 / 2 , y > 0, t > 0,

dv •—,

0 <x

< 1 / 2 , y > 0, t > 0,

y>0,

t > 0,

v(0,y,t) J F (y,t), a

w(0,y,f)

m

y>o,

'J»Q (y>t), m

v(x,y,0) v{\/2,y,t) w{\/2,y,t)

0 <x

hfmiy't),

y > 0, f > 0, unknown,

/ <7 (y,f),

y > 0, ί > 0, unknown,

0,

0

m

Without loss of generality, we will seek boundary t e m p e r a t u r e function J f or Jq in the unit square D = [0,1] x Consider a uniform grid in the (x, y, t) s

s

m

i>o,

0,

s

v(x,0,t)

m

(4.12)

< 1 / 2 , y > 0,

<Λ: <

1 / 2 , t > 0.

to reconstruct the unknown mollified mollified boundary heat flux function [0,1] of the (y,t) plane χ = 1/2. space:

{(*,- = ih, y = js, t„ = nk), i = 0 , 1 , . . . , N, Nh = 1/2; }

j = 0 , 1 , . . . , M, Ms = L; η = 0 , 1 , . . . , Ρ, Pk = C), where L and C depend on h, s, and A: in a way to b e specified later, L , C > 1. Let the grid functions V and W be defined by

0 < i < N, 0 < j < M, 0 < η < P.

114

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

W e notice that Ki

= J F (yj, S

t„),

m

0 < j < Μ, 0 < η < Ρ,

wSj = J Q (yj>'»)> s

ο < y < μ , ο < « < p,

m

V"o = 0,

0 < i < Ν, 0 < η < Ρ,

ν°ΐ = 0,

0
W e approximate t h e system of partial differential equations (4.12) with the consistent finite-difference schemes W" , • = W i+\J

i,j

V

y

h , (V

-\ ^

n

n

+ 1

h

- V"' ] 1

1

" i + l J '

« = o,ί,.,.,Ν-

1,7 = 1 , 2 , . ..,Mj = 1 , 2 , . . .,M-

W" =

j = 1 , 2 , . .,.,Μ

V" = o,

i = 0 , 1 , . .,.,Ν-

n

'1,1

, - 2V" + V"

(V

i - 1, η = 1 , 2 , . . . , P - i - l , l . i i = 1,2,..., - Ι,η

= 1,2,..., Ρ

~ *' (4-13) = 1 , 2 , . . . , Ρ - ι,

Ι,η

i = 1 , 2 , . .• -,N,j =

o,

Ρ -1,

1,2,...,Μ.

Notice that as we march forward in the χ direction in space, we must drop the estimation of the interior temperature from the highest previous point in time and the associated rightmost point in the y direction. Since we want to evaluate {V£ ) a n d {Wfij} at the grid points of the unit square D = [0,1] X [0,1] after Ν iterations, t h e minimum initial length C of the data sample interval in the time axis needs t o satisfy t h e condition C = Pm = 1 - k + k/h. Similarly, t h e minimum initial length L of the data sample interval in the y direction satisfies L = Ms = 1 - s + s/h. Now, in order to analyze t h e stability of the finite-difference scheme (4.13) and the convergence of the numerical solution to the solution of the mollified problem (4.7), we use the discrete Fourier transform method once again. T h e definition of a two-dimensional discrete Fourier transform of a grid function {i/ "}"„_j—similar t o the one-dimensional discrete Fourier transform introduced in Definition A.16—is given by ;

U(w w ) u

2

sic = —

50

Σ

°° Σ

Uj e n

i a w i s + n w

i

k )

,

NUMERICAL PROCEDURE AND ERROR ANALYSIS

115

Applying this transformation to (4.13), we get

W = W, + •^r(v e ^ i

i+l

V

i+i

2,/c

=

k

h - - ( Ϋ ^ s

- V -^ ) k

i

ie

21>· +

-

Ϋ ^ ' ) ,

V^hW

i+v

Equivalently, we have

=

I

1 + ha a

h) l)

where Ah ι ws \ a = — 5 - sin s \ 2 ι

2

x

+

h

i—sinw k. 2

k

Thus, \V \<

{1 +h +h\a\)

i+l

max(lK-|, |W>|)

and \W \< i+l

{1 + |a|} m a x ( | l / . | , W l ) .

Consequently, max(|K

< + 1

| , | ^ . | ) < {1 + \a\) max(|J/|, i ^ l ) + I

and max (\V \,\W \) N

N

< {1 4-lel}^ max(|K |, l^ol) 0

= {1 + \a\)

max(\p V \,\p W \)

N

g

= {l +

0

t

0

\a\} \p \max(\F \,\Q \). N

s

m

m

F r o m t h e last inequality it follows that \V \ <{l 2

N

+ \a\} \p \ (\Fj 2N

2

+ \Qj )

2

(4.14a)

2

s

and \W \

2

N

< {1 +

|α!} |ρ | (ΐ4| 2

2Ν

5

2

+ \QJ ). 2

(4.14b)

116

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

A similar analysis to the one done in Sec. 3.3, using Poisson's summation formula in (4.14a) and (4.14b), integrating with respect to w, and w , and taking square roots, gives 2

max(||K„|| , ||H^|| ) < 4 2

sup

2

\a\f

{1 +

0<|w,|STr/fc 0<\w )
\

2

+

xexp

< 4

2

2

»(HFJ| + | l G J l 2 )

4

2

4Λ (1 w , |

1 +

sup

w )s

,)

2

\w,\k)

4

s

2

0<|νν,|£7Γ/λ 0<\w \&ir/s

1 / 2

2

(w

Xexp

<4

sup 0 < | > v

2

+ νν )δ 2

2

»(ll/JI + IIQjl2)

4

| { l + ^(IwJ +

2

\w \)}

N

2

| < i r / 5

(w

2

xexp

<4

2

sup

+

w )8 2

2

\(\\Fj2

+

\\Qj2)

>(HFJI2 +

IIGJI2)

j e x p [ M i ( | w j | + |w |)] 2

0<|νν,|<7Γ/Α; \ 0<\w \
(w

2

Xexp

< 4

sup

\

j exp

sup 0<|lV |S7!-/S

[

2

x(\\FJ

+

2

< 16 exp provided that δ > m2<x(^J k/^τ, Γ

2

< exp

0<|w,|<17//t

x 4

+ ν,2\s2 ν )δ

\\QJ ) 2

ι 4δ

2

(II/JI2 +

y/s/π).

IIGJI2),

2

NUMERICAL RESULTS

117

Thus,

< 16 exp

\\VJ

2

1 4δ

2

(II FJ

+

(II FJi

+

2

iieji )

(4.15)

WQjz)

(4.16)

2

and

||wgi < 16 exp 2

by Parseval's equality.

1 4S

2

•

T h e estimates (4.15) and (4.16) show that for a given δ > 0, the finitedifference scheme (4.13)—consistent with the partial differential equation J xx (Js h ~ h —i unconditionally stable and the finite-difference solution converges to the solution of the mollified problem (4.7). u

=

u

u

s

yy

s

Remarks 1. T h e radius of mollification, δ, can b e selected automatically as a function of the level of noise in the data. T h e selection criterion discussed in Sec. 3.5 extends naturally to the two-dimensional case. In fact, for a given ε > 0, there is a unique δ > 0, such that ym F

s

- F J D = ε-

(4.17)

2. From a more practical point of view, it is possible to replace the discrete two-dimensional mollification of the data functions by two successive one-dimensional mollifications in time and y space. In this manner, the data filtering task can be executed as a parallel process, eliminating the most time-consuming aspect of the algorithm.

4.4

NUMERICAL RESULTS

Two-Dimensional Strip W e test the accuracy and the stability properties of the m e t h o d on two model problems. Problem 1 T h e approximate reconstruction of a surface t e m p e r a t u r e / ( y , i ) and a surface heat flux q(y, t) at Λ; = 1 / 2 are investigated for a two-dimensional strip exposed to a heat flux data function at the free surface χ = 0,

118

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

given by «,(0,y,0 -G(y,t) 1 + 2 £ (-l)"exp(-« 7r /) n= l 2

2

erfc(^),

0,

y±0,

t

>

0

.

otherwise

and a t e m p e r a t u r e data function «(0, y, i ) = F(y,

t) = 0

for a l i i and y.

T h e exact temperature solution to be approximately reconstructed at the surface χ = 1/2 has equation " ( y ν,ί) = / ( y , 0

- Hi lh^k^ - } (^r)' 2n 1)V2i] erfc

+

y

^ °''" °'

\0, otherwise, and the exact heat flux solution at the surface χ = 1 / 2 is given by "'(2'V'')

=q(y,t) i(l+2E(-l)"exp[-(2n) 77 i]Jerfc(^), 2

=

2

\0,

y > 0, t > 0, otherwise.

Problem 2 We attempt to approximately reconstruct a surface temperature / ( y , i ) and a surface heat flux q(y, t) at χ = 1/2 for a two-dimensional slab subject to a nonlinear diffusion process. The I H C P is described as follows: u, = u (0,y,t)

χ - 1 u t

u

xx

u

+ u, yy

x

= -2te'e-

u(0,y,t)

= te'e~

= Q{y,t),

y

y

= F(y,t),

K(x,y,0)=0,

x

u(x,0,t) u(x,y,t)->0

y > 0,0

= q(y,t), = ( 1 -xfte' as


y > 0,0 < t < T, 0 <x

« ( 1 / 2 , y , i ) =f(y,t), u (l/2,y,t)

0 < χ < 1 / 2 , y > 0, 0 < t < T,

x

< 1 / 2 , y > 0,

unknown,

y > 0, 0 < t < T,

unknown,

y > 0, 0 < t < T,

= h(x,t), y -» 00,

0 <x

< 1 / 2 , 0
(4.18)


0 < χ < 1 / 2 , 0 < t < T.

NUMERICAL RESULTS

119

T h e unique t e m p e r a t u r e solution of the associated direct problem is u(x,y,t)

-x) te'e- .

= (1

2

y

H e n c e , the exact solutions for the nonlinear I H C P are given by f{y,t)

= \te e-\

y > 0,0

t


and q{y,t)

= -te'e~ ,

y > 0,0

y


In order to approximate the nonlinear partial differential equation in (4.18), we rewrite the first equation of the finite-difference scheme (4.13) as W" , · = W '

+

1

·'

1

>

+ —

/

jc - -

J

f

W"- Γ h — —(V 1 Vfr [2k^ '·> n

h

-

+ l

V"~ \ > l

'

l ]

(V

, - 21/"· + V

n

n

Λ

1

W e assume the data functions F and Q to be discrete functions measured at equally spaced points in the ( y , t) domain [0, L] X [0, C ] , where L = 1 - s + s/h, C = 1 - k + k/h, Nh = 1/2, h = Δ.Χ, s = Ay, and k = At. T h e ( M + 1) X (P + 1) sample points in this rectangle have coordinates (v>, f„) = (js, nk), 0 < < M, 0 < η < Ρ, Ms = L, Pn = C. After extending the data functions in such a way that they decay smoothly to 0 outside [0, L] X [0, C], we consider the data functions F and Q defined at equally spaced sample points on any rectangular domain of interest in the ( y , t) plane. Once the radii of mollification S and 8 , associated with the data functions F and Q , respectively, have b e e n obtained—after solving the discrete version of (4.17)—and the discrete filtered data functions m

m

m

F

m

Q

m

V ( y , , Ο = νϋ ,Ρ η

m

m

J Q (yj,t ) Se

m

n

= ws ,

< Μ , ο<*

o<j

ti

< />,

are determined, we apply the finite-difference algorithm described in Sec. 4.3, marching forward in the χ direction. T h e values V£ j and Wfij, 0 <j < Μ - Ν, 0 < η < Ρ - Ν, so obtained, are then taken as the accepted approximations for the boundary t e m p e r a t u r e and heat flux histories respectively at the different y locations in the domain D = [0,1] X [0,1] of the ( y , t) plane at χ = 1/2. In Problem 1 we use h = s = 0.1 and k = 0.01. In Problem 2 we choose h = 0.1, s = 0.015, and k = 0.01. If the exact data t e m p e r a t u r e (heat flux) is denoted by F(y, t) [Q(y, t)], the noisy data F ( y , f„) [<2 (y , f„)] is obtained m

y

m

y

120

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

by adding a random error to F

m

Q (yj,t ) m

[Q ]\

that is, for every grid point (y

m

p

= Q(yj,t )

n

+

n

/„),

e „, 2Ji

where ε j and ε j« Gaussian random variables of variance ε = (ε,) = (ε ') . To study the numerical stability of the algorithm, we use different average perturbations for ε = 0, 0.001, 0.002, 0.003, 0.004, and 0.005. T h e solution errors for the discrete t e m p e r a t u r e and heat flux functions at χ = 1/2 in the discretized time interval / = [0,1], at the y = 1/2 location, are respectively given by a r e

χ

n

2

2

2

2

2

[ W

I

» - > - M ' - \ P -

P-N+\

N

+

1

£

\ ( ^ - - / " )

2

'/

2

J

and

"β*./ - Φ

ί 1 " \ p N

~^ Σ

P

+

l

i\ ~ If) ]

+l

β

i / 2

-

1 js = 2 ·

Tables 4.1 and 4.2 show the results of the numerical experiments associated with Problems 1 and 2, respectively. T h e qualitative behavior of the reconstructed surface t e m p e r a t u r e for Problem 1 (at y location 1/2) is illustrated in Fig. 4.1a where the numerical solution for ε = 0.005 is plotted. T h e corresponding plot for the heat flux function is shown in Fig. 4.1b. Figure 4.2 a shows the reconstructed surface t e m p e r a t u r e ( y location 1/2) for Problem 2 corresponding to the noise level ε = 0.005. T h e associated heat flux is plotted in Fig. 4.2b.

TABLE 4.1. Error Norm as a Function of the Amount of Noise in the Data for the Surface Temperature and Surface Heat Flux in Problem l a t x = y = l / 2

Temperature ε 0.000 0.001 0.002 0.003 0.004 0.005

0.01 0.04 0.04 0.04 0.04 0.04

Heat Flux

Error Norm

ε

0.003580 0.003580 0.003581 0.003583 0.003585 0.003588

0.000 0.001 0.002 0.003 0.004 0.005

ΰ 0.01 0.04 0.04 0.04 0.04 0.04 δ

Error Norm 0.017352 0.017356 0.017364 0.017374 0.017388 0.017404

NUMERICAL RESULTS

121

TABLE 4.2. Error Norm as a Function of the Amount of Noise in the Data for the Surface Temperature and Surface Heat Flux in Problem 2 a t x = y = l / 2

Temperature

ε 0.000 0.001 0.002 0.003 0.004 0.005

0.01 0.04 0.04 0.04 0.04 0.04

Heat Flux

Error Norm

ε

0.010779 0.010831 0.011099 0.011341 0.011343 0.011345

0.000 0.001 0.002 0.003 0.004 0.005

Error Norm 0.01 0.04 0.04 0.04 0.04 0.04

0.043116 0.043327 0.043462 0.044398 0.044428 0.045371

Fig. 4.1a Reconstructed temperature function—Problem 1: χ = y = 1/2, e = 0.005, δ = 0.04, h = s = 0.1, k = 0.01. Exact solution ( * * * ) ; computed solution (—).

122

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM 0.10-1

β.00

0.2S

0.60

0.75

1.00

t I me (b) Fig. 4.1ft Reconstructed heat flux function—Problem 1: χ = y = 1/2, ε = 0.005, S = 0.04, h = s = 0.1, k = 0.01. Exact solution ( * * * ) ; computed solution (—).

Bounded Rectangular Domain W e now investigate the two-dimensional I H C P when the space domain in the (x, y) plane is restricted to the b o u n d e d prototype rectangle R = [ 0 , 1 / 2 ] x [0,1]. W e must add the " b o u n d a r y " condition—from the point of view of the I H C P and the x-space marching scheme— u(x,l,t)

= Λ,(*,ί),

0 <x

< 1 / 2 , t > 0,

and, at the same time, we shall not require the homogeneity of the boundary condition (4.1g) that should now read u(x,0,t)

= h (x,t), 0

0<x0.

NUMERICAL RESULTS

123

e.«-i

0.00

0.25

0.E0

0.75

1.00

t Ime (a)

Fig. 4.2a Reconstructed temperature function—Problem 2: χ = y = 1/2, ε = 0.005, δ = 0.04, h = 0.1, s = 0.015, k = 0.01. Exact values ( * * * ) ; computed solution (—).

However, since the functions / , h , h and F uniquely determine the solution of the direct problem—including the heat fluxes Q and q at χ = 0 and χ = 1/2, respectively—it is clear that, for the inverse problem, the data functions F and Q possess all the necessary information for the recovery of the t e m p e r a t u r e and heat flux functions / and q at the surface JC = 0 and also the boundary conditions h and h at y = 0 and y = 1, respectively. Consequently, the functions u(x, 0, t) = h (x, t) and u(x, 1, t) = h^x, t) will b e treated as unknowns and their recovery becomes a natural task for the two-dimensional I H C P . Mathematically, the new inverse problem can be stated as follows: 0

0

u

l

0

u

u(0,y,t)

t

=

xx

u

+

=F(y,t),

yy>

u

0 <x < 1 / 2 , 0 < y < 1, t > 0, 0 < y < l , i > 0 ,

(4.19a) (4.19b)

124

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM 2.00-i

1.75

1.2ΒΗ

Ο *>

Ο Ι.ββ

Η

3

0.76

Η

0.60

Η

Ο Ο Φ

0.26 Η

0.00

f*i r ι ι | ι

0.00

ι

0.2Ε

ι

ι

|

ι

ι

ι

ι

ι

0.60

ι

ι

ι

0.76

ι

| 1.00

t lm©

(b) Fig. 4.2ft Reconstructed heat flux function—Problem 2: χ = y = 1/2, ε = 0.005, δ = 0.04, Λ = 0.1, s = 0.015, k = 0.01. Exact values ( * * * ) ; computed solution (—).

with corresponding approximate data function F (y, m

« , ( 0 , y , f ) = Q(y,t),

t\

00,

with corresponding approximate data function Q {y, m

t),

0 <x < 1 / 2 , 0
w(*,y,0)=0, «(l/2,y,0 =/(y,0»

(4.19c)

0 < y < l , i > 0 ,

(4.19d) (4.19e)

the desired but unknown t e m p e r a t u r e function, u {\/2,y,t) x

= q(y,t),

0 < y < l , f > 0 ,

(4.19f)

NUMERICAL RESULTS

125

the desired but unknown heat flux function, u(x,0,t)

= h {x,t),

0 <x

0

< 1 / 2 , t > 0,

(4.19g)

0 < * < 1 / 2 , ί > 0,

(4.19h)

unknown boundary t e m p e r a t u r e function, u{x,\,t)

= h,(x,t),

unknown boundary t e m p e r a t u r e function. A few words about t h e uniqueness of the solution of system (4.19) are appropriate. T o that end, consider t h e direct problem for t h e two-dimensional heat conduction system given previously, with boundary data t e m p e r a t u r e F(y, t) = u(0, y, r), h (x, t) = u(x, 0, i), ft,(x, t) = u(x, 1, f), and / ( y , i ) = M ( 1 / 2 , y, i). This problem has a unique t e m p e r a t u r e solution and, in particular, an induced heat flux solution at χ = 0 that we denote Q(.y, 0 = u (0, y,t). O n the other hand, the same direct problem with data temperature F(y, t) = u(0, y, t), h (x, t) = v(x, 0, t), h,(x, t) = v(x, 1, f), and / ( y , f) = i ^ ( l / 2 , y, f) also has a unique solution and the induced heat flux at χ = 0 will be denoted Q(y, t)_== υ (0, y, t). Is it possible to have Q = Q if / Φ f a n d / o r h Φ h a n d / o r h Φ h(! By superposition, the_solution of the direct problem with boundary data temperature AF = F - F = 0, Δη = h - h , Ah = h - Ji and Δ / = / - / is such that t h e heat flux at χ = 1/2, Δ Q = Q - Q, is identically 0. W e can now imbed the original rectangular domain symmetrically into t h e square [ 1 / 2 , 1 / 2 ] X [0,1] in the (x, y) plane—for every time t—and seek some relationship (if any) among the boundary data functions Δ / , Δη , Ah AF, AH , and AH such that AF = AQ = 0. Pictorially, with Τ = u - υ, t h e situation looks like this: 0

x

0

χ

0

0

0

0

1

0

x

x

v

0

Q

v

{

AH,

Ah, AF =

τ = Τ t

+Τ

xx

0

T, = Τ

yy

AQ =

AH

0

+ Τ

xx

yy

Ah

0

0

Given that we do not consider interior sources in our heat transfer model, the condition AF = 0 can be achieved if and only if " t h e effect of each of the two symmetric boundary data temperatures annihilates each other at every point (x, y, f ) , " that is, if and only if Af=-AF,

Ah, = -AH , t

and

Ah

0

= -AH . 0

(4.20)

126

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

The same argument implies that for t h e heat flux condition AQ = 0 to occur, we must have Af=AF,

Ah = ΑΗ , l

and

λ

Ah = AH . 0

0

(4.21)

From equalities (4.20) and (4.21), it follows that Af=

Ahi = Ah = 0 , 0

and we conclude that AF = AQ = 0 ii and only if the t e m p e r a t u r e data o n the boundary is identically 0. • This argument shows that the I H C P (4.19) has a unique solution. We once again approximate t h e system (4.19) with t h e consistent finitedifference scheme W"

= W

+ — (V

- V ~\

n+i

n

- — (V

l

, - 2 F " . + V" , A

/ = 0 , l , . . . , i V - 1 , ; = 1 , 2 , . . . , Μ - 1, η = l,2,...,P-i K"j = (J F )h

; = 0 , 1 , 2 , . . . , M , η = 1 , 2 , . . . , P - 1,

W j = (J Q )h

J = 0 , 1 , 2 , . . . , Μ , η = 1 , 2 , . . . , Ρ - 1,

S

m

n

0

s

m

K°, = 0,

- 1,

ί = 1,2,...,AU = 1,2,...,Μ.

A s we march forward in the χ direction in space, we still drop t h e estimation of the interior t e m p e r a t u r e from the highest previous point in time but t h e discrete heat fluxes rV" and W" are computed by linear extrapolation from t h e already estimated values W" , W" a n d W" _ , ^ + I , M - I > introducing an extra local error of o r d e r O(s): +U0

+l M

+lji

+l<2

+l M

2

w i, -2ivr -wr , 0

+

^ " ΐ , Μ

+hl

= 2^"

+ 1

+U2

,Λ/-1-^" 1,Λ/-2+

T h e corresponding unknown boundary t e m p e r a t u r e s a r e then calculated directly using t h e usual formulas

and Vt x,M= n

+

vr + hwr , . <M

+l M

NUMERICAL RESULTS

Fig. 4.3

127

Exact temperature function at χ = 1 / 2 .

This process is continued for η = 1 , 2 , . . . , Ρ - i - 1 until the approximate solution is computed at all the grid points of the domain in the plane x . T h e n the entire cycle is repeated for t h e discrete points of the plane x , and so on. Figures 4.3 to 4.5 show the exact t e m p e r a t u r e solution restricted to t h e grid points, the discrete computed solution—obtained with t h e numerical scheme introduced previously with h = 0.1, s = k = 0.01, δ = 0.04, ε = 0.005—and the corresponding discrete error surface at χ = 1/2 and 0 < t < 0.8, respectively, for the following I H C P : i + 1

i + 2

u, = u

xx

H ( 0 , y,t)

u(x,

x

u(x,0,

1 = -j=e

y, t) = q(y,

0 < χ < 1 / 2 , 0 < y < 1, / > 0, 0 < y < 1, t > 0,

=0,

« ( 1 / 2 , y , r ) =f(y,t), u (l/2,

yy

y ' sm-j=, y χ y,0) = s i n - y s i n - y ,

u (0,y,t) x

+ u,

unknown, t), unknown,

0 < y < l , i > 0 , 0 < χ < 1 / 2 , 0 < y < 1, 0 < y < 1, ί > 0, 0 < y < 1, t > 0,

t) = h (x,

t), unknown,

0 < χ < 1 / 2 , t > 0,

u(x, 1, t) = h,{x,

t), unknown,

0 < χ < 1 / 2 , f > 0.

0

(4.22)

128

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

Fig. 4.4 Reconstructed temperature function at χ = 1/2: h = 0.1, s = k = 0. δ = 0.04, ε = 0.005.

T h e unique exact t e m p e r a t u r e solution for problem (4.22) is

u(x,y,t)=e

y χ 'sin-p-sin-^, v2 V2

0 < χ < 1 / 2 , 0 < y < 1, f > 0,

and, consequently, the unknown functions for the I H C P are given by

f(y,t)

y 'sm-τ-sm

= e

ΊΪ

1

1

1 ~"/2 h (x,t)

= h (x,t)

h^x.t)

= e

0

0

0 < y < 1, ί > 0,

:

2 ^ '

= 0,

'sin^sin-y

2^2'

0 < y < 1, t > 0,

0 < x < 1 / 2 , t > 0,

0

<

JC <

1 / 2 , t > 0.

EXERCISES

129

is * e

t. ® ■o

0 t. Φ

I

<&> <9> <8> <* s,

~J?

·«*?

'«b·

Fig. 4.5 Temperature error function at x = 1/2: h = 0.1, s = k = 0.01, <S = 0.04, ε = 0.005. 4.5

EXERCISES

4.1. Show that given ε > 0, there exists a unique radius of mollification 8 > 0, such that Ι Ι / β ^ - .FIID = e ,

(4.23)

where || ■ || 0 indicates the L2 norm in the unit square D = [0,1] X [0,1] in the (y, t) plane and JSF represents the two-dimensional convolution of the Gaussian kernel pa(y,t) with the locally integrable function FU, y). Here 1 Psi ' y) = ^ 2 e x P 1

y2 + t2

ITO

4.2. Discuss at least two possible implementations for solving (4.23) in Exercise 4.1. 4.3. If the two-dimensional convolution in Exercise 4.1 is replaced by two successive one-dimensional convolutions in t and y, how can we accelerate the data filtering process? Should we modify the selection criterion of the radius of mollification in this case?

130

TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEM

4.4. Show that the uniqueness result for the two-dimensional I H C P on a bounded rectangular d o m a i n — a n d all times t—(4.19) holds in the more general case of a b o u n d e d domain Ω c U with smooth boundary dfl if the t e m p e r a t u r e and the normal heat flux are specified in a portion of ΘΩ X [0, 0 . 2

4.6

R E F E R E N C E S AND C O M M E N T S

T h e following references and comments serve to expand the basic material covered in the corresponding sections. T h e first analytical solution—requiring exact d a t a — t o the inverse heat conduction problem that is applicable to two-dimensional conduction systems for geometries of arbitrary shape was presented in M. Imber, Temperature extrapolation mechanism for two-dimensional heat flow, AIAA J. 12 (1974), 1089-1093. 4.1. Most of the literature related to the two-dimensional I H C P is based on different ways of combining finite-element realizations with the future temperatures method introduced in C h a p . 3. For some of the early applications of these ideas, see B. R. Bass and L. J. Ott, A finite element formulation of the two-dimensional inverse heat conduction problem, Adv. in Comput. Tech. 2 (1980), 238-248. More numerical experimentation can be found in T. Yoshimura and K. Ituka, Inverse heat conduction problem by finite element formulation, Internat. J. Systems Sci. 16 (1985) 1365-1376. A comprehensive exposition was presented later in J. Baumeister and H. J. Reinhart, On the approximate solution of a two-dimensional inverse heat conduction problem, Inverse and Ill-Posed Problems, H. G. Engl and C. W. Groetsch (eds.), Academic, Orlando, 1987, pp. 325-344. N. Zabaras and J. C. Liu, An analysis of two-dimensional linear inverse heat transfer problems, using an integral method, Numer. Heat Transfer 13 (1988) 527-533. 4.2-4.4. H e r e we follow in part the work of L. Guo and D. A. Murio, A mollified space-marching finite difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry, Inverse Problems 7 (1991) 247-259.

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

5 APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

T h e r e are several different physical situations where the space marching approach for the solution of the I H C P allows for a more general and natural point of view of the original problems, partially eliminating the sharp dichotomy between the mathematical treatment of ill-posed and well-posed tasks. For instance, in the identification of boundary source functions and nonlinear boundary radiation laws, the original well-posed problem is replaced by a related stabilized ill-posed o n e — a daring suggestion—gaining great advantages in terms of location and collection of the data necessary to solve the problem. This is also the case in other applications related to solidification p h e n o m e n a in phase-change materials, when the evolution of the solid-liquid interface is sought (Stefan problem). T h e space marching scheme can also be quite naturally applied—with no significant c h a n g e s — t o the direct solution of the ill-posed inverse Stefan problem where the front evolution is approximately known and one wants to find the t e m p e r a t u r e and heat flux histories at a fixed liquid boundary location. Finally, in another quite different application, we suggest how to approximately recover the initial t e m p e r a t u r e distribution in the one-dimensional I H C P while marching in space, illustrating an interesting relationship between the backward-in-time heat conduction problem and the I H C P .

5.1

IDENTIFICATION O F BOUNDARY S O U R C E FUNCTIONS

In this section we investigate the numerical identification of surface transient heat sources in one-dimensional semi-infinite and finite slab mediums when 131

132

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

the active surface radiates energy according to a known nonlinear law. Alternatively, if the active surface is heated by a source at a rate proportional to a given function, the nonlinear radiating boundary condition is then numerically identified as a function of the interface temperature. These two tasks can be viewed as suitable generalizations of the classical problem of attempting to determine the interface temperature between a gas and a solid with a nonlinear heat transfer law. Indeed, the nonlinear radiation law and the transient boundary source are supposed to be known in order to determine the interface temperature. Consequently, if the new task consists of the identification of the transient boundary source function, a different approach must be used. We consider a one-dimensional I H C P in a semi-infinite or finite slab, in which the temperature and heat flux histories fit) and q(t) on the left-hand surface (x = 0) are desired and unknown, and the temperature and heat flux at some interior point χ = x or on the right-hand surface (x = 1) are approximately measurable. Note that, equivalently, the data temperature histories might be measured at two interior points. For the semi-infinite medium, 0 < x , and for the finite slab, 0 < χ < 1. W e assume linear heat conduction with constant coefficients and normalize the problem by dimensionless quantities. Without loss of generality, we consider x = 1 in all cases. T h e problem can be described mathematically as follows. For the semi-infinite or finite slab, the unknown temperature u(x, t) satisfies, respectively, 0

0

0

u,{x,t)

=

«(1,0

u {x,t), xx

=F(t),

t > 0, 0 < * < o o o r 0 < x < l ,

(5-la)

i>0,

(5.1b) F (t),

with corresponding approximate data function -«,(1,0

m

t > 0,

=Q(t),

with corresponding approximate data function u(x,0)

= w (χ), 0

u(0,0=/(0,

(5.1c)

Q (t), m

0 < χ < oo or 0 < χ < 1, ί

>

0

(5.1d) (5.1e)

'

the desired but unknown temperature function,

-u (0,0 x

=e(0 = £(«(0,i)) -

G(0,

t > 0

(5.1f)

the desired but unknown heat flux function. T h e nonlinear boundary condition (5.If) indicates that the active surface radiates energy at a rate proportional to Ε and is heated at a rate propor-

IDENTIFICATION OF BOUNDARY SOURCE FUNCTIONS

133

tional to the function G. O u r aim is to obtain more detailed information about the boundary condition at the interface χ = 0. More precisely, we want to estimate the function Ε if G is known or, reciprocally, we want to identify the source function G if the radiation law Ε is given. As usual, we assume that all the functions involved are L functions in any time interval of interest and hypothesize that the exact data functions F(t) and Q(t) and the measured data functions F (t) and Q U) satisfy the L data error bounds 2

2

m

WF - FJ < ε

m

< ε.

\\Q - QJ

and

W e recall that under the conditions stated in T h e o r e m 3.4, the t e m p e r a t u r e and heat flux boundary functions—corresponding to the mollified problem associated with (5.1)—satisfy on the bounded time interval D = [0, T]: 11/ - JJJ

<{-Μδ

D

+ ε exp[(2S) "

2 / 3

]

(5.2)

and Wq-J qJ a

< ϊΜδ + 2ε e x p [ ( 2 S )

D

_ 2 / 3

].

(5.3)

Once the mollified temperature and mollified heat flux functions have been evaluated at the interface, it is feasible to attempt to identify the source function G or the radiation energy function Ε given in formula (5.If). Identification of the Source Function G Assuming that the radiation law at the active surface is known, according to (5.If), the exact source function is given by G(t)

= £ ( / ( * ) ) ~Q(t).

(5.4)

T h e approximate source function, denoted G {t\

is defined by

a

G {t)=E(J f (t))-J q (t), a

s m

a

(5.5)

m

and in order to estimate the error, we suppose that the surface radiates energy at a rate proportional to [f(t)] . H e r e ρ is a positive integer, the value ρ = 1 corresponding to Newton's law of cooling, and ρ = 4 to Stefan's radiation law. T h e difference (5.4) - (5.5) gives p

G(t)

- G (t) a

= [f(t)V-

[J f (t)]

- q(t)

P

s m

+J q (t). s

m

From the identity x" - b" = (x - bXx ~ + x"' b + ••• +xb"~ + taking norms and introducing Μ = max{||y / |U, D , ll/IL, D }, we get n

]

2

g

ε

WG - G J |

0

< ρΜΓ

' 1 1 / - J fJ a

D

2

m

+ Wq ~

J qJ . t

D

b ~ ), n

l

134

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

Combining the last inequality with the u p p e r bounds (5.2) and (5.3), we obtain the estimate IIG - G \\ a

< (ρΜΓ

~ l){iMS

1

D

+ 2eexp[(25)

_ 2 / 3

]}.

(5.6)

This shows that the identification of the source function G is stable with respect to errors in the data functions F and Q, for fixed ρ and δ > 0. Remarks 1. Notice that the approximate source function G is actually a function of the radius of mollification δ, the amount of noise in the data ε, and the exponent ρ in the radiation model E. a

2. From a more theoretical point of view, inequality (5.6) can be used to show the convergence of G to G in the L norm for a suitable choice of the radius of mollification δ. In fact, setting δ = [ 1 η ( 1 / ε ) ] ~ , after replacing this quantity in (5.6), we obtain 2

a

Ι / 2

IIG - GJ\

< (ρΜΓ

+ l){W[HWs )]~

1

D

l/2

+

3/2

3 / 2

2ε^}.

This last inequality implies that, for the special selection of the radius of mollification indicated previously, | | G - G | | - > 0 as ε -> 0, for any value of p. 0

o

Identification of the Radiation Law Function Ε From (5.If) it follows that the exact function E, assuming that the source function G is given, satisfies E(u(0,t))=E(f{t))

= G(t)+q(t).

(5.7)

T h e approximate law, denoted E , is defined by a

E (J f (t)) a

s m

= G(t)

+J q (t). 5

(5.8)

m

Subtracting (5.7) from (5.8), taking norms, and using inequality (5.3), we immediately have ll£ - E \\ a

D

< {MS + 2ε e x p [ ( 2 5 ) "

2 / 3

].

(5.9)

This estimate also shows that the identification of the radiation law—as a function of time—is stable with respect to perturbations in the data functions F and Q, for a fixed δ > 0, provided that the source function is known. However, this information is clearly not sufficient to identify the physical process at the interface. Nevertheless, since at each time r, we know the

NUMERICAL PROCEDURE

135

o r d e r e d pairs (f,, / / ( f , ) ) and (i,, E U )), it is possible to collect t h e coordinates U f (tj), E U)) for t in a discrete subset of D and obtain a graph of the approximate functional relationship between the radiation law and t h e t e m p e r a t u r e at the interface. This is certainly always the case if the cardinality of the range of temperatures U f t(i)} is sufficiently large. Similar remarks to the ones in the previous paragraph, about the p a r a m e t e r dependency of E and convergence in the L norm of E to Ε as the quality of the data functions improve, ε -» 0, also apply here. T h e computational details are presented in the next section. a

s m

m

a

t

a

a m

a

5.2

2

a

NUMERICAL P R O C E D U R E

W e approximate the mollified partial differential equation associated with system (5.1) with the consistent finite-difference scheme (3.47) introduced in Chap. 3: wr-,

=

w r - ^ { v r

- v r %

x

V"-i = V? - hW/Lx,

(5.10)

i = Ν, Ν - 1 . . . . . 1 , η = 1,2,..., Notice that VjiXj, t ) = J u{Xj, n

s

f„), as before, but

»tf instead of d/dxJ u(x s

Μ - 1.

=

-d/dxJ u( ,t ) s

Xi

n

t ). Also,

h

n

VN = hF { m

t ),

W$ = J Q (

n

s

m

0
t ), n

and V? = u (x,),

0
0

Once the t e m p e r a t u r e J f and the heat flux J q have been reconstructed, we proceed with the approximate identification of the source function G or the radiation law function E as explained in Sec. 5.1. 8 m

s

m

a

a

Numerical Results In Problem 1 the approximate reconstruction of a source function G ( r ) and a nonlinear radiation law £(w(0, i)) are investigated for a one-dimensional finite slab exposed to a heat flux data function at the free surface χ = 1 given by - Η , ( Ι , ί ) = G(0

=fj>

f>o,

136

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

and a t e m p e r a t u r e data function u(l,t)

= F(T) 1

(t - 0.2)

2

( - 1)

00

- 3 nΣ= \

η — e x p [ - n n ( t - 0.2)], 2

2

t > 0.2, 0 < t < 0.2.

lo,

T h e exact source solution to b e approximately reconstructed at t h e interface χ = 0 has equation G ( i ) = £ ( « ( 0 , f)) -
u(0,O = / ( f ) 2 " 1 (f - 0.2) + - - — Σ — e x p [ - « 7 7 ( i - 0.2)], 1

2

2

t > 0.2,

n= \ L

0,

0 < t < 0.2,

and

-« (o,O = « ( O = z

{J;

t > 0.2, 0 < t < 0.2,

With this information we generate t h e exact functions £ ( « ( 0 , r)) and G ( r ) for our model problem. In Problem 2 we attempt t o approximately reconstruct t h e transient source function G ( f ) for a semi-infinite body initially at zero t e m p e r a t u r e with data functions

erfc

u ( l , f ) =F(t) =

1 2(t - 0.2) 1/2

t > 0.2, 0 < t < 0.2,

.0, and -u (l,t) x

= Q{t) '[ΙΓ(Γ-0.2)]

,0,

'

/ 2

e x p { - [ 4 ( i - 0.2)]

'},

t < 0.2,

0
NUMERICAL PROCEDURE

137

T h e unique t e m p e r a t u r e solution at the interface is

«(o,o-/
0

<°-
and the corresponding heat flux at the interface is

- Μ 0 , Ο - ί ( Ο - ί

[

π

(

' - ° ·

2

)

"

1

,

/

'

2

ί

\θ,

>

·

0

2

'

0
In this case, we do not attempt the identification of the radiation law at the active boundary. T h e energy as a function of the interface t e m p e r a t u r e is either 0 or 1 for any value of ρ making its identification impossible. T h e r e is not enough information in the range of boundary temperatures which in this example is reduced to just two t e m p e r a t u r e values. O n c e the radii of mollification 8 and S associated with the data functions F and Q , respectively, and the discrete filtered data functions hFm^r) = N and J Q (t ) = W$, 0 < η < Μ, are determined with δ = m a x ( 5 , S ), we apply the finite-difference algorithm described previously in this section, marching backward in the χ direction. T h e values V " and W ", 0 < η < Μ - Ν, so obtained, are then taken as the accepted approximations for the interface t e m p e r a t u r e and heat flux histories, respectively, at the different time locations at χ = 0. Finally, we identify the approximate transient source function G or the approximate radiation law function E at t h e grid points of the time interval / = [0,1] using (5.5) and (5.8). In all cases, we use h = Ax = 0.01 and k = At = 0.01. T h e noisy data are obtained, as in the previous models, by adding a random error to the exact discrete data function at every grid point of the extended data interval. If the discretized computed transient source function component is denoted by G£ and the true component is G" = G(i„), the weighted I n o r m of the solution error is then given, in the time interval / = [0,1], by F

m

Q

m

V

a

f

m

n

g

0

0

a

a

2

1

\\G - G||/ = a

V

/2

M-N ( g ; - g - )

Σ

2

TABLE 5.1. Error Norm as a Function of the Level of Noise in Problem 1

ρ = 1 (Newton)

ρ = 4 (Stefan)

ε

δ

Error Norm

ε

δ

Error Norm

0.000 0.002 0.005

0.04 0.06 0.06

0.0866 0.0921 0.1014

0.000 0.002 0.005

0.04 0.06 0.06

0.0867 0.0929 0.1038

138

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

TABLE 5.2. Error Norm as a Function of the Level of Noise in Problem 2 ρ = 4 (Stefan)

ρ = 1 (Newton) e

δ

Error Norm [0, l]/[0.3,1]

ε

δ

0.000 0.002 0.005

0.04 0.06 0.06

0.5208/0.0183 0.5560/0.0631 0.5879/0.1108

0.000 0.002 0.005

0.04 0.06 0.06

Error Norm [0, l]/[0.3,1] 0.5135/0.0373 0.5673/0.0675 0.5935/0.1375

2.26-1

-0.26-|—ι—ι—ι—ι—[0.00

0.26

0.60

Τ I me

voluee

Fig. 5.1 Reconstructed boundary source function. Newton's law—Problem 1: e = 0.005, δ = 0.06, h = k = 0.01. Exact ( · · · ) ; computed (—).

NUMERICAL PROCEDURE

139

If the discretized computed radiation law function component is denoted by E" = E (t ) and the true component is E" = E(t ), after evaluating the ordered pairs (Vq, Ε"), 0 < η < Μ - Ν, we obtain a graph of the approxim a t e functional relationship between the radiation law and the t e m p e r a t u r e at the interface. This plot is then compared with the exact graph corresponding to the values ( / ( i „ ) , E(t )) of the model problem. Tables 5.1 and 5.2 show the results of our numerical experiments associated with Problems 1 and 2, respectively, when attempting to identify the transient source function at the interface. T h e uniformly smaller error norms in Problem 1 are expected since at time t = 0.2 the exact source solution has a finite-jump discontinuity whereas in Problem 2 the exact source solution has an infinite j u m p at time t = 0.2. For this reason, we have added an extra column in Table 5.2 indicating the error norms in the time interval [0.3,1], after the discontinuity. It is clear that the method rapidly dissipates the effect of the singularity, a very desirable feature. a

a

n

n

n

-β·25

I β.ββ

ι

ι

ι

ι

ι 0.25

ι

ι

ι

ι

ι

ι

ι

ι

ι

0.50

ι 0.75

ι

ι—ι—r—ι 1.00

Tlme-voIues

Fig. 5.2 Reconstructed boundary source function. Stefan's law—Problem 1: « = 0.005, 8 = 0.06, h=k = 0.01. Exact ( · • · ) ; computed (—).

140

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP 1.75-1

ΤθΓηρθΓ o t u r e Fig. 5.3 Reconstructed Stefan radiation law. Problem 1: e = 0.005, δ = 0.06, h = k = 0.01. Exact (—); computed ( * * * ) .

T h e qualitative behavior of the reconstructed transient source function for Problem 1 is illustrated in Figs. 5.1 and 5.2 where the numerical solution for an average perturbation ε = 0.005 (full line) is plotted for ρ = 1 (Newton's cooling law) and ρ = 4 (Stefan's radiation law), respectively. In Fig. 5.3 we show the graph associated with the reconstructed nonlinear radiation law as a function of the approximate temperature at the interface for ρ = 4 (starred symbols) and the exact boundary radiation law (full line). Figures 5.4 and 5.5 show the computed source functions (full lines) for ρ = 1 and ρ = 4, respectively, for Problem 2 and for the noise level ε = 0.005. In both problems the source functions to be identified have discontinuous histories and in one case an infinite jump. T h e algorithm restores stability with respect to the data, which is essential for the introduction of the inverse problem approach, and good accuracy is obtained, even for small time sample intervals and relative high noise levels in the data.

IHCP WITH PHASE CHANGES

141

4.60-

c 0

3.60-

- J

*J

0 c 3

3.00-


2.58-

XI Φ

*-> o 3 £.

*J

2.00-

_

CO 1 . 5 0 c 0 Ü

Φ

&.

_ -

1.00-

0.50-

0.00-

kM

0.00

f/**

■ ■

i

I

0.26

i i

i

I

0.60

i

I

i

i

I i

0.76

Time-ve lues

1.00

Fig. 5.4 Reconstructed boundary source function. Newton's law—Problem 2: ε = 0.005, δ = 0.06, h = k = 0.01. Exact ( · · · ) ; computed (—).

5.3 IHCP WITH PHASE CHANGES In many applications related to solidification phenomena in phase-change materials, it is important to know with good accuracy the evolution of the solid-liquid interface. It is possible to consider suitable inverse heat conduction problems defined only on the liquid domain or on the solid domain to approximately determine the transient location of the interface. In this setting, the onedimensional Stefan problem is simply considered as a one-dimensional IHCP with an unknown moving boundary location which is assumed to be an isotherm. To solve the associated IHCP, the temperature and heat flux history functions are needed at one location of the liquid (solid) domain or, alternatively, two temperature histories at two different locations of the liquid (solid) domain must be measured. This point of view amounts to

142

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP 4.00-1

- 0 . 5 0 —I ι—ι—ι—ι—|—ι—ι—ι—r—]—ΓΊ—I 0.00 0.25 0.50

Tlme

I | I—I—Γ Γ ] 0.76 1.00 -

-

values

Fig. 5.5 Reconstructed boundary source function. Stefan's law—Problem 2: ε = 0.005, δ = 0.06, h = k = 0.01. Exact ( • · · ) ; computed (—).

replacing Stefan's well-posed problem by the I H C P , an inherently ill-posed problem: small errors in the input data might induce large errors in the computed interface temperature which is needed to locate t h e front and, consequently, special methods are necessary in order to restore continuity with respect to t h e data. W e consider the implementation of the mollification method, originally developed for the solution of the I H C P as a front-tracking space marching procedure to determine t h e location of the solid-liquid interface in o n e dimensional melting problems. The space marching method is also applied to the solution of the inverse Stefan problem: if the front evolution and t h e heat flux at t h e melting interface are approximately known, the task consists now of finding the t e m p e r a t u r e and heat flux histories at the fixed liquid (solid) boundary location. It is quite natural to attempt to solve this problem with some variant

DESCRIPTION OF THE PROBLEMS

143

of the I H C P since it is well known that the inverse Stefan problem is ill posed with respect to perturbations in the data.

5.4

DESCRIPTION O F T H E P R O B L E M S

Stefan Problem Mathematically, the one-dimensional one-phase Stefan problem can be defined as follows. Assuming linear heat conduction with constant coefficients and dimensionless quantities, the unknown t e m p e r a t u r e in the liquid phase u(x, t) satisfies u,{x,

t) = u (x,

t),

xx

u(x,0)

0 < χ < s(t),Q

= u (x),

0 <x

0

u(0,t)=F(t), u(s(t),t)

<s(0),

0 < t < T,

ds = —(t)=q{t),

x

<s(0),0 0
= 0,

-u {s(t),t)

< t < T,

0
where the initial temperature distribution u (x) and the fixed boundary t e m p e r a t u r e history F ( i ) are known and one has to determine the transient interface location s(t) and the heat flux at the front q(t). For the application of the I H C P techniques to the approximate solution of this problem, it is necessary to also measure the heat flux history at the fixed boundary χ = 0 or, alternatively, to measure the t e m p e r a t u r e history at another location JC,, 0 < JC, < s(0). Thus, we consider the problem 0

u (x,t) s

= u (x,t),

0 <x

= u (x),

0 < x < s(0) > 0 ,

(5.12b)

0 < i < 7 ,

(5.12c)

xx

u(x,0)

Q

u(0,i) = F(r),

< s(t),0

with corresponding approximate data function -u (Q,t) x

= Q(t),

-u {s{t),t) x

(5.12a)

F (t), m

0
with corresponding approximate data function u{s(t),t)

< t < T,

(5.12d)

Q (t), m

=0=/(r),

0 < r < 7 \

(5.12e)

ds = -{t)=q(t),

0
(5.12f)

144

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

We also assume that all the functions involved are L functions in any time interval D of interest and that the exact data functions F(t) and Q{t) and the measured data functions F (t) and Q U) satisfy the L data error bounds 2

m

I I F - FJD

m

< ε

2

\\Q - QJD

and

< ε.

Stabilized Problem Liquid Phase Mollifying system (5.12), we obtain the following associated problem: attempt to find J f U) = J$u(s (t), t) and J q U) = -J u (s (t), t) at some point / of interest and for some radius δ > 0, given that J u(x,t) satisfies in the liquid phase s m

s

s

m

s

x

s

s

(J u), J u(x,0) s

J u(0, s

( S»)xx>

0 < χ < s (t),

0 < t

J u (x,0),

0 <x

> 0,

hF {t),

0 < t < T,

s

J

s

s

t)

0

m

s

0 < t < T.

-J u (s (t),t) s

x

(5.13)

0 < t < T,

J u(s (t),t) s

s

0 < t < T,

W o , ' )

-

<s (0)


s

T h e normalized version of this problem and its solutions satisfy the following special version of T h e o r e m 3.4. Theorem 5.1 Suppose that 0 < s (0) < s (t) ε, and \\Q - Qj < ε. T h e n e

a

< 1, 0 < t < T, \\F - FJ\

D

<

D

(i) Problem (5.13) is a formally stable problem with respect to perturbations in the data. (ii) If the exact boundary temperature function fit) and the exact heat flux function q(t) have uniformly bounded first-order derivatives on the b o u n d e d domain D = [0, T], then J f and J q verify s

m

s

I I / - / / J I D < \M8 + ε ε χ ρ [ ( 2 δ ) S

m

2 / 3

]

(5.14)

and \\q - J q \\ s

m

D

< \Μδ + 2ε ε χ ρ [ ( 2 δ )

_ 2 / 3

].

(5.15)

Once the mollified t e m p e r a t u r e function has been evaluated, it is feasible to attempt to identify the interface function s (t). s

DESCRIPTION OF THE PROBLEMS

145

Identification of the Interface Function s(t) In general, the recovered mollified temperature J f will be 0 at the approximate isotherm s (t) which is different from the exact isotherm s(t). However, since the function J u(s, t) is strictly increasing with respect to the first variable s for 0 < s(0) < s(T) < 1, u n d e r the condition \\dJ u/ds\\ > k > 0, it is possible to estimate the error \\s — s|lo. From s

m

a

s

a

a

D

a

- ν * ( > · ) Ι Ι θ ^ sh 5

\\ 8 ( 8'-) J

Κ

U

S

~

k

S \\ , S

D

- 4D * ( ^ β Γ Ί ΐ ' β Φ β . · ) ~ " ( * . ' ) + Φ,') = (*e)

- ,

ll«( »") s

-

V (

»

J

-

J u(s (t),t) a

=J f (t)

s

D

t) = 0.

a

and

a m

a

) L »

taking into account that u(s, t) = 0 and J u(s , Hence, using the estimate (5.14) with s

-J u(s,-)\\

iiO(f),r)

=/(f),

we get lis, - s\\

< (k y [±M8 l

D

a

+ £exp{(25)"

2 / 3

}].

(5.16)

Remarks 1. Notice that the approximate interface function s is actually a function of t h e radius of mollification δ and the amount of noise in the data ε. s

2. Inequality (5.16) can b e used to show the convergence of s t o s in the L norm as in R e m a r k 2 in Sec. 5.1. a

2

Solid Phase Clearly, it is also possible to state a similar problem for the solid phase: attempt to find J f U) = J u(s (t\ t) = 0 and J q U) = -J u (s (t), t) at some point of interest t and for some radius δ > 0, given that J u(x, t) satisfies in the solid phase s m

s

s

e

m

s

x

a

a

(^")

= (h*)xx>

s (t)

=J u (x,0),

s (Q) <x < 1,0 < s ( 0 ) < 1,

(

J u(x,0) a

a

o

s

x

a

a

a

a m

-J u (s (t),t)=J q (t), s

x

a

a

m

0
m

J u(s (t),t)=J f (t)


0
m

-J u {\,t)=J Q {t), a


a

J u(\,t)=J F {t), a

<x <1,0

a

=0,

{

·

0T, 0
Remark. W e notice that the same stability bounds developed for the "forward in space" problem in the liquid phase apply also to the solutions of the Stefan problem in the solid phase.

146

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

Inverse Stefan Problem In the formulation of problem (5.11), assume that a given initial t e m p e r a t u r e distribution u (x), a prescribed interface of zero t e m p e r a t u r e s(f), and the transient heat flux at the front -u (s(t), t) = ds/dt{t) = q(t) are known. T h e inverse Stefan problem consists on finding the t e m p e r a t u r e and heat flux history functions at the fixed boundary χ = 0. This problem can be immediately reinterpreted in terms of the "backward in s p a c e " I H C P in the liquid phase, with an "initial" transient boundary. 0

x

Stabilized Problem Proceeding as in the previous sections, we obtain the associated well-posed normalized problem: attempt to find J F (t) = J u(0,t) and J Q (t) = - 7 ^ ( 0 , t) at some time t of interest and for some radius of mollification δ > 0, given that J u(x, t) satisfies in the liquid phase S

m

&

s

m

s

( s")xx>

0 <X < s(t)

J u (x,0),

0 <X < s ( 0 ) , 0 < s ( 0 ) < 1,

J

J u(x,0) s

s

0

J u(0,t)

0 < t < T, unknown,

s

-J u (0,t) s

0 < /

x

J u(s(t),t) s

0 < t

= o,

hf (t) m

-J u (s(t),t) s

< 1, 0 < t < T,

0 < t

x

(5.18) < T, unknown, < T, < T.

Remarks 1. T h e data function q U) represents the measured heat flux function at the interface and we assume ||<7 - q\\o < ε. m

m

2. T h e conclusions of Theorem 5.1 also apply here with the obvious changes in notation, that is, the corresponding estimates now read \ \ F - J

S

F J

< \Μδ + ε ε χ ρ [ ( 2 δ Γ

D

2 / 3

]

(5.19)

and WQ-JSQJD

< ΪΜδ + 2 ε ε χ ρ [ ( 2 δ ) "

2 / 3

].

(5.20)

T h e computational details are presented in the next section.

5.5

NUMERICAL PROCEDURE

W e approximate t h e mollified partial differential equation associated with system (5.11) with t h e consistent finite-difference scheme (3.47), introduced

NUMERICAL PROCEDURE

147

in Chapter 3, with ^ ( x , , i„) = J u(x t ) and W^JC,-, t ) = -d/dxJguix^ r„), marching forward in the χ direction. At each space step, once the t e m p e r a t u r e V" and the heat flux W" have been reconstructed for 0 <, η < « , we proceed with the approximate identification of the front s by first collecting the pair of indices (i, η - 1), (ί', η) for which V"' ' V" ^ 0, 1 < η < « (vertical search). Thus, for the space location ih, the time location of s is obtained by linear interpolation. T h e horizontal search proceeds similarly: after collecting the pair of indices (i - 1,«),(:', n) for which V"_ · V" < 0, i < η < « , for the time location ik, the space location of s is then obtained by linear interpolation. s

h

n

n

m a x

s

1

m a x

s

x

m a x

s

Remarks 1. While marching forward in space, the horizontal search for the location of the space coordinate of the interface S at the time grid points requires the storage of the computed time vector t e m p e r a t u r e solution at the previous space location. s

2. T h e implementation of the numerical procedure for the solution of the normalized Stefan problem in the solid domain proceeds as explained previously except that the data functions are given at the fixed location χ = 1 and we march backward in space. Numerical Results Stefan Problem: Liquid Phase In order to test the accuracy and the stability properties of our method, in Problem 3 the approximate reconstruction of an interface function s (t) is investigated for a one-dimensional one-phase Stefan problem exposed to a heat flux data function at the liquid (fixed) surface χ = 0 given by s

-«(0,0 = g(0

= 2-

1 / 2

exp(l - 2 - '

/ 2

+ t/2),

t > 0,

and a t e m p e r a t u r e data function « ( 0 , 0 = F(t)

= exp(l - 2 "

1 / 2

+ r / 2 ) - 1,

/ > 0.

If the initial t e m p e r a t u r e distribution is u(x,0)

= exp(l - 2 ~

1 / 2

- 2~

1 / 2

0 < χ < s(0),

Λ : ) - 1,

then the exact temperature and heat flux functions in the liquid phase are given respectively by u(x,t)

= exp(l - 2 - ' /

2

+ t/2

- 2~ x) l/2

- 1,

0 < χ < s(t)

< 1, 0 < t < T,

148

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

and -u (x,t)

= (u(x,t)

x

+ 1)2~

1 / 2

0 <x

,

< s(t)

< 1,0 < t < T.

The exact interface solution function to be approximately reconstructed has equation sit) = 2~ t + 2 - 1 and the exact t e m p e r a t u r e and heat flux functions at the melting surface are, respectively, u(s(t),t) = fit) = 0, 0 < t < T, and -u isit\ t) = qit) = 2 ' , 0 < t < T. In Problem 4 we attempt t o approximately reconstruct the transient interface function sit) and the heat flux -u isQ), t) = qit) at the melting front for a normalized one-dimensional Stefan problem with exact data functions 1/2

1 / 2

1 / 2

x

x

w ( 0 , i ) = Fit)

= -t

- 0.5,

0
- « , ( 0 , 0 = Q(/) = - 2 ,

0 < / < 7 \

and initial temperature distribution, in the liquid domain, given by u(*,0) = -0.5x

2

+ 2x - 0.5,

0<x
T h e exact temperature solution in the liquid phase is uix,t)

= -0.5x

+ 2x - t - 0.5,

2

< s(0

0 <x

< 1, 0 < t < T,

and the exact heat flux has equation -u ix, x

t) = χ - 2,

0 <x < sit)

< 1,0 < t < T.

It follows that the exact transient interface location is given by sit) = 2 — (3 - 2 / ) , 0 < t < T, and the t e m p e r a t u r e and heat flux solutions at the liquid-solid interface are, respectively, uisit), t) = fit) = 0, 0 < t < T, and -u isit), t) = qit) = - ( 3 - 2 ? ) , 0 < t < T. Once the radii of mollification 8 and 8 associated with the data functions F and Q , respectively, and the discrete filtered data functions h U) = V£ and J Q it„) = W£, 0 < η < Μ, are determined with δ = m&xiB , 5 Q ) , we apply the finite-difference algorithm described previously in this section, marching forward in the χ direction and activating the horizontal and vertical search techniques, as explained before, to approximately determine the location of the melting front. The coordinates ix , t ) so obtained are t h e n taken as the accepted approximations for the interface location. Finally, the discrete heat flux history qix , t ) is evaluated at those coordinates by linear interpolation. In all cases, we use h = Ax = 0.01 and k = At = 0.01. The noisy data are obtained by adding a random error to the exact discrete data function at every grid point of the extended data interval. 1 / 2

1 / 2

x

F

m

Q

m

F

m

n

B

m

F

s

s

s

s

NUMERICAL PROCEDURE

149

If the discretized computed transient front function component is denoted by s% and the true component is s" = s(t ), the weighted I norm of the solution error is then given by 2

n

-.1/2

lis* -

(5.21)

s\\p

where L indicates the number of points of the discrete approximation s in the time interval D of interest. T h e qualitative behavior and the error norm of the reconstructed transient front function for Problem 3 is illustrated in Fig. 5.6 where the numerical solution for an average perturbation ε = 0.005 (full line) and the exact s

0.90-1

0.Θ0

x-vaIues

Fig. 5.6 Reconstructed front for Problem 3: ε = 0.005, δ = 0.04, h = k = 0.01, Hi - i || = 0.0041. Exact ( * * * ) ; computed (—). 4

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

150

1.10-1

x-vaIues

Fig. 5.7 Reconstructed front for Problem 4: ε = 0.005, δ = 0.04, h = k = 0.01, lis - s \\ = 0.0032. Exact ( * * * ) ; computed (—). s

solution are plotted. The computed front function (full line) and the corresponding discrete error norm given by (5.21), for Problem 4 and for the noise level ε = 0.005, are shown in Fig. 5.7.

Stefan Problem: Solid Phase. To emphasize the versatility of the method, in Problem 5 we consider a one-dimensional semi-infinite body, initially at zero temperature, exposed to a heat flux data function at the surface χ = 0 given by

-",(0,0 =

[ir(/ - 0 . 2 ) ] " 0,

1 / 2

,

/>0.2, 0 < t < 0.2.

NUMERICAL PROCEDURE

151

Hence, the exact temperature distribution has equation (

u

x

t

) = |erfc{jc/(2[r - 0 . 2 ]

1 / 2

χ > 0, t > 0.2,

)},

\θ,

x>

0,0

< t

<0.2,

and in order to model the Stefan problem in the solid domain, we measure the data temperature function

M

(

l

j

O

=

/

(

O

(erfc{l/(2[ -0.2r)},

ί > 0.2,

i

=

[θ,

0
and the data heat flux function =q(t)

-u,(l,i) =

- 02)]- exp{-[4{t

l[ir(t

l/2

- 0.2)]- }, 1

t > 0.2, 0 < t < 0.2,

\θ,

and attempt to determine the location of an isotherm u(x, t) = e r f c [ x / ( 2 r ) ] = constant, 0 < JC < 1, / > 0.2. For instance, if we set u(x,t) = 0.257899, we find x/[2(t )] = 0.8 and the exact location of the simulated "melting" front is given by I / 2

1/2

Λ

)

\0,

2, 0 < i < 0.2.

T h e computed front s for this example, obtained by using the finite-difference scheme (5.10)—with the vertical and horizontal front-tracking proced u r e s — n o w marching backward in space, is plotted in Fig. 5.8 (full line) together with the corresponding exact front location and the associated discrete I error norm. Notice that the noisy data functions are now given by a

2

/„('„) = / ( ' „ ) + * „ , , and

where t indicates a grid point in the data time interval χ = 1 and ε ε„ , are Gaussian random variables of variance σ = ε. n

η

2

, and

152

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP 0.66-

0.60-

0.65-

0.60

Η

0.46H C

>

0.40·

CD

ε

I-

0.36H 0.30-

0.26

Η

0.20

0.16-j—ι—ι—|—ι—j—ι—ι—|—|—|—|—|—|—|—|—|—|—|—r 0.00 0.25 0.60 0.75 1.00

X-VOIues

Fig. 5.8 Reconstructed front for Problem 5: ε = 0.005, δ = 0.04, h=k lis - ί ί | | = 0.0043. Exact ( * * * ) ; computed (—).

= 0.01,

Inverse Stefan Problem For our last example, Problem 6, we utilize the inverse Stefan problem associated with Problem 3. W e recall that in this case the exact location of the front is given by s(t) = 2 " f + 2 - 1, 0 < t < T, and for the formulation of t h e inverse Stefan problem, t h e two data functions are described by 1 / 2

u(s{t),t)

1 / 2

= / ( i ) = 0,

0
and -u (s(t),t) x

= ds/dt(t)

= q(t)

= 2~

1 / 2

,

0 < t < T,

with initial temperature distribution given by K ( J C , 0 ) = exp[l - 2 " / 1

2

- 2"

1 / 2

Λ:] -

1,

0 <x

<s(0).

NUMERICAL PROCEDURE

153

In this problem, the task consists of recovering the t e m p e r a t u r e and the heat flux functions « ( 0 , i ) = F(t)

= e x p [ l - 2'

+ t/2]

1/2

- 1,

0 < t < 1,

and -u (0,t) x

= Q(t)

= 2"

1 / 2

exp[l - 2"

1 / 2

+ t/2],

0 < t < 1,

at the fixed liquid boundary χ = 0, marching backward in space. In this case, the vertical and horizontal search procedures for the numerical tracking of the melting front are no longer necessary. However, instead of starting at the front location s(t), we first introduce the transformation r = x/s(t), t > 0. In this manner, the parabolic problem in the new variables 1.30-,

0.00

0.2E

0.60

0.76 Τ

1.00

1.26

tme

Fig. 5.9 Reconstructed temperature in Problem 6: ε = 0.005, 8 = 0.04, h = k = 0.01, WF ~ hF W = 0.0038. Exact ( * * * ) ; computed (—). m

154

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

for the inverse Stefan problem becomes urr κ(1,/) - u r (l, 0 «(r,0)

= -rsur ds/dt + s2u,, 0 < t < T,0 < r < 1, s = s(t), =/(f) =0, 0
The finite-difference scheme consistent with the new partial differential equation is updated as follows: W? = [l + (i + I)h2-i/2(nk2-l/2

+ 21/2 + 1 ) ] ^ ,

-ink2-^ + 2>/2 - ιγ^ιν

n

Vi = Vi"+1-hWi",

- vr+v),

« = 1 , 2 , . . . , M - 1,

i=

N-l,...,l,0.

Figures 5.9 and 5.10 illustrate the qualitative behavior and the discrete error norms of the computed temperature and heat flux grid solutions {F0n} 1.6β-ι

0.00

I I I I | I I I I | I I I I | I 1 I I I I I I ' I 0.2E 0.60 0.75 1.00 1.2E

Tim©

Fig. 5.10 Reconstructed heatfluxin Problem 6: e = 0.005, δ = 0.04, h = k = 0.01, Ilo - hQmW = 0.0041. Exact (***); computed (—).

SEMI-INFINITE BODY

155

and [W "}, respectively, at the boundary location χ = 0 (full line) for a noise level ε = 0.005. T h e exact t e m p e r a t u r e and heat flux solutions are also plotted for graphical comparison and we notice that the initial oscillations in the approximate solutions—notably for the reconstructed heat flux—disappear almost immediately as the method recovers very quickly. In all cases, the numerical experiments corroborate the theoretical stability properties of the algorithm; it effectively restores stability with respect to perturbations in the data, which is essential, in particular, to justify the introduction of the inverse problem approach for the numerical solution of the Stefan problem, a well-posed problem. {

5.6 IDENTIFICATION O F T H E INITIAL T E M P E R A T U R E DISTRIBUTION We claim that the usual data for the one-dimensional I H C P discussed in C h a p . 3 uniquely define the initial condition for the direct problem in the finite slab case. Moreover, it is also possible to analyze a suitable model problem for the semi-infinite conductor that permits us to solve the I H C P numerically a n d — a t the same t i m e — t o identify the initial t e m p e r a t u r e distribution. These tasks are achieved by slightly modifying the usual space marching finite-difference scheme with the addition of an extrapolation "at the boundary" procedure.

5.7

SEMI-INFINITE BODY

W e consider a one-dimensional semi-infinite conductor in which the temperature history on the active boundary surface χ = 0 is desired and unknown, the initial temperature distribution is also desired and unknown in the space interval 0 < χ < 1, and the t e m p e r a t u r e history at the interior point χ = 1 is approximately measurable together with the initial temperature distribution for χ > 1. Physically, this corresponds to the situation where there is no information whatsoever regarding a region—presumed inaccessible—surrounding the active boundary of the conductor. Mathematically, assuming linear heat conduction with constant coefficients, the new inverse problem can be described as follows: the t e m p e r a t u r e solution w(x, t) satisfies

w,{x,t)

w(l,0

w (x,t),

t > 0, χ > 0,

(5.22a)

a(t),

t > 0,

(5.22b)

xx

156

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

a {t),

with corresponding approximate data function \ν(χ,0)=β(χ),

m

ί >0,

(5.22c)

unknown for 0 < χ < 1, with corresponding approximate data function χ > 1, t>Q,

w(0, ( ) = / ( ' ) >

β„(χ),

(5.22d)

the desired but unknown t e m p e r a t u r e function, w(x,t)^>0

as

χ -> °°, t > 0.

In order to be able to identify / ( i ) , f > 0, and β(χ), first the intermediate direct problem z,{x,t)

= z (x,t), xx

z(0,f)=0,

as

0 < χ < 1, we consider

t > 0, χ > 0,

(5.23a)

f>0,

(5.23b)

' < * • » > - " > " Ζ(Λ:, 0 ^ 0

(5.22e)

'

1

(

Λ: - » oo, t > 0.

5

Λ

)

(5.23d)

By superposition, the function u(x, t)—of t h e new inverse problem obtained by subtracting t h e solution of problem (5.23) from the solution of problem (5.22)—satisfies u,(x,

t) = u (x,

t),

xx

«(1,0

=F(t)

=a(t)

-2(1,0,

t > 0, χ > 0,

(5.24a)

f>0,

(5.24b)

with corresponding approximate data function F (t)

= a (t)

m

«(*,0) =

° * * !' χ > l, x

lo,

κ(0,0=/(0, Μ(Λ:,0^0

u

n

k

n

o

i > 0 , unknown, as

- z(l, 0,

m

χ - » =o, ί > 0.

w

n

'

(5.24c) (5.24d) (5.24e)

In what follows, system (5.24) will always b e considered our prototype model, that is, we shall assume that problem (5.23) has been solved and the original data for t h e inverse new problem (5.22) have been u p d a t e d accordingly, so that the inverse problem (5.24) is now feasible.

FINITE SLAB SYMMETRY

157

Remarks 1. Notice that the interior history t e m p e r a t u r e function «(1, t) = Fit) in problem (5.24) can be thought of as " i n d u c e d " by the initial data function βix) and the boundary data function fit) of the associated direct problem, and, as such, it has all the necessary data information for the solution of the inverse problem. 2. T h e prototype system (5.24) is somewhat intermediate between the classical backward-in-time heat conduction problem and the inverse heat conduction problem. W e recall that for the backward-in-time problem, u(x, T) = gix), Τ > 0, 0 < χ < 1, «(0, t) = fit), 0 < t < T, and « ( 1 , t) = Fit), 0 < t < T, are given and the initial t e m p e r a t u r e distribution uix,0) = βix), 0 < χ < 1, is sought. For the I H C P instead, uix,0) = βix), 0 < χ < 1, and « ( 1 , i ) = Fit), t > 0, are given and the boundary condition u(0, i ) = fit), t > 0, needs to be identified. Uniqueness. Before attempting to solve problem (5.24), it is important to show that the solution—assumed to exist on physical grounds—is unique. T o that effect, suppose that problem (5.24) admits two different solutions w,(jt, t) and u ix, t), analytic functions of their arguments χ and t. T h e n the difference function υ = u — u is such that 2

x

v,(x,t)

2

= v (x,t),

t>0,x>0,

xx

ν(0,ί)Φ0,

t > 0,

v(x,0)

=0,

χ > 1,

υ(χ,0)

Φ0,

0 <x

v(l,t)

= 0,

t > 0.

< 1,

Consequently, in the region χ > I, t > 0, vix, t) = 0 and by the uniqueness of the analytic continuation procedure, vix, t) = 0 everywhere in the q u a r t e r plane χ > 0, t > 0. •

5.8

FINITE SLAB SYMMETRY

In the case of a finite-length conductor, the corresponding new inverse problem can be described as follows: the t e m p e r a t u r e solution uix, t) satisfies u,(x,

t) =u (x, xx

«(1,0=^(0.

t),

0<*<1,ί>0,

(5.25a)

' > 0,

(5.25b)

158

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

F (t),

with corresponding approximate data function

-ιι (1,0 = β ( 0 ·

m

ί

>

'

0

χ

- « , ( 0 , / ) = q(t), u(x,0)

= β(χ),

5

Q U),

with corresponding approximate data function u(0,t)=f(t),

( -25c)

m

i>0,unknown,

(5.25d)

t > 0, unknown,

(5.25e)

0 < χ < 1, unknown.

(5.25f)

Remark. System (5.25), when the initial temperature distribution β(χ)—in (5.25f)—is known, constitutes the usual statement of the I H C P for the one-dimensional finite slab geometry. Uniqueness. Once again, we need to show that the system (5.25) has a unique solution. Suppose that problem (5.25) admits two different solutions u (x, t) and u (x, t). T h e n the function υ = u - u satisfies x

2

x

v,(x,t) v(\,t) -v (l,t) x

v(0,t)

2

= v (x,t),

0 <x
=0,

t > 0,

= 0,

f>0,

xx

= Af(t)

Φ 0,

x

= Aq(t)

Φ0,

v(x,0)

= Αβ(χ)

- v (x,t)

#0,

> 0,

t>0, t > 0, 0 <x < 1.

If we imbed the slab domain [0,1] X [ 0 , 0 symmetrically into the slab domain [0,2] X [0, 0 , the relationship among the " b o u n d a r y " d a t a functions v(0, t) = Δ / , fX2, Ο = Δ / , - u ( 0 , Ο = Aq, -v (2, t) = Aq, v(x,0) = Αβ, 0 <x < 1, and v(x, 0) = Αβ, 1 <x < 2, is uniquely determined by the conditions that the function ν must satisfy at χ = 1 for all t > 0. Given that we do not consider interior heat sources, the condition v(l,t) = 0, t > 0, can be achieved if and only if Δ / = - Δ / , Δ # = - Aq, and Αβ = -Αβ. O n the other hand, the condition -v {\, t) = 0, t > 0, can be obtained if and only if Δ / = Δ / , Aq = Aq, and Αβ = Αβ. Thus, it follows that x

x

x

Af=Aq

=

Aβs0

and this implies that the inverse problem (5.25) has a unique solution.

•

STABILIZED PROBLEMS

5.9

159

STABILIZED PROBLEMS

Mollifying system (5.25), we obtain the following associated problem: attempt to find J f (t) = J u(0,t), J q U)= -J uJS),t\ and J u(x,0) = β(χ) at the points of interest and for some radius δ > 0, given that J u(x, r) satisfies e m

a

s

m

s

a

a

ih ), u

=

J u{\,t)

( S J

=

s

U

)

X

X

t>Q,x>0,

>

t > 0,

W ) .

-•WM)

t > 0,

J u(0,t)

t > 0, unknown,

-•Wo,0

t > 0, unknown,

s

J u(x,0)

(5.26)

0 < χ < 1, unknown.

a

T h e computational details and the necessary extrapolation procedure n e e d e d to numerically identify the initial temperature distribution β(χ) are presented next. Numerical Procedure With ν = J u and ζ = -dv/dx,

system (5.26) is equivalent to

a

dv

dz

Tt ~

dx'

t > 0, 0 <x

dv

t > 0, 0 < χ < 1,

dx ' v(l,t)

=J F (t),

/ > 0,

2(1,0

=J Q (0,

t > 0,

s

m

s

m

=hf {t),

t > 0, unknown,

=JsQ (0,

t > 0, unknown,

«(o,0

m

z(0,0

< 1,

m

v(x,0)

=Wx),

(5.27)

0 < χ < 1, unknown.

Without loss of generality, we seek to reconstruct the unknown mollified boundary temperature function J f and the mollified boundary heat flux function J q in the unit interval /, = [0,1] of the time axis χ = 0 and the mollified initial temperature distribution J β (x in the unit interval I = [0,1] of the space axis t = 0). Consider a uniform grid in the (x, t) space: a

a

m

m

a

{(x = t

x

in, t = nk), i = 0 , 1 , . . . , N, Nh = 1; n

η = 0 , 1 , . . . , Μ , Mk = L = 1 - k +

k/h).

160

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

Let the grid functions V and W b e defined by V" = v(x

W? = z ( * , , 0 >

t ),

h

n

0


<M.

Notice that VN=hF {t ) m

and

n

WZ=J Q {t ), s

m

0
n

We approximate the partial differential equation in system (5.27) with the consistent finite-difference scheme

vr-^vr-nwr^, i = N,N

-

(5.28)

Ι,.,.,Ι,η = M -

Ι , Μ -

2,...,1,

obtaining a local truncation error that behaves as 0(h + k ) for the discrete mollified t e m p e r a t u r e and Oih + k ) for the discrete mollified heat flux as h, k -> 0. W e assume that the data functions F and Q are discrete functions measured at equally spaced sample points t-•.= jk, j = 0 , 1 , . . . , M, in I = [0, L]. W e recall that in o r d e r to compute the discrete mollified functions J F (tj) and J Q Uj), we n e e d to extend the data functions in such a way that F and Q decay smoothly to 0 in I ^ = [-8 ,L + 8 ], k < 8 < 0 . 1 , and both are 0 in R - I . In what follows, we consider the extended discrete data functions F and Q defined at equally spaced sample points on any time data interval of interest. T h e selection of the radius of mollification is implemented as usual, solving the discrete equations 2

2

2

m

m

L

s

m

s

m

m

m

LS

max

L

mm

m3X

δ

m

m

and WhQ

m

~ G J k ,

w

= *,

using, for example, the bisection method. T h e space marching technique discussed here requires the replacement of the discrete t e m p e r a t u r e and heat flux data functions F and Q at χ = 1 by their filtered—and suitably extended—discrete mollified versions J F and JQ , respectively. In most cases, the "compatibility" condition β(1) = J F (0) will be lost in the process, even if the initial t e m p e r a t u r e function β(χ) is known a n d / o r replaced by J / 3 ( x ) after mollifying the initial condition function in the χ direction. As we march backward in space, the initial discrepancy might propagate generating a " b o u n d a r y layer" effect that inm

m

s

s

m

a

m

s

m

NUMERICAL RESULTS

161

fluences the initial values of the computed solutions J f and J q on the time interval /, = [0,1], at the active interface χ = 0. However, it is possible to attempt to control this situation in a practical m a n n e r by replacing—after each step in s p a c e — t h e unknown mollified initial t e m p e r a t u r e 7 /3(//ι) = V®, i = N,N - 1 , . . . , 0, by the linear extrapolation value obtained from the already estimated quantities V* and V , introducing an extra local error of order O(k) in the sequel. T h e unknown approximate mollified initial t e m p e r a t u r e is calculated directly using the formula s

m

h

m

8

2

V

= 2V -V ,

a

l

i

i = N,N

2

i

- 1,... ,0.

Once the radii of mollification 8 and S associated with the data functions F and Q , respectively, and the discrete filtered data functions J F (t ) = V£ and J Q (t„) = W$, 0 < η < Μ, are determined, we apply the finite-difference algorithm (5.28)—modified " a t the b o u n d a r y " as explained previously—marching backward in the χ direction with δ = max(8 , 8 ). F

m

s

m

n

s

F

Q

m

m

Q

Remark. For the semi-infinite conductor, the numerical scheme remains exactly the same if the temperature and the heat flux histories are measured at the interior point χ = 1. T h e implementation is quite similar with obvious changes if, instead, two t e m p e r a t u r e histories are measured at two different interior point locations χ = 1 and χ = 1 - h. 5.10

NUMERICAL RESULTS

In o r d e r to test the accuracy and stability properties of the method, in Problem 7 the approximate reconstruction of an initial t e m p e r a t u r e distribution function u(x, 0) = β(χ), 0 < χ < 1, is investigated for a one-dimensional finite slab exposed to a t e m p e r a t u r e data function at the free surface χ = 1 given by « ( 1 , i ) = F(t) = 1 + e~'(cos 1 + sin 1), t > 0, and a heat flux data function -u {\, t) = Q(t) = 0. T h e exact t e m p e r a t u r e solution for this problem is x

u(x, t) = 1 + e ~ ' ( c o s χ + sinx),

0 < χ < 1, t > 0,

and the initial t e m p e r a t u r e distribution to be approximately reconstructed has equation u(x, 0) = β(χ) = 1 + cos χ + sin x. T h e t e m p e r a t u r e and heat flux history functions at the χ = 0 boundary are also unknowns for this inverse problem and the exact solutions are u(x, t) = / ( f ) = 1 + e ' and —u = q(t) = e"'(sin χ - cos x). In Problem 8 we attempt to approximately reconstruct the initial temperature function β(χ) for a finite slab with data functions « ( 1 , t) = F(t) = 0, t > 0, and -u (l, t) = Qit) = ne'" ', t > 0. T h e unique t e m p e r a t u r e solution is given by u(x, t) = e ' sin vx. Thus, the exact unknown functions for the inverse problem are u(x,0) = β(χ) = -

x

2

x

_ 7 r

162

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

sin πχ, 0 < x < 1, «(0, f) = fit) = 0, t > 0, and -«,(0, t) = q(t) = -ire-"1', t > 0. The noisy data are obtained by adding a random error to the exact data at every grid point /„ in the interval IL s : Fm(*n) = Ht„) + en,i> Qm(tn)=Q{tn)

+e„2,

where εη { and εη2 are Gaussian variables of variance σ2 = ε2. In all cases, we use h = Ax = 0.01 and k = Δί = 0.01. If the discretized computed initial temperature function component is denoted by j8£ e and the true component is /3" = ßU„), we use the sample 2.4B-I 2.40

1.90 X-VQIU9S

Fig. S.ll Reconstructed initial temperature in Problem 7: e = 0.005, δ = 0.04, h = k = 0.01, ||0 a - β\\ = 0.0421. Exact (---); computed (—).

NUMERICAL RESULTS

163

1.26

-0.60

x- να Iu

Θ

s

Fig. 5.12 Reconstructed initial temperature in Problem 8: ε = 0.005, δ = 0.04, h = k = 0.01, \\β - β\\ = 0.0931. Computed (---); exact (—). &

ε

root mean square norm to measure the error in the discretized interval I = [0,1]. T h e solution error is then given by x

ι Μ - Ν

1/2

M-N

Σ

η= 1

(βΙ.-β"Υ

T h e solution error for the reconstructed t e m p e r a t u r e and heat flux functions in the discretized interval /, = [0,1] is evaluated similarly. T h e qualitative behavior of the computed initial t e m p e r a t u r e functions for Problems 7 and 8 are illustrated in Figs. 5.11 and 5.12, respectively, for an average perturbation—level of noise—ε = 0.005 and radius of mollification

164

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP 2.10-1

2.00

-4

1.30-j—ι—ι—ι—ι—ι—r—ι—ι—ι—[—ι—ι—ι—r—ι—ι—ι—r—ι—ι 0.00

0.25

0.60 t - V

β IUθ

0.76

1.00

S

Fig. 5.13 Reconstructed boundary temperature in Problem 7: ε = 0.005, δ = 0.04, h = k = 0.01, | | / - / I I = 0.0245. Exact (- - -); computed (—). 6>ε

δ = 0.04. For completeness, in Fig. 5.13 we also show the reconstructed boundary t e m p e r a t u r e for Problem 7. Remarks 1. T h e space marching computational procedure for the solution of the IHCP—modified at each step in space by performing a linear extrapolation "at t h e b o u n d a r y " — c a n be implemented to evaluate—in a stable m a n n e r — t h e unknown initial t e m p e r a t u r e . 2. In certain situations, t h e modified algorithm might constitute a viable alternative for the solution of t h e backward-in-time inverse heat conduction problem.

EXERCISES

5.11

165

EXERCISES

5.1. Consider system (5.1) with t h e condition (5.If) replaced by -u (0,t)

= q(t)

x

= a(t)E(u(0,t))

- G(t),

t > 0,

where a(t) is t h e transient heat transfer coefficient at t h e surface χ = 0. Define t h e approximate transfer coefficient a (t) by s

,

hQmit)

λ

+

e

G(t)

'

WM)

'

> 0

'

(

"

9 )

U n d e r t h e conditions of T h e o r e m 3.4, if E(u(0, t)) > λ > 0 and Ε h a s a Lipschitz constant L with respect to the L norm, show that in any time interval of interest, D = [0, T], 2

\\a - a J , s

2

< C[\M8

D

where C = C(A, L , \\J qJL,D, _

a

_ 2 / 3

]},

(5.30)

IIG|U, , | | £ | L , ) . D

s

5.2. Prove that H« s,sh,D

+ 2ε e x p [ ( 2 5 )

it is possible to choose -» 0 as ε -> 0.

D

δ = δ ( ε ) such that in (5.30),

5.3. Implement the mollification method to numerically solve t h e I H C P u, = {e'uu ,

0 < x < l , 0 < f < l ,

xx

- u {\,t) x

= Q{t)

= -3e',

0 < ί < 1,

«(1,0

= F{t)

= 4e',

0 < ί < 1 ,

M(X,0) = χ

2

+ χ + 2,

0<JC<1.

Set h = Ax = k = At = 0.01 and use discrete random perturbations to obtain F and Q with ε = 1 0 and 5 X 1 0 ~ . In each case, find t h e discrete mollified t e m p e r a t u r e and heat flux functions J f and Jq , respectively, at t h e active surface χ - 0. - 3

m

3

m

s

m

s

m

5.4. If t h e exact heat flux function at χ = 0 in Exercise 5.3 is given by q(t) = -u (0, t) = a ( r ) u ( 0 , i ) + G(r) and x

/0,

G

«>-L

0<ί<1/2,

l/2

i < S

l,

166

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

compute the discrete version of a from (5.29). C o m p a r e your results with the exact values—at the grid points of the unit time interval—given by s

α

5.12

('

)

=

ε

1/2, <0,

0
R E F E R E N C E S AND C O M M E N T S

T h e following references and comments serve to expand the basic material covered in the corresponding sections. 5.1. The existence and uniqueness of a strictly increasing solution for the semi-infinite body version of the problem of determining the interface temperature between a gas and a solid under nonlinear monotone Lipschitz radiation laws was first considered by W. R. Mann and F. Wolf, Heat transfer between solids and gases under nonlinear boundary conditions, Quart. Appl. Math. 9 (1951), 163-184. T h e Lipschitz condition on the nonlinear heat transfer law was removed in the subsequent work of J. H. Roberts and W. R. Mann, On a certain nonlinear integral equation of Volterra type, Pacific J. Math. 1 (1951), 431-445. A constructive proof for the existence and uniqueness of the interface temperature in the presence of a positive integrable transient source by the method of lower and u p p e r solutions can be found in J. B. Keller and W. E. Olmstead, Temperature of a nonlinearly radiating semi-infinite solid, Quart. Appl. Math. 30 (1972), 559-566. The numerical solution of the Volterra integral equation characterizing the active surface t e m p e r a t u r e history was successfully implemented, using the method of successive approximations, by P. L. Chambre, Nonlinear heat transfer problem, J. Appl. Phys. 10 (1959), 1683-1688. More recently, combined Abel inversion formulas and β-spline approximations together with product integration quadrature techniques appeared in C. W. Groetsch, Convergence of a numerical algorithm for a nonlinear heat transfer problem, Z. Angew. Math. Mech. 65 (1985), 645. C. W. Groetsch, A simple numerical model for nonlinear warming on a slab, in Proc. of the International Symposium on Computational Mathematics, T. Yamamoto (ed.), North-Holland, Amsterdam, 1991, pp. 149-156.

REFERENCES AND COMMENTS

167

More general problems of the same kind, combining the effects of convection and radiation at the interface, are described in A. Friedman, Generalized heat transfer between solids and gases, under nonlinear boundary conditions, J. Math. Mech. 8 (1959), 161-183. V. Saljnikov and S. Petrovic, Heating problem of a horizontal semi-infinite solid by natural convection, Z. Angew. Math. Mech. 68 (1988), 58-59. 5.2. H e r e we follow in part the work of D. A. Murio, Numerical identification of interface source functions in heat transfer problems under nonlinear boundary conditions, Comput. Math. Appl. 24 (1992), 65-76. 5.3. T h e I H C P approach for the numerical solution of Stefan's problem was initially implemented, using space marching moving finite elements, by M. A. Katz and B. Rubinsky, An inverse finite element technique to determine the change of phase interface location in one-dimensional melting problems, Numer. Heat Transfer 7 (1984), 269-283. Space marching

finite-difference

techniques were utilized in

M. Raynaud, Comparison of space marching finite difference technique and function minimization technique for the estimation of the front location in nonlinear melting problems, Fifth Symp. on Control of Distributed Parameter Systems, A. El Jai and M. Amoroux (eds.), Universite de Perpignan, France, 1989, pp. 215-220. where a nonlinear version of the problem is investigated and some numerical experiments are presented. For the inverse Stefan problem, a combined numerical procedure based on the integral method and the future temperatures algorithm was investigated by N. Zabaras, S. Mukherjee, and O. Richmond, An analysis of inverse heat transfer problems with phase changes using an integral method, ASME J. Heat Transfer 110 (1988), 554-561. T h e numerical solution of the inverse Stefan problem was also implemented using nonlinear approximation techniques of optimal control in a series of notes: P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math. 34 (1980), 411-429. P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory, J. Approx. Theory 30 (1980), 81-89. 5.4. For the one-dimensional Stefan problem, there are n u m e r o u s results available concerning the existence, uniqueness, and stability of the solution.

168

APPLICATIONS OF THE SPACE MARCHING SOLUTION OF THE IHCP

For a comprehensive review, see L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, 1971. A detailed description of several direct methods for the approximate solution of the Stefan problem and some selected algorithms for the numerical solution of the inverse Stefan problem can be found in the classical book of J. Crank, Free and Moving Boundary Problems, Clarendon, Oxford, 1984. 5.5. In this section we follow in part the article D. A. Murio, Solution of inverse heat conduction problems with phase changes by the mollification method, Comput. Math. Appl. 24 (1992), 45-57. 5.6-5.8. For specific information about the numerical solution of the backward-in-time inverse heat conduction problem, see the interesting article B. L. Buzbee and A. Carasso, On the numerical computation of parabolic problems for preceding times, Math. Comp. 27 (1973), 237-266. A discussion of more general backward-in-time parabolic problems can be found in K. Miller, Efficient numerical methods for backward solution of parabolic problems with variable coefficients, Improperly Posed Boundary Value Problems, A. Carasso and A. P. Stone (eds.), Pitman, London, 1975, pp. 54-64. Basic ideas about analytic continuation are discussed—at an elementary but rigorous level—in the well-known textbook R. V. Churchill and J. V. Brown, Complex Variables and Applications, McGraw-Hill, New York, 4th ed., 1984. 5.9-5.10. In these sections we follow in part the original presentation given in D. A. Murio and D. Hinestroza, On the space marching solution of the inverse heat conduction problem and the identification of the initial temperature distribution, Comput. Math. Appl., 25 (1993), pp. 55-63.

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

6 APPLICATIONS OF STABLE NUMERICAL DIFFERENTIATION PROCEDURES

In this chapter we investigate several ill-posed identification problems in ordinary and partial differential equations—appearing in various important problems in science and engineering—that can be solved numerically by utilizing methods based on the stable computation of derivatives when the experimental information is obtained through measured data. In all cases we use the mollification m e t h o d — a s introduced in Chap. 1—to develop completely automated algorithms for the approximate reconstruction of the unknown functions in a suitable compact subset of the domain where the solution is observed. T h e compact subset is automatically determined by the amount of noise in the data. W e begin by attempting to reconstruct the unknown forcing term in a system of ordinary differential equations—a problem related to perturbation theory and validation of physical structures—and then proceed with the numerical identification of the transmissivity coefficient in linear and nonlinear elliptic and parabolic partial differential equations, a quite general problem arising, for example, in porous media and related to ground water modeling and reservoir simulation. For clarity, we restrict our attention to the one-dimensional case in space. References to the extension of these techniques to two-dimensional problems are provided at the end of this chapter. 6.1

NUMERICAL IDENTIFICATION O F F O R C I N G T E R M S

W e consider a system of ordinary differential equations of the form %{t)=f{t,y{t))+p(t)

(6.1) 169

170

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

together with t h e corresponding initial a n d / o r boundary conditions ( I C - B C ) , where

y (0

y(0 =

2

{yi{t),y {t),...,y {t)) , T

2

n

f{t,y{t))={f {t,y{t)),f {t,y(t)),...,f {t,y{t))) , T

l

2

n

P(0 = ( P i ( 0 . P ( 0 . - - - . P n ( 0 ) ' . 7

2

and

^ .

Λ

(

My.

. ί

)

=

( ^

. dy

dy

2

(

ί

)

' ^

(

ί

)

' - ' ^

\ (

ί

)

T

)

In general, / and ρ are known functions and we are required to find a solution function y that satisfies (6.1) and the ( I C - B C ) . This constitutes the direct problem. However, it is possible to attempt to gain some knowledge about the structure of the system, for example the forcing term pit), from experimental information given by the approximate knowledge of the solution y ( r ) at a discrete set of points in some interval of interest. This problem belongs to a general class of inverse problems, known as system identification problems, and, in particular, it is an ill-posed problem because small errors in the data function y ( r ) might cause large errors in the computation of t h e derivative function dy/dt(t) which is needed in order to estimate the forcing term function p ( r ) . O u r method begins by attempting to reconstruct a mollified version of the derivative function. Once this has b e e n computed, the filtered data function is used to evaluate the function / and estimate the unknown forcing term. In the process we need to generalize the parameter selection criterion—introduced in Sec. 1.4 for the scalar c a s e — t o the vector case.

6.2

STABILIZED P R O B L E M

In what follows we assume that the solution function y ( i ) is measured in the interval / = [0,1]. O n the basis of this information, we discuss the problem of estimating the forcing term function pit) in some suitable compact set K, Kczl.

STABILIZED PROBLEM

171

If C ( / , R") denotes the set of continuous vector functions over / with ||ylL.i=

y, e C ° ( 7 ) ,

max Hy.-L,/, ι <( <«

ι = 1,2,. . . . n ,

where C ° ( / ) represents the set of continuous scalar functions over / with lly.-L,/ = m a x | y , ( i ) | , we assume that pit) e C°(A:,R"), d y(t)/dt ε C ° ( K , R ) , and the function /(f, y ( i ) ) has a Lipschitz constant L with respect to the norm || • IL W e also assume that instead of t h e function y ( i ) , we know some data function y (t) e C ° ( / , R " ) such that 2

2

n

m

|y

-y\U.i

m

(6.2)

<e.

In order to stabilize the differentiation problem, if δ > 0 is smaller t h a n t h e distance from Κ to the boundary of / , dl, we introduce the function /•°°

/

p (r-i)y,(j)& e

^ —00

hy{t)

J

= (p *y)(0 = s

p (t - s)y(s) s

ds.

ρ ( ί - i ) y „ ( s ) ds β

J y(t) is a C°° function in R" a n d for fixed ί e Κ it has compact support in the interval / . T h e following two lemmas are needed for our stability analysis. Their proofs, given in Sec. 1.2 for the scalar case, extend naturally to the vector function situation. s

L e m m a 6.1 (Consistency)

If \\d y/dt \U, 2

d

κ ^ M , then 2

d

dt ** -Jt {p

Lemma 6.2 (Stability)

2

y)

y

< 3δΜ,.

», κ

If y ( f ) e C ° ( / , R " ) a n d \\y

m

m

Ο,κ W e immediately have the following result.

- yh,

K

δ\ίπ

< ε, then

172

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

Theorem 6.1 (Error Estimate)

U n d e r the conditions of Lemmas 6.1 and 6.2,

d

d

(6.3)

We can now state our main theoretical result. Theorem 6.2 U n d e r the conditions of Lemmas 6.1 and 6.2, if f(t, y) has a Lipschitz constant L, then

\Pa,e ~ Ρ\\*,κ ^ 38M

+ L

+ ε

2

(6.4)

where

Ρ β

''

=

d t (

P

S

*

y

m

^

(6.5)

" A ' ^ m i O )

is the approximate reconstructed forcing term. Proof. F r o m (6.1) and (6.5), we have

p(t)-PS.AO

= j y(0 t

~ j (Ps*y )(0 t

m

+

/(',y (0)-fO,y(0)m

Taking norms, using the Lipschitz property of / and inequality (6.3), we obtain

I p , ~p\L,K e

< 3δΜ + - = 2

+L||y

m

OVTT

By (6.2),

\Ρδ,ε -Ρ\\°°.κ

^ 3δΜ

2

+ ε

+ L

-y\l. . K

NUMERICAL RESULTS

173

We observe that the right-hand side of inequality (6.4) cannot, in general, be minimized because M is not known. Instead, we apply the p r o c e d u r e described previously in Sec. 1.4 to each component y,, 1 < i < n, and set 8 = min, δ,·, where δ, is the unique solution of the equation 2

s

/

s

n

II*

y

m

j

-

y,lk/

=

£

,

ι < ζ < «.

It is clear—from the monotonicity property (Lemma 1.3)—that δ is such that ll^-y

m

- y j u

< ε.

(6.6)

Notice that the choice of δ = δ" automatically defines the compact subset Κ = [38,1 - 3δ] where we seek to reconstruct the unknown forcing term pit). W e also notice that (6.6) justifies replacing fit, y (t)) by f(t,(p * y X i ) ) , when attempting to reconstruct the approximate forcing term p using formula (6.5). m

8

m

s

6.3

e

NUMERICAL RESULTS

T h e numerical m e t h o d is implemented exactly as explained in Sec. 1.5 with the updates given previously. As usual, the discrete noisy data function is obtained by adding a random error to the exact data function, that is,

y .,(tj) m

= y,(i,) + ε0, , 7

where i, = j Δ / , j = 0,1,..., Ν, Ν At = 1, / = 1,2,..., n, and 0, is a Gaussian variable with values in [ - 1 , 1 ] . If the discrete numerical approximation to the forcing term components ρ,(ί,) are denoted P/Ltj) = Pe, ,,-(iy), 1 < i <. n, after computing d/dt(p * y X i y ) , we use the equation y

e

a

m i

W i ) = j {Pi*y ,i){tj) t

-fi(tj,(ps*y j)(tj))

m

m

on the sample points of the interval K - = [38,1 - 3δ]. T h e solution error is then given by s

lift-^112,*,=

i = 1,2,..., M

Jmm

where Μ indicates the number of sample points in

n,

174

STABLE NUMERICAL DIFFERENTIATION PROCEDURES 0.7B-1

0

-

2 0

I II 0.20

ιιι ιιιιι ι ιιιι ι ιιιι ι ιι 0.30

0.40

0.50

0.60

ι ι I I I I j 0.70 0.80

1 1 me Fig. 6.1 Reconstructed forcing term in Example 1: ε = 0.005, At = k = 0.01, ||p - P|| = 0.00097. Exact ( * * * ) ; computed (—).

S = 0.06,

E X A M P L E 1 As a first example we consider the scalar initial value problem dy /dt = y + sin t, y(0) = 1. T h e exact solution is y ( i ) = \[3e' cos t - sin t ]. Figure 6.1 shows the solution obtained with the mollification method and the exact forcing term function pit) = sin /. With ε - 0.005 and Δ ί = 0.01, it follows that δ = 0.06 and \\p - Ρ \\ ,κ = 0.00097. x

χ

2

έ

E X A M P L E 2 T h e second example is an initial value problem given by the system dy /dt = 5y, - 6 y + t + 3, dy /dt = 3y, - 4 y + t + 2, y^O) = 1, y (0) = 1. The exact solution is y = e ' + e~' + t - 1, y ( i ) = \e + e~' + t — \. Figures 6.2a and b show the numerical solutions and the exact forcing functions p (t) = t + 3 and p (t) = t + 2, respectively, corresponding to ε = 0.005, Δ ί = 0.01, and δ = 0.06. T h e associated error norms are given by \ \ = 0.00101 and \\p - P \\ ,K = 0.00099. x

2

2

2

2

2t

x

2

x

P l

2

2

2

2

2

S

TRANSMISSIVITY COEFFICIENT IN 1-D ELLIPTIC EQUATION

175

3.Θ0-

3.75-

3.70-

3.66-

Q-3.60H

ε «

3.65-1

0)3.6βΗ Ο

3.45

Η

Ο ^3.40-1

3.36-

3.30-

3.26-

3.20 0.20

ι ιιι

I I

0.30

IIIιιιι ι ι ιι ιι ι ιιιι ι ιιιι 0.40

0.60

t Ime

0.60

0.70

0.80

(a) Fig. 6.2a Reconstructed forcing term p, in Example 2: ε = 0.005, δ = 0.06, Δ/ = k = 0.01, Hp, - Ρ,|| = 0.00101. Exact ( * * * ) ; computed (—).

6.4 IDENTIFICATION O F T H E TRANSMISSIVITY C O E F F I C I E N T IN T H E ONE-DIMENSIONAL E L L I P T I C EQUATION T h e identification problem can be described as follows. If the solution function u(x) and the source function fix) are measured in the space interval / = [0,1], it is desired to identify the transmissivity coefficient a{x) such that d die

(α(*)^(χ))

du «(0)^(0)-c,

=/(*),

0 < * < 1 , (6.7)

176

STABLE NUMERICAL DIFFERENTIATION PROCEDURES 2.Θ0-1 2.75-

2.702.65CM

°-2.é0-

ε

C αι 2 . 5 5 -

0)2.60Ü 2.46C

ο

^ . 4 0 Η 2.36-

2.302.262.20

I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | 0.20 0.30 0.40 0.60 0.60 0.70 0.Θ0

t Ime fb) Fig. 6.2b Reconstructed forcing term p2 in Example 2: e = 0.005, δ = 0.06, M = k = 0.01, \\p2 - P2\\ = 0.00099. Exact (***); computed (—). in the linear case, and du ( d -\^a(u(x))—(x)\=f(x),

0<*<1, (6.8)

du «("(0)) — ( 0 ) = c ,

in the nonlinear case. We study these two problems under each of the following assumptions, valid over a suitable compact set K c /: inf K

-dx-U{x)

>0

(Al)

STABILITY ANALYSIS

177

or du

inf { max K

\

du 2

(x)

'~d?

(A2)

>0

Assumption (A2) eliminates the possibility that du/dx(x) and d u/dx (x) vanish simultaneously and also the vanishing of du/dx(x) at more than one point in K. W e also assume that the exact and measured data functions satisfy error bounds of the form 2

ε,

(6.9)

< ε.

(6.10)

11/-/Jl.,/ <

\\u - uJUj

6.5

2

STABILITY ANALYSIS

Under assumption ( A l ) , (6.7) and (6.8) can be integrated trivially, obtaining the unique solutions

ff(s) a(x)

ds + c

=

(6.11) dx-

U{x)

and

ff(s) a(u(x))

ds + c

= -£-3

,

(6.12)

respectively. Since both formulas are similar, we devote all our attention to the linear case. T h e approximate transmissivity coefficients at (x) and <x (χ) are respectively defined by s

sε

178

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

and ff {s)ds

+ c

m

(6.14) dx We now establish some preliminary results. Lemma 6.3 (Consistency) d

If d

2

dx

u(x)

2

< M

and

7

> λ > 0,

χ e K

s

c /,

then a - a L,K, < s

(6.15)

C8,

for some constant

C = C

λ,Μ , 2

Proo/. From (6.11) and (6.13), for χ e K , we have s

/(s) ώ + c \-u(x)

f(s)ds

1

d + c | | — Jgu(x) dx

< ^ ( k l + ll/L,/) 1 < ^(kl + = C3 by Lemma 1.1.

-Jgu(x))

d dx

\\f\L,,)38M

s

2

-

d —u dx

d

-^u(x)

)

STABILITY ANALYSIS

Lemma 6.4 (Stability) u(x)

dx

If u

and f

m

m

179

verify inequalities (6.9) and (6.10),

> A > 0 , x e i , c / ,

δ >

and

then CE

"'··"-•*· * λ ( λ δ ν ΐ - 2 ε ) ( λ δ

\a

vV}.

where C = max{2(|c| + | | / L , / ) , /Voo/. For Λ: e \a (x) s

~

(6.16)

+ 1),

- at*

x

from (6.13) and (6.14), we have

a (x)\ Sie

1

1

[j*f(s)ds + c

\ dx~

-f (s))ds\ m

dx- »>

JsU(X)

r

/*(/(*)

JsU

ix)

x

ds + c

/m

-£j,u (x)

d d —J u(x)—J u (x) s

(|c|+

< — 1

(

s

-

m

-/,«(,) +

d \— 6 m(x) J

m

dx'

U

2ε

11/11.,/)

s—:

— +

δνπ)

λ

Ce

(λδ + 1),

r-

λ ( λ δ ν π - 2e)

using L e m m a 1.2 and setting C = max{2(|c| + H/L,/), -fir).

•

W e immediately have the following result. Theorem 6.3 (Error Estimate)

U n d e r the conditions of Lemmas 6.3 and 6.4,

Ha-a JU,K
s

+

ε ( λ δ + 1)

λ(λδνττ - 2ε)

(6.17)

Remarks 1. Notice that if we set δ = 0 ( v T ) , for instance, we get uniform convergence, that is, Hoc — a \\^ -» 0 as ε -» 0. Se

Ks

180

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

2. Similar bounds for the approximate transmissivity coefficient are obtained in the nonlinear case. Under assumption (A2), where u has a unique critical point x* at an interior point of K , for all χ in K , χ Φ χ*, we have s

s

ff(s)ds a(x)

= ^

,

(6.18)

«a(*) = - f ' ^-/ w(x)

< - )

dx-

U(x)

ff(s)ds 6

1 9

a

and ff (s)ds m

««..(*) = ~

(6-20)

d

in the linear case. Also in the linear case, at χ = χ*, we have fix*) «(**)

= ~J2 dx

·

u(x*)

(6-21)

2

For the nonlinear case we proceed similarly. We observe that under assumption (A2) no boundary conditions are needed. Moreover, we can select an open interval V(x*) = (x* — y, x* + y) such that \d/dxu(x)\ > \ * > 0 for all χ e K — V(x*). In the next section we will explain how to approximate the value of α at any point in V(x*) and avoid the computation of the second derivative in (6.21). x

6.6

a

NUMERICAL M E T H O D

In what follows we assume that the data functions f and u are discrete functions in the interval / = [0,1], measured at the JV + 1 sample points m

m

1

X: = ih.

i=

0.1

TV. h =

—

NUMERICAL METHOD

181

Once the radius of mollification 8 and the discrete function J u are determined, we use centered differences to approximate d/dxJ u at the sample points of the interval K = [38,1 - 35], T h e n u m b e r of sample points in K is Μ = int{(l - 68)/h} + 1, and we consider the following partition of the set / : s

s

m

m

s

s

x = 0, Q

x, = 38 + h,

x = 38 + 2h,...,

x -2

2

M

= 1 - 3δ - Α,

XM-\

1

=

Under assumption ( A l ) , introducing the quantities d

in (6.14), at the sample points of the interval / , we obtain

VeU)

W e define the

= &«..(*.--.) +

finite-difference Bh B'

tA

B

/»(*)

d S

-

( · 6

2 2

)

equation

h-i+ f ,„

=

Γ

/ = 2,...,M-2,

h

m

= c + (38 + h)f ,

(6.23a) (6.23b)

mA

Β',,ο = c,

(6.23c)

and begin by establishing a basic stability result for these coefficients. Lemma 6.5 If f stant L , t h e n

is uniformly Lipschitz on / = [0,1], with Lipschitz con-

m

||S|m.^<2!|/JU

/

+

|c|

and \K

e

- BlL,

< L(98

2

K/i

+ 68h + h).

Proof. According to (6.23), taking absolute values, we have \B!J =

+ h\f J

< ΙΒΙ,,Ι + (i -

m

1)A||/JL,/

< |c| + ( 3 5 + A)ll/ IL,/ + (i - l)AH/*lk/ m

<\c\ + 38\\f Lj m

+


This proves the first part of the lemma.

ih\\f Lj m

i = 2 , 3 , . . . , M - 2.

182

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

Subtracting (6.23a) from (6.22), we obtain, for 2 < i < Μ - 2, \b Jx,)

-BiJ

a

-β ,,-ιΙ + Γ

<\b Jx^ ) s

\fM

ε

δ

x

< ( ι - \)h L

t

2


t

L{x f

< ( i - l)fc L +

x

+ 68h +h).

2

m

- 51,, I

+\b Jx )

2

-f .i\ds

•

In order to compute a {x ), we approximate dJ u /dx(Xj) differences. W e introduce the notation &

Pi(0 =

Λ Μ * .

e

t

a

-

+ ι)

J

s U m ( * i - \ )

u

m {

x

z )

by centered

2 < i < Μ - 3,

2h h

m

-

2h and Pi(M-2)

2h

These discrete approximations are such that d

0(/i ), 2

dx

/ = 1,2,...,A/-2.

We recall that u n d e r the conditions of Lemma 6.4, > λ

2ε

* e

Κ,

Thus, by the mean value theorem, we also have 2ε

i = 1 , 2 , . . . , Μ - 2.

(6.24)

T h e finite-difference approximations for the coefficients a ,,(*,) are defined s

by

NUMERICAL METHOD

183

T h e stability and convergence results for these coefficients are given in the following lemma. Lemma satisfy

6.6

The

finite-difference

Sifc(\c\

\A h, ,< a

A,

coefficients

2\\f h,,)

+

δλ]/π

K

i = 1 , 2 , . . . , Μ - 2,

Si

m

— 2ε

and SVVL(95

H«e.« ~ οΚκ Α

δ

<

+ A)

+ 68h

2

T= λδι/ττ — 2ε

(Id + 2 | | / J L , / ) 8 π

0(h )

2

+

(δλνπ

-

2

2ε)

Proof. T h e first part of the lemma follows immediately from definition (6.25), inequality (6.24), and L e m m a 6.5. For the second part of the proof, we write Β - J u ( x )

«<'">

P

1

\bl{*i)-Bl,i\

1

- J u ( x ) dx

^

P

h U m { X i )

5vV||fc , e

<

J=

B%h,

(|c| + 2\\fjl,,)

Ks

λ δ ν ν - 2ε

+

5 v V L ( 9 5 + 6δΛ + A) 2

λ δ ν ^ - 2ε

(

lla -A\\L

Klt

< C8 1 +

0(h )

2

ς: _

δ λ ν

2 ε

)

2

ζ

( I d + 2 | | / t , / ) δ ττ Q{ A ) 2

2

w

(

using inequality (6.24), equation (6.25), and L e m m a 6.5. Theorem 6.4 (Error Estimate) and 6.6,

δ ττ

5 A v

^

_ 2 )

2

ε

•

U n d e r the conditions of Lemmas 6.3, 6.4,

e(2 + δ) λδ(λδ -

δ\Ζΐτ£(9δ

2

2ε)

+ 6δΑ + A)

Κ8-\ίπ — 2ε

(|c| + 2 H / J L , / ) δ π 0 ( Α ) 2

(δλνπ

7

-

2ε)

2

184

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

Proof. It follows directly from Lemmas 6.3, 6.4, and 6.6 and the triangle inequality. • W e now consider the case when u has a critical point at x , for some k. Under assumption (A2), the coefficients a (x ), i = 1 , 2 , . . . , Μ - 2, i Φ k, are obtained from (6.20) and the corresponding b . coefficients are given by k

s

e

t

s

t

<··*/

s,A i)

b

= I f(s)ds,

x

i = 1 , 2 , . . . , Μ - 2, ιΦ

k.

In this case, the finite-difference scheme is defined as follows: B»,i = * a . / - i + Λ/™.,·. Bli-x=Bli

i = k + \, k + 2,...,

+ hf , mJ

Μ - 2,

(6.26a)

i = k,k-\,...,l,

(6.26b)

= 0.

(6.26c)

and B!,

k

T h e basic stability results are obtained as before. Lemma 6.7 If f stant L, then

is uniformly Lipschitz on / = [0,1], with Lipschitz con-

m

\\BlL, <2\\f Lj Ks

m

and \\b

StB

- BI\l,

< L(98

2

Ks

+ 68h + h).

Proof. Similar to the proof of Lemma 6.5, setting c = 0. To approximate the coefficients a (χ^, - h, x + h) and introduce s

(x

k

ε

we consider the interval V(x ) k

k

/

X = mini h

W e define

du

•

du

=

NUMERICAL RESULTS

185

with PgU) satisfying 2ε

TO)

*

h

(6.28)

i Φ k.

δι/π '

For these finite-difference approximations, the stability result is as follows. Lemma satisfy

6.8

T h e finite-difference

i = 1 , 2 , . . . , Μ - 2,

Si

23v^ll/JU,/

U' \l,K z s

A,

coefficients

-τ—7=—r-

t

and

δ λ ν π - 2e Λ

\ 8^rr — 2ε δ ν π £ ( 9 δ + 6δΛ + A) 2||/JU,/3 ffO(/i ) h

2

2

+

( δ λ ^

2

- 2ε)

+ 0(A ). 2

2

JVoo/. Similar to Lemma 6.6. Theorem 6.5 (Error Estimate) and 6.8, < C5 1 +

\\a-A%\L,

Ki

U n d e r the conditions of Lemmas 6.3, 6.4,

ε(2 + δ)

2

+

λ δ(λ„δ-2ε) Η

+•

\ 8yfir h

2||/„ΙΙοο,/δ ττΟ(Λ ) 2

δ / τ τ " £ ( 9 δ + 6δΑ + h) — 2ε

2

+

0(h ). 2

(δλ νπ~ - 2 ε ) Α

Proof. Similar to Theorem 6.4.

6.7

•

NUMERICAL RESULTS

In this section we discuss the implementation of the method to some test problems of interest. In all the examples, the noisy data functions are obtained by adding an ε random error to / and u at the sample points of the interval / = [0,1], that is,

/«(*/) = / ( * / ) + ( i)

u

x

m

»

εθ

= "(*,) + 1 η ε

i =

Ο,ί,.,.,Ν,

186

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

where 0, and η, are Gaussian random variables with values in [ - 1 , 1 ] . T h e selection of the radius of mollification is implemented as usual and the discrete weighted I norm on K is used to measure the solution error: 2

S

Iι MS-all2,*,=| EXAMPLE 3

- M-2 _-

Σ [A' - a(xj)]'

M-2

~2

,

| 1 / 2

tJ

In this example the exact data functions are u(x)

=χ +

f(x)

= 4x(\

JX , 3

+x ) 2

and the initial condition is given by du « ( 0 ) - ( 0 ) = l. T h e exact transmissivity coefficient is a(x) = 1 + x . Table 6.1 shows the error norm of the solution as a function of 8 and ε for h = 0.01. T h e qualitative behavior of a typical reconstructed transmissivity coefficient is shown in Fig. 6.3. 2

E X A M P L E 4 This is a nonlinear example. T h e exact data functions and the initial condition are u(x)

" e,

f{x)

= 3e *,

x

3

« ( " ( 0 ) ) ^ " ( 0 ) = 1. T h e exact transmissivity coefficient is a(u) = u . In this case, after computing A, i = 1 , 2 , . . . , Μ - 2, we collect all the pairs («„,(*,), A\ ,·) to obtain the approximate functional relationship between the data function u and the transmissivity coefficient a. Table 6.2 illustrates the numerical stability of the method. A plot is shown in Fig. 6.4. 2

si

TABLE 6.1. Error Norm as a Function of ε in Example 3

ε

δ

Error Norm

0.000 0.002 0.005

0.04 0.04 0.04

0.0178 0.0195 0.0269

NUMERICAL RESULTS

I I I I I I 1I I I I I I I 1I I I I I I I I I I II 0.2B 8.38 8.48 0.60 8.68 8.70

187

I I 0.80

x-velu©s Fig. 6.3 Reconstructed coefficient in Example 3: e = 0.005, δ = 0.04, Ax = h = 0.01, \\A - α|| = 0.0269. Exact (***); computed (—). TABLE 6.2. Error Norm as a Function of ε in Example 4 ε δ Error Norm 0.000 0.04 0.0363 0.002 0.04 0.0372 0.005 0.06 0.0458

EXAMPLE 5 Here we consider the case where d/dxu(x) = 0 at a single point in the interval / = [0,1]. The input data functions are u(x)

= - s i n TTx,

f(x)

= - ì r e o s TTX + (1 + χ)π2 sin trx.

The unique exact transmissivity coefficient to be reconstructed is a(x) =

188

STABLE NUMERICAL DIFFERENTIATION PROCEDURES 4.50-

4.25^

4.00^

3.75-;

3.B0C 5

3

-

2

B

~

ο ^

: 3.00-


2.60-

2

-

A

B

~

2.00^

1.75·^

1·ββ j 1.20

I I I I | I I I I | I I I I | I I I I | I I I I | 1.40 1.60 1.80 2.00 2.20 U - V 8 I U 8 8

Fig. 6.4 Reconstructed coefficient in Example 4: ε = 0.005, δ = 0.06, Δ* = h = 0.01, Μ - α|| = 0.0329. Exact ( * * * ) ; computed (—).

1 + χ a n d the singularity occurs at χ = \, w h e r e du/dx{\) = 0. Figure 6.5 shows a graph of the reconstructed coefficient for ε = 0.005 and δ = 0.06. T h e numerical stability properties of this example are summarized in Table 6.3. Remark. W e observe that in this case t h e initial condition is not needed; it is replaced by the "information" given at the singularity. In fact, once the centered differences Ρ / Ο ) , i = 1 , 2 , . . . , Μ - 2, have b e e n computed and the singularity located at the n o d e x , if P (k) Φ 0, the discrepancy is used t o correct (update) the values of in a neighborhood of the singularity. These updated values are then utilized in the evaluation of the approximated transmissivity coefficients A\ in (6.27) after determining B\,, / = l , 2 , . . . , M - 2 , from (6.26). s

k

t

s

TRANSMISSIVITY COEFFICIENT IN 1-D PARABOLIC EQUATION

189

1.80-3

1.76-^

1.70-

1.66^

1.60-

: c •

1.55-

O

I

H_

1.60-

Φ O

: 1.46-


u

:

:

1.40-

1.36-

1.30·^

1.25-^ Τ·

2 0

\ ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι ι 1 I 1 11 0.20 0.30 0.40 0.60 0.60 0.70 0.Θ0 x - v a l u e s

Fig. 6.5 Reconstructed coefficient in Example 5: ε = 0.005, δ = 0.06, Αχ = h = 0.01, \\A - all = 0.0361. Exact ( * * * ) ; computed (—). TABLE 6.3. Error Norm as a Function of ε in Example 5

ε

δ

Error Norm

0.000 0.002 0.005

0.04 0.06 0.06

0.0212 0.0494 0.0502

6.8 IDENTIFICATION O F T H E TRANSMISSIVITY C O E F F I C I E N T IN T H E ONE-DIMENSIONAL PARABOLIC EQUATION T h e parabolic problem is essentially solved as a sequence of elliptic problems. For each time t, we apply the marching scheme in the χ direction—as explained in the previous sections—and then repeat the process for a new updated time value.

190

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

The identification problem is now described as follows. If the solution function u(x, t) and the source function f(x,t) are measured in the domain I = [0,1] X [0,1], identify the transmissivity coefficient a(x, t) such that ( du -(x,t)

d I

du

-\ a(x,t)-(x,t)\=f(x,t),

+

[

(6.29)

du —

a(0,t)

(0,t)=g(t),

in the linear case, and ( du du

d I

du

\

- x , 0 + -[e(«(x,/))^(x,0 (

-/(*,0.

(6.30)

du a(u(0,t))—(0,t)=g(t),

in the nonlinear case. As in the elliptic case, we study these two problems under each of the following assumptions: du

inf κ

Τχ ' { Χ

> 0

ΐ )

(ΑΙ')

or inf {max

du dx

3u ,J^(*,t)\) 2

(x,t)

(Α2')

>0,

where the arguments (x, t) are taken in some suitable compact set Κ c I and, for each t, K, = [x: (x, t) e K). We also assume II/-/™

6.9

,/ < ε,

\\u - u„

,ι < ε,

and

\\g-gJl,l<ε.

(6.31)

STABILITY ANALYSIS

Under assumption (AY), (6.29) and (6.30) can be integrated with respect to x, obtaining the unique solutions f(/(M) -

«<*.') - —

-^(s,t))ds+g(t)

d

or T x ^

(· ) 6

l

)

32

STABILITY ANALYSIS

191

and f if(s,t)a(u(x,t))

-(s,t)\ds+g(t) ΊΓ,

= —

"

du

,

(6-33)

0 , 0

dx

respectively. Once again, both formulas a r e similar, and, consequently, we devote our attention to t h e linear case. T h e approximate transmissivity coefficients a ^x, t) and a (χ, t) a r e respectively defined by &

,xl

f \f(s,t)

S[&2

dj u

\ -j^ ,t)\d +g(t)

x

-

{s

τJ η ,

« (*. 0 = —

d

M2

dx

S

•

u

Si

ε

(x,t)

(· ) 6

34

and rx(

(

t

/

\

dt

;

m

-M

s

dj* U

/ ο , ο - - dt^ 0 . 0 ds + ( t ) Μ

dx

g m

(x,t)

H e r e , dJ u/dt and dJ u/dx indicate one-dimensional mollifications in t h e t and χ directions, respectively, and, in order to simplify some calculations, in what follows we write a and a with δ = m a x f ^ , δ ) . F o r t h e parabolic identification problem, t h e preliminary results a r e as follows. B

s

s

s

Lemma 6.9 (Consistency)

ε

2

If dU 2

max

2

—?o> 0 dx

du

^ T ( * , 0 \<M

2

'

lK

and

\

dU

> λ > 0, (χ, i ) e i

T x ^

8

c /,

then ΙΙα

_

«J»,K

for some constant C = C(A, M , ||Vw|L 2

F I

iJCe

^

C8,

, l l / I U , / , HglL,/).

(6.36)

192

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

Proof. From (6.32) a n d (6.34), for (x, t) e K , we have s

\a(x,t)

1 < - j f \f(s,t)-—(s,t)}ds

-a (x,t)\ s

x{

dJfU

r

ί\

r (

f

x

f * | / ( 5 , 0 l * s

—

du

+ /„ dt

ds

'

x

1

a*

dJ u

du

dx

~ Hx

s

+ 2

(Λ:

dJ U s

du

dx


3ΛΜ (jc.i) le(0l — v ;

3Μ δ

-

dt

ί

+

1 du

du

dJ u

+

(x,t)

O j * +i ( 0 ^ ( * . 0 3«

( * . 0

dx •'ο

s

dx

\

~ "dJ Ui

f(s,t)

dJ u

+ g(t)

Q

5M (JC, ί )

'

in

dx /Ik*.

+ llglkp.

+

du

dJ u s

dt

, 0

~

Έ

a*

( l l / l k ^ + 2 | | V L , ^ + ||glU, , ) M

[0

1]

< C8 by Lemma 1.1, with C = C(A, A / , l l / l k x , , llVulU,^, llglk[
Lemma 6.10 (Stability) 3u dx

If u , g , a n d f m

> λ > 0,

(x,t)

m

m

•

verify inequalities (6.31),

(x,t)eK cI,

and

s

δ>

2B

then Ce »

β

' -

β

· . - ^ *

A(A6vV-2s)

where C = C ( M , | | / | | „ , κ , | | V H | U . l l s l k p u ] ) . 2

δ

Proof. W e leave this as an exercise.

(λδ + 1 ) ,

(6.37)

STABILITY ANALYSIS

193

Combining the last two lemmas, we obtain the following result. Theorem 6.6 (Error Estimate) 6.10,

U n d e r the conditions of Lemmas 6.9 and

ε ( λ δ + 1) s e n

l« - « « , . l k * .

λ(λδνπ

-

(6.38)

2ε)

Remark. Similar bounds for the approximate transmissivity coefficient a are obtained in the nonlinear case.

s

e

Under assumption (Α2'), where for each fixed time t, t < t < i , u has at most one critical point (JC*, f) in K , for (x, t) e K , (x, t) Φ (χ*, t), we have min

s

du

a(x,t)

ds

(x,t)

dx

dJ U s

ds

dt dJ u

a

+g(t) (6.39)

du

a (x,t)

m a x

s

s

+g(t) (6.40)

(x,t)

dx

and

f

U,t) ds+g (t)

fm(s,t)

m

(6.41)

dx

(X,t)

in t h e linear case. Also in the linear case, at (x, t) = ( J C * , r), we have du

ot(x*,t)

=

- —

f(x*,t)

u*,o

(6.42)

du 2

dx

-(x*,t)

For the nonlinear case we proceed similarly. W e observe that u n d e r assumption (Α2') no boundary conditions are needed. Moreover, for each fixed t, we can select an open interval V (x*) = t

194

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

{(*, ί): χ* - γ < χ < χ* + γ], such that \du/dx(x, t)\ > λ * > 0 for all (x, t) e V,(x*)r\K . We will explain later how to approximate the value of a at any point in V (x*) and avoid the computation of the second derivative in (6.42). χ

t

t

6.10

NUMERICAL METHOD

In what follows we assume that the data functions f and u are discrete functions in the interval / = [0,1] X [0,1], measured at the (N, + 1 X N + 1) sample points m

m

2

(*,·.'*)

=

(ih kh ), u

2

i = 0,l,...,N ,* = 0,l,...,W ,A = 1

2

I

1 —,A

=

2

1 —.

Once the radii of mollification δ,, δ and the discrete function J u for δ = m a x ^ δ ) are determined in the χ and t directions, we use centered differences to approximate the partial derivatives of J u at the sample points of the compact set Κ = [3δ, 1 - 3δ] Χ [3δ, 1 - 3δ]. In the sequel we set A, = A = A to simplify some analysis, and we consider the partition of the set / defined by the pairs {x , t ), 0 < i, k < Μ - 1 such that s

2

1 ;

m

2

s

m

δ

2

t

0,

x = 38 + A,

1,

t

k

x

x

0

= 3δ + 2 A , . . . , x _

2

M

2

= 1 - 3δ - A,

t = 3δ + A,

= 0,

x

3δ + 2 A , . . . , t _2

= 1 — 3δ — A,

M

f _,

= 1.

M

Under assumption (ΑΓ), introducing the quantities dJ u s

m

0 = «»..(*· ο-^-(·«.ο in (6.35), at the sample points of the set / , we obtain

= 6e..(-«i-i.i*) +

b , {Xi,t ) & e

k

/

dt

ds.

(6.43)

W e define the finite-difference equation BU

=

k

Bh, -g , k

m k

B».o = gm,k,

+ A ( / , „ , - T '{i,k)), s

m

+ (3δ + A ) ( /

m i l

- Γ/(1,*)),

/ = 2,...,M-2,

(6.44a) (6.44b) (- ) 6

44c

NUMERICAL METHOD where T (i,k) and satisfies

dj u /dt(x ,t )

is the central-difference approximation to

e

s

hm

195

s

m

i

k

u

a

0(h ). 2

γ / ( Ι \ * ) - - ^ ( * , · , ' * )

W e now establish a basic stability result for these coefficients. Lemma 6.11 If f and u are uniformly Lipschitz in the χ direction on I = [0,1], with Lipschitz constants L , and L , respectively, then m

m

x

2

llflllkjc, < 2

ll/Jk,+

+ llglk[0,i]

dt

and

\K*

^ i l l - . ^

-

^

(

L

<

+

y

8

) (

9

δ

2

+

6

δ

+

η

h

)

+

° (

h

2

) -

Proof. According to (6.44), taking absolute values, we have, for each /' 1 , 2 , . . . , Μ - 2 and k = 1 , 2 , . . . , M - 2, \BU \ k

= \Bl _ \ i

<\Bi \ uk

hk

+

h\f ^ -T *(i,k)\ m

k

s

ΙΙγ;ιι.,*,)

ι)α(ΙΙ/„ΙΙ.,/ +

+ (i-

* ll*Jk[o.i] + (3δ + A ) ( | | / J k / +

lir/iu.^)

+ ( ' ' - i ) A ( l l / l k / + H71flkic ) m

4

< lli„lkp).i] + 2 l l / J k / + by t h e m e a n value theorem. This proves the first part of the lemma. Subtracting (6.44a) from (6.43), we obtain, for 2 < i < Μ - 2,

+ /*' I / « ( Μ * )

•r:

-f (Xi,t )\ds m

hm

d

k

U

a/

a?

ds + C»(A ) 3

v

7

K /

a/

ar

ώ +

0(h ). 3

196

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

W e estimate the integral term as follows.

hm

d

dt

U

ds

-('·'*)-—(*"'.)

u (s,z) m

dt L |,-*,|/

t

ds

m Xi

dz\ ds <

J Ps(h~z)

2

- u ( ,z)\dz)

Lh2

2

Therefore,

*/-!.'*)

<

-Bh-x, \

+L h

+ -L |(95 2

2

+ Lh-

2

k

+0(h )

2

x

3

2

+ 68h + h) +

0(h ). 2

In order to compute a (x , t ), we approximate djgu/dxix^ tered differences. W e introduce the notation a

e

t

k

t ) by cenk

2 < i < Μ - 3,

2Λ -^"m(*2.'fc)

- sU (^S,t ) J

m

k

2h and

P!(M-2,k)

=

1

2A

< M -

These discrete approximations are such that

= 0(A ), 2

l < i , ; t < M - 2 .

2.

NUMERICAL METHOD

197

W e recall that u n d e r the conditions of Lemma 6.4, for fixed t,

dx

(x,t)

2ε

> λ

(x,t)

eK . s

Thus, by the mean value theorem, we also have for each k, 1 < k < Μ — 2, 2ε

1 < i < Μ -

δ^ττ '

2.

(6.45)

T h e finite-difference approximations for the coefficients a {x ,t ) fined by Se

Β S,i,k

l

1 < i, k < Μ -

p;(i,k)'

i

are de-

k

2.

(6.46)

T h e stability and convergence results for these coefficients are given in the following lemma. Lemma 6.12 satisfy

A% , , 1 < i, k < Μ - 2,

T h e finite-difference coefficients

k

^ ( l l s J L . [ o , i ] + 2||/ L,/ +

dx

m

\A h. < e

s

oo,

K

t

δλ\Ρπ — 2ε

Ks

and

δ]/ττ

L\

+ g L | ( 9 δ + 68h + h) + 2

2

0(h ) 2

λδι/π — 2ε l*L,io.i] + 2 ( | | / J L , / +

+

(SAVTT

Vδ ττ 0(h )

dj u s

m

2

dx -

2ε)

2

2

Proof. T h e first part of the lemma follows immediately from definition (6.46), inequality (6.45), and Lemma 6.11.

198

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

For the second part of the proof, we write

dx 1

1

+ 15, d

J

&

u

dx §vV|iy -g|IU,x e

m

(

t \

x

*

(''·*)

p

8

tm

dJ u

HiL.[o,i] + 2J||/J|.,/ +

dx

(δλ\ίπ

dx

+

8 g 2 |(9S L

λδτ/π

s

2

y δ ττΟ(Η )

m

2

dx

HSL.[o.I] + 2 I I / J I . . / +

2

(δλ/ττ - 2ε)

Theorem 6.7 (Error Estimate) and 6.12,

Ks

Q(h )

dJ u

using (6.45) and (6.46) and Lemma 6.11.

\\a-AlL,
+

— 2ε

ί

+

- 2ε)

+ 6«A + h)

2

κ.

oo,

•

U n d e r the conditions of Lemmas 6.9, 6.10,

ε(λδ + 1) δ + \ " ' λ(λδνιΓ - 2 ε ) L

\ +

| ( 9 δ + 6δΑ + h) + 2

λδι[π ll*IU.[o.i]+

2III/JL/

Q{h ) 2

— 2ε +

\ δ ττΟ(Λ ) 2

dx

(δλνπ - 2ε)

2

2

NUMERICAL RESULTS

199

Proof. It follows directly from Lemmas 6.9, 6.10, and 6.12 and the triangle inequality. • Under assumption (Α2'), if dJ u /dx(x , t ) = 0 for (x , t ) e K , finite-difference scheme is defined as follows: s

BUk

= Bl,i-u

k

+ Kf ,,, m

k

m

lo

k

=

+ A(/

m i i

,

a

the

(6.47a)

ι" = i ,i

(6.47b)

0

- Ρ (/, Α)), β

4

k

Ρ&Λ)), i = i + 1,2 + 2 , . . . , M - 2, 0

B'sj-uk

if)

β

0

- 1.....2,

0

and = 0.

(6.47c)

T h e basic stability results are obtained as before. T o approximate the coefficients a ( ,t ), we consider for each fixed time t , the interval V, (x ) = {(x, t Y. x - h <x < x + h) and introduce Se Xi

k

k

io

k

k

io

ifj

W e define (6.48a) (6.48b) with P g ( i , k) satisfying ^h, ~ k

''^'"ο·

(6-49)

For these finite-difference approximations, the stability and convergence results are similar to the ones stated in Lemmas 6.9 and 6.10 and T h e o r e m 6.7. 6.11

NUMERICAL RESULTS

In this section we discuss the implementation of the method by numerically solving some test problems of interest over the unit square. In all the

TABLE 6.4. Error Norm as a Function of ε in Example 6

e

δ

Error Norm

0.000 0.002 0.005

0.04 0.06 0.06

0.0001 0.0065 0.0162

200

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

examples, the noisy data functions are obtained by adding an ε random error to / and u at the sample points of the square I = [0,1] x [0,1], that is,

" m i * / . * * ) = u(Xi,t )

+

k

0


{


where 6 and -n are Gaussian random variables with values in [ - 1 , 1 ] . T h e selection of the radius of mollification is implemented as usual and with N = 7V = N, the discrete weighted I norm on K , δ = max(£,, <5 ), is used to measure the solution error. Thus, we have t

k

ik

2

l

s

2

M-2

I

Ms

EXAMPLE 6

A

LL2,A:

S

(Μ -

,V2 L)

η

Σ =ι

2

1/2

M-2

Σ 7 =

i

[A , -a(xj,t )]' SJ n

n

In this example the exact data functions are u(x, t) = xt, f(x,t)

=x

Fig. 6.6 Reconstructed transmissivity coefficient in Example 6: e = 0.005, δ = 0.06, Δί = Αχ = 0.01.

NUMERICAL RESULTS

201

and the initial condition is given by

a(0,t)

du — (0,t)

=t.

The exact transmissivity coefficient is a(x, t) = 1 and the parabolic equation reads

du —

dt

du 2

+

— Ϊ = Χ ,

dx

2

0

<

χ

<

1,

0

<

t

<

1,

with exact solution u(x, t) = xt. Table 6.4 shows the error norm of the solution as a function of 8 and ε for A, =h = h = 0.01. Figure 6.6 illustrates the qualitative behavior of the numerical solution for ε = 0.005 and 8 = 0.06. 2

(a) Fig. 6.7a

Exact transmissivity coefficient in Example 7: singularity at χ = 0.4.

202

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

(b)

Fig. 6.7ft Reconstructed transmissivity coefficient in Example 7: e = 0.005, 8 = 0.06, Δ< = Δχ = 0.01; singularity at χ = 0.4.

E X A M P L E 7 H e r e we consider the case where du/dx(x ,t) times t. T h e differential equation is given by

= 0 for all

ifj

du Β I du\ ^^J- [^' )=2\-te-' x-0A) x

Tx

,

{

+

{

t - 0 A )

+

e - ^

0 < * < 1,0 < t < 1, with exact solution u(x, t) = (x - OA) + (/ - 0.4) . The exact input data functions are 2

u(x,t)

= {x-

OA)

f(x,t)

= 2[-te~'(x

2

+ (t -

2

OA) , 2

- OA) + (t - OA) +

e~ ']. x

T h e unique exact transmissivity coefficient to be reconstructed is a(x, t) = e~ and the singularity occurs at x = 0.4. Figures 6.7A and b show graphs of the exact transmissivity coefficient and the reconstructed coefficient for ε = 0.005 and δ = 0.06, respectively. T h e numerical stability properties of this example are summarized in Table 6.5 for A, = h = h = 0.01. T h e xl

t

2

EXERCISES

203

TABLE 6.5. Error Norm as a Function of ε in Example 7

ε

δ

Error Norm

0.000 0.002 0.005

0.04 0.06 0.06

0.0051 0.0228 0.0445

remark at the end of Sec. 6.7—singular elliptic case—applies also to this parabolic example: for each value of t, the starting information for t h e marching scheme is obtained at the location of the singularity and no initial condition at t h e boundary is needed. E X A M P L E 8 This is a nonlinear example. T h e exact data functions and the initial condition are u(x, t) = xt, f(x,t)

= x +

du a(«(0,f))—(0,f)

t, 2

= 0,

corresponding to the partial differential equation du d I du — + —\u—I = JC + i , dt dx\ dxj 2

0 < χ < 1, 0 < t < 1,

with solution u(x, t) = xt. T h e exact transmissivity coefficient is a(u) = u. In this case, after computing A \ , 1 < i, k < Μ - 2, we collect all the pairs (.A ,u (Xj,t ))— about 4000 with h = 0.01 and δ = 0.06—to obtain the approximate functional relationship between the data function u and the transmissivity coefficient a. A plot is shown in Fig. 6.8 corresponding to t h e noise level ε = 0.005. Table 6.6 illustrates the numerical stability of the method. i

6.12

k

Stitk

m

k

EXERCISES

6.1. Consider the initial value problem dy 2

dt y ( 0 ) = 1,

2

( 0 = y ( 0 +P(0,

dy ^ - ( 0 ) = 1 - 5ir,

and

ο < ί < ι, p(t)

= (1 + 25ττ ) sin57ri, 2

204

STABLE NUMERICAL DIFFERENTIATION PROCEDURES 0.00-1

Η

0.66

0.46-

0.40-

c

_5

0.35-

/

O

Η

^

0.30

<»Φ Ο

0.25-

U

0.20-

0.16·^

0.10-^

0.05-

ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—Γ

0.00 0.00

0.25

0.50

Tempere

0.76

1.00

ture

Fig. 6.8 Reconstructed transmissivity coefficient in Example 8: ε = 0.005, δ = 0.06, Δί = Δχ = 0.01.

TABLE 6.6. Error Norm as a Function of e in Example 8

ε

δ

Error Norm

0.000 0.002 0.005

0.04 0.06 0.06

0.0030 0.0033 0.0048

which has exact solution y(?) = e' - sin57ri, 0 < f < 1. Using the mollification method, numerically identify the forcing term pU) by (a) reducing the second-order differential equation to a first-order system. (b) directly approximating the second derivative. Consider as data for the inverse problem the exact solution y(t) perturbed with r a n d o m noise of amplitude ε = 0.005 and sample step Δ ί = 0.01. C o m p a r e the relative error norms in both cases.

REFERENCES AND COMMENTS

205

6.2. Prove Lemma 6.10. 6.3. Introduce Pa(x)

= Ps,( i)Ps ( 2) ··" Ps„( J> x

x

x

2

where

Ps,K) =

^ v V

e

X

" (

P

*„)>

δ

= (δι,δ .···.*»)• 2

Show that if J u = ( p * wXx) = / „ p ( x - s ) n ( s ) d s and ||Vu|L, ^ < Λ/, then s

6

\\J$u - « l k

< 4ηΜδ,

R

e

δ =

max ( δ ) < d i s t ( Κ $ , d l ) , (

with Kg a suitable compact subset of η times

/ = [ 0 , 1 ] X [0,1] X · · · X [ 0 , 1 ]

cR".

6.4. With the same notation as in Exercise 6.3, show that if u, u e C ° ( 7 ) (continuous functions over 7) and \\u / < ε, then | | V 7 - M V7 -M|L, K ^ 8 ε / δ , where δ = πνίη(δ,, δ , . . . , δ„). m

m

s

6.13

6

S

M

2

R E F E R E N C E S AND C O M M E N T S

T h e following references and comments serve to expand the basic material covered in the corresponding sections. 6.1-6.3. H e r e we follow in part the article D. A. Murio and D. Hinestroza, Numerical identification of forcing terms by discrete mollification, Comput. Math. Appl. 17 (1990), 75-89. 6.4. Several algorithms have been proposed for the solution of the p a r a m e t e r identification problem in the steady-state flow. A n important review of p a r a m e t e r estimation for ground water hydrology was presented in W. W. G. Yeh, Review of parameter identification procedures in ground water hydrology: the inverse problem, Water Resources Res. 22 (1986), 95-108. The output least squares minimization method was first considered by E. Frind and G. Pinder, Galerkin solutions of the inverse problem for aquifer transmissivity, Water Resources Res. 9 (1973), 1397-1410.

206

STABLE NUMERICAL DIFFERENTIATION PROCEDURES

T h e method has been recently studied by K. Kunish and L. W. White, Identifiability under approximation for an elliptic boundary value problem, SIAMJ. Control Optim. 25 (1987), pp. 279-297. and, combined with regularization techniques, by C. Kravaris and J. H. Seinfeld, Identification of parameters in distributed parameter systems by regularization, SIAM J. Control Optim. 23 (1985), 217-241. Stability results have been given by R. S. Falk, Error estimates for the numerical identification of a variable coefficient, Math. Comp. 40 (1983), 537-546. Further applications of this method to different parameter problems can b e found in

identification

G. Chavent, Identification of distributed parameter systems: about the output least squares method, its implementation and identifiability, Proc. Fifth IFAC Symp. on Identification and System Parameter Estimation 1, R. Iserman (ed.), Pergamon, Darnstadt, 1980, pp. 85-97. Η. T. Banks and K. Kunish, Parameter estimation techniques for nonlinear distributed parameter systems, Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham (ed.), Academic, New York, 1982, pp. 57-67. A new interesting variational approach based on the minimization of a convex functional was introduced in R. V. Khon and B. D. Lowe, A variational method for parameter estimation, Math. Modeling Numer. Anal. 22 (1988), 119-158. Singular perturbation techniques have been proposed in G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl. 145 (1986), 265-296. G. Alessandrini, On the identification of the leading coefficient of an elliptic equation. Boll. U.M.I., Analisi Funzionale Applicazioni, Ser. VI 4-C (1985), 87-111. Finite-difference " m a r c h i n g " methods, integrating along the lines of steepest descent or characteristics, were first developed—for noiseless data—by G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math. 41 (1981), 210-221. G. R. Richter, Numerical identification of a spatially varying diffusion coefficient, Math. Comp. 36 (1981), 404-427. 6.5-6.7. In these sections we follow closely the treatment in D. Hinestroza and D. A. Murio, Identification of transmissivity coefficients by mollification techniques. I. One-dimensional elliptic and parabolic problems, Comput. Math. Appl. 25 (1993), 59-79.

REFERENCES AND COMMENTS

207

6.8. T h e most commonly used method for the approximate determination of unknown coefficients in the one-dimensional transient flow case is the output least squares approach indicated in the previous references. Some alternative approaches including asymptotic imbedding and adaptive control techniques have been investigated in H. W. Alt, Κ. H. Hoffmann, and J. Sprekel, A numerical procedure to solve certain identification problems, Internal. Ser. Numer. Math. 68 (1984), 11-43. J. Baumeister and W. Scondo, Asymptotic imbedding methods for parameter estimation, Proc. 26th. Conf. on Decision and Control, Los Angeles, 1987, pp. 170-174. For spatially varying coefficients, a finite-difference scheme is discussed in R. E. Ewing and T. Lin, Parameter identification problems in single-phase and two phase flow, Intemat. Ser. Numer. Math. 91 (1989), 85-108. A new method that combines the output least squares and the error equation, resulting in an optimization problem solved by an augmented Lagrangian approach, can be found in K. Kunish and G. Peichel, Estimation of temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique, Numer. Math. 59 (1991), 473-509. 6.9-6.11. H e r e we follow the article of D. Hinestroza and D. A. Murio referenced previously. For the application of the mollification method to the two-dimensional identification of transmissivity coefficients, see D. Hinestroza, Identification of transmissivity coefficients by mollification techniques. II. Two-dimensional elliptic and parabolic problems, to appear.

The Mollification Method and the Numerical Solution of Ill-Posed Problems by Diego A. Murio Copyright © 1993 John Wiley & Sons, Inc.

APPENDIX A MATHEMATICAL BACKGROUND

T h e r e are several basic concepts concerning properties of Hubert spaces, mollifiers, Fourier transforms, and discrete functions that play an important role in the topics developed in this appendix. We state all the necessary definitions and propositions but offer only a selected set of demonstrations. For the remaining proofs, the interested reader should consult the references indicated at the end of the chapter. Throughout this appendix, the space of real numbers (the real line) is denoted R, the complex numbers C, and R" = R x R x - - - X R ( n times) denotes the η-dimensional Euclidean space. T h e length of the vector χ = ( x χ , · · · , x ) e R" is represented by 1 ;

2

n

and Ω always denotes an open, bounded domain in U". In the sequel additional notation and conventions will be defined where they first appear.

A.1

L" SPACES

DEFINITION A.1 L (fL), ρ > 1, denotes the space of all equivalence classes of real-valued (or complex-valued) Lebesgue-measurable functions / , defined almost everywhere on an open, bounded domain flcR", whose p

208

L"SPACES

pth powers \f\

are Lebesgue integrable over Ω , that is,

p

/

χ = (x

u

209

x ,...,

\f(x)fdx
x ) s W and dx = mesidx,,

2

dx ,dx ).

n

2

n

Any pointwise property attributed to a function / e L (il) will thus be understood to hold in the usual sense for some function in the same equivalence class. W e recall that the function / is equivalent to the function g, if / ( J C ) = gix) for almost all χ e Ω, that is, except for a set of n-dimensional measure 0. W e notice that if fix) and gix) belong to L (Cl), using the inequalities p

p

f(x)

+g{x)\

<2 msx{\f(x)\ ,\g(x)\ }

P

p

P

<2"{\f{x)\

P

+\g{x)\ ),

P

P

it follows that (/ + *)(*)

+g(x)^L {Sl) p

and (cf)(x)=cf(x)^L (D.)

for every real (complex) n u m b e r c.

p

T h e set of functions L (Cl) with the addition and scalar multiplication defined previously is a vector space. p

DEFINITION A.2

T h e real function

is a norm in the vector space L {£i); satisfied:

that is, the following axioms are

p

(Nl)

11/IUn £

(N2)

||/||„

(N3)

V/eL'(ft),

0,

= 0

> η

if and only if

/ = 0,

V c scalar, V / e

l|c/IUn = lc|||/|| ,n, p

( N 4 ) l!/ + g ! U n < l l / l ! p . n + ! l g ! ! , n ,

L {il),

\ff,geL'(Cl).

P

From property (N4), the triangle inequality, we find (Ν4')

ll/-gll ,n>lll/llp,n-|lgllp, il, P

i

V/,geL'(n).

p

210

MATHEMATICAL BACKGROUND

Proof l l / l l p . n = l l ( / - 8) +g\\ M P

Ϊ \\f-8\\p.n

+

WsWp.a

and ll/IUn-W

P

.n^ll/-ilUn.

Similarly, H s I L n - l l / l l p . n < Hi - / l l p . n = l l ( - i ) ( / - i ) l , n = l l / - « I U n .

•

We shall use t h e notation | | / | | for | | / | | and | | / | | for | | / | | when there is no ambiguity. ( Ζ Λ Ω ) , || · briefly Ζ Λ Ω ) — i s a normed space. T h e distance, as usual, is defined by t h e real function (functional) p

Ρ > Ω

2

= ll/-gllp.n,

d(f,g) which satisfies properties (dl)

d(f,g)

>0,

V/,?eL («),

(d2)

d(f,g)

=0

if and only if

(d3)

d(f,g)

(d4)

d{f,h)

p

ν/,«εί (Ω),

=d(g,f),
/ = g,

ρ

+d{g,h),

Theorem A.1 (Riesz and Fischer)

\ff,g,h^L (il). p

T h e normed space L iii) p

is complete.

In other words, L (Ci) is a Banach space. This means that for every sequence {fX of functions in L iCl) such that l l / „ - / l l 0 , there exists / e L"iil) such that \\f„ - f\\ ^ „ ^ 0. For ρ = oo, Ζ,°°(Ω) denotes the Banach space of b o u n d e d functions on Ω with the norm | | / | | » , Ω = s u p _ , . | / ( j c ) | . In this context, the supremum (infimum) of t h e function / should b e understood as t h e essential s u p r e m u m (infimum). Note: T h e minimum essential u p p e r b o u n d σ of fix) is called t h e essential supremum of fix); σ is an essential upper bound of fix) if fix) < σ for almost every x, that is, there exists a set Ρ of measure 0 iP c Ω ) such that fix) < σ if χ £ P. p

p

m

=l

p

P

0 O

en

Theorem A.2 (Holder) If l/p + l/q = 1, ρ ^ 1, q > 1 and if / e Ζ Λ Ω ) , g e L iVt\ then fg is integrable, that is, fg e L ' ( O ) and q

THE HILBERT SPACE L ( Ω)

211

2

Observe that for ρ = 2, we have q = 2 and Holder's inequality reduces to the Cauchy-Schwarz inequality. (See T h e o r e m A.3.) E X E R C I S E A.1 1. Show that L {0) p

c L (n\

ρ > 1.

l

2. Prove Minkowski's inequality

j/j/(;r)f

+g{x)\ dx<

Qf(x)

P

ώ:} "

+ {/j g ( * ) Γ V/,geL'(n).

3. Show that if / e L ( f t ) and g e Ζ Λ Ω ) , where 1/p + l/q / g e Ζ,'(Ω). p

= 1, then

4. Show that 1 ( Ω ) is a metric space—satisfies ( d l ) - ( d 4 ) — w i t h Ρ

d(f,g)

V/,ge^(n).

=\\f~g\\ , P

5. For what values of ρ does the function f(x)

= x~

6. Prove that if / „ / in L (il) Exercise 3), then l i m „ ^ „ j f {x)g{x)

n

p

a n

A.2

belong to

x/1

and g e L«(n) dx = f f(x)g(x)

L ((0,1))?

( p and dx.

p

as in

L (Cl)

T H E H I L B E R T SPACE

2

We observe that if / e L (n) and g e ί " ( Ω ) with ρ Φ 2, 1 < ρ < c ° , Holder's theorem is not applicable, that is, in general fg will not be integrable. However, for ρ = 2, it is possible to introduce a geometrical structure: the bilinear form p

(f,g)

= [f(x)gjxldx,

V/,geL (n), 2

(A.l)

defines an inner product in L ( 0 ) , that is, it verifies 2

(HI)

(/,/)

=||/|| ,

V/eL (il),

(H2)

(/,*)

= ΰΓΠ,

V/,geL (n),

Cg

=c(f,g),

VceC,V/,geL (n),

(f + g,h)

= (f,h)

(H3) (H4)

( ,f)

2

2

2

2

+ (g + h),

Vf,g,h

eL (n). 2

H e r e , ( g , / ) stands for the complex conjugate of the complex number ( g , / ) .

212

MATHEMATICAL BACKGROUND

From these properties it follows that (H5)

(f,cg)=c(f,g),

VceC,V/,geL (il), 2

and (H6)

(f,g

+ h) = (f,g)

V/,i,A6L (0).

+ (f,h),

2

Proof (f,cg)

= (cg,f)

= c(g,f)

=c{f,g).

Also, (f,g

+ h)=

(g + h,f)

=

+ (/,,/)

(g,f)

= ( / , g ) + (/,A).

•

DEFINITION A.3 A H u b e r t space Η is a Banach space—complete normed vector space—with an inner product ( · , · ) that verifies properties ( H I ) to (H4). Note: T h e space £ ( Ω ) with the inner product ( A . l ) is a H u b e r t space. 2

DEFINITION A.4 Two functions / and g in Ζ Λ Ω ) are said to be orthogonal, and we write / 1 g if ( / , g) = 0. In a Hubert space H,

Theorem A.3 (Cauchy-Schwarz Inequality) | ( u , v ) | < llullllvll,

Vu,ve//.

Proof. Consider ν Φ 0. From 0 < ||u - c v | | = ( u - c v , u - cv) 2

= ( u , u ) + ( u , - c v ) + ( - c v , u ) + (— cv, - c v ) = ||u|| - c(u,v) - c ( u , v )

+|c| ||v|| ,

2

2

with c = ( u , v ) / | | v | | , we get |(u,v)| < | | u | | | | v | | . 2

2

2

2

2

•

We observe that the inner product with properties ( H I ) to (H4) determines t h e norm in t h e space. Again, if ρ Φ 2 we cannot talk about orthogonality or angles in L (fl) and we can only consider distances. L ( f i ) is the only L ( i l ) space that is a Hilbert space. A function / that belongs to Ζ. (Ω) is said to be square integrable in Ω. Notice that if a function fix) is continuous ϊη_Ω (the closure of Ω ) , then from jnfix) dx = 0 it follows that fix) = 0 in Ω. If / e Ζ. (Ω), without the p

2

p

2

2

THE HILBERT SPACE L (Cl) 2

213

continuity assumption, our conclusion should be that fix) = 0 almost everywhere, that is, except perhaps on a set of measure 0. For instance, the function (0,

0<x<\/2,

1,

/(*)

x = l/2,

0,

1/2<χ<1,

is square integrable in (0,1) but it is not identically 0 there. At χ = 1 / 2 — o r at any other set of measure 0 — t h e function need not even be defined. Two functions fix) and gix) which are equivalent in L iSl) are characterized by the property 2

f [f{x) n

-g(x)] dx

= 0.

2

J

E X E R C I S E A.2 1. Show that if / e L iCl) which is dense in L iil),

is orthogonal to all the elements of a subset D then / = 0.

2

2

2. Let f ix) n

= n/i\

+ ni

1 / 2

) for 0 < χ < 1, η = 1 , 2 , . . . .

(a) Is t h e sequence {/„}^ , a Cauchy sequence? =

(b) Does /„ e L ((0,1)) V «? 2

(c) Does l

i

m

„

e

L ((0,1))? 2

3 . Let {/3)7= ι be an orthonormal sequence of functions in L ( ( a , b)), that is, iβ ,β ) = 0 V i,j, ΪΦ), \\β \\ = 1 V j . Let S„ix) = E J _ , c j 8 ( j c ) , where c = )α^χ)β^χ) dx. Show that for any function / e L ( ( a , b)), 2

;

i

j

}

2

A

t

2

k

WS„-fh

(The coefficients c

=

f \fix)\ dxh

2

η 2 Zc kk=\

are called Fourier coefficients.)

k

4. U n d e r the conditions of Exercise 3, show that °°

b

Σ l c

^ j

\f( )\ x

dx

2

(Bessel's inequality).

5. U n d e r the conditions of Exercise 3, prove that if l i m „ _ J | S „ - / I I 2 = 0, then b

J \f(x)\

°°

2

dx =

Σ k c

(Parseval's identity).

214

MATHEMATICAL BACKGROUND

A.3

APPROXIMATION O F FUNCTIONS IN

L ((l) 2

DEFINITION A.5 C ° ( i l ) denotes the set of all functions which are continuous together with their derivatives of all orders in Ω. DEFINITION A.6 T h e support of a function_/(x) in Ω, written s u p p / , is the closure in U" of the set of points χ e Ω for which f(x) Φ 0. DEFINITION A.7 support in Ω.

CRSl)

is the set of functions from C"(iQ) with compact

Observe that if / e Q(n\ then / e CXH) and supp / c Ω. Moreover, since s u p p / is a closed set contained in an open domain Ω, there is a positive distance from any point in supp / to the boundary δ Ω of the domain Ω. W e say, briefly, that fix) is 0 in a neighborhood of the boundary. For instance, the function - 2 < χ < 2, otherwise, is C°XU) and, relative to the domain Ω = ( - 5,5), for example, supp / = [2,2]. Thus, / e With more generality, we can consider the function

CRSl).

- δ < χ < 8, otherwise, where δ is a fixed positive number.