Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics

Подождите немного. Документ загружается.

426 Chapter 8 Other problems

IMPLICIT REAL

8 ( A-H, O-Z )

DIMENSION F(N1,N2), YD1(N1), YD2(N2), YD3(N1), YD4(N2)

+ ,YY1(N1), YY2(N2), YY3(N1), YY4(N2)

DOI=1,N1

DO J = 1,N2

F(I,J) = 0.D0

END DO

DOI=3,N1-2

F(I,2) = 2.D0

(YD1(I) - YY1(I)) / H2

F(I,N1-1) = 2.D0

(YD3(I) - YY3(I)) / H2

END DO

DOJ=3,N2-2

F(2,J) = 2.D0

(YD2(J) - YY2(J)) / H1

F(N2-1,J) = 2.D0

(YD4(J) - YY4(J)) / H1

END DO

F(2,2) = (H1+H2)

(YD1(2) - YY1(2)) / (H1

H2)

F(2,N2-1) = (H1+H2)

(YD4(2) - YY4(2)) / (H1

H2)

F(N1-1,2) = (H1+H2)

(YD3(2) - YY3(2)) / (H1

H2)

F(N1-1,N2-1) = (H1+H2)

(YD3(N1) - YY3(N1)) / (H1

H2)

RETURN

END

SUBROUTINE BNG (U, X1, X2, N1, N2)

C FIRST-KIND BOUNDARY CONDITION

IMPLICIT REAL

8 ( A-H, O-Z )

DIMENSION U(N1,N2), X1(N1), X2(N2)

DOI=1,N1

DOJ=1,N2

U(I,J) = 1.D0 + X1(I)

END DO

RETURN

END

SUBROUTINE BNGDER (U, H1, H2, N1, N2, YD1, YD2, YD3, YD4)

C MESH FUNCTION AT THE NEAR-BOUNDARY NODES

IMPLICIT REAL

8 ( A-H, O-Z )

DIMENSION U(N1,N2), YD1(N1), YD2(N2), YD3(N1), YD4(N2)

DOI=2,N1-1

YD1(I) = U(I,2)

YD3(I) = U(I,N2-1)

END DO

DOJ=2,N2-1

YD2(J) = U(2,J)

YD4(J) = U(N1-1,J)

END DO

RETURN

END

SUBROUTINE GRAD (U, XI, H1, H2, N1, N2, GR)

Section 8.5 Coefﬁcient inverse problem for elliptic equation 427

C MESH FUNCTION AT THE NEAR-BOUNDARY NODES

IMPLICIT REAL

8 ( A-H, O-Z )

DIMENSION U(N1,N2), XI(N1,N2), GR(N2)

DOJ=2,N2-1

SUM = 0.D0

DOI=2,N1-1

SUM = SUM + U(I,J)

XI(I,J)

END DO

GR(J) = SUM

END DO

RETURN

END

In solving the direct problem, the coefﬁcient c(x

) is set in the subroutine-function AC.

In the subroutine BNG, the boundary condition (8.166) is set. The coefﬁcient array for

the difference equation is generated in the subroutine FDS. The difference problem for

the conjugate state (8.190), (8.192) differs from the difference problem for the ground

state (8.180), (8.181) in the right-hand side and in the boundary conditions only (see

the subroutine RHS).

8.5.6 Computational experiments

In solving the problem on identiﬁcation of the lower coefﬁcient, the input data were

taken from the solution of the direct problem (8.165), (8.166) in the unit square 

= l

= 1) with

c(x

) = 10x

,ϕ(x) = 1 + x

The problem was solved on the uniform grid with h

= h

= 0.02. Difference-

solution contour lines spaced 0.05 apart are shown in Figure 8.27.

From the approximate solution of the direct problem, the values at the near-

boundary nodes of the calculation grid are extracted; in this way, a condition of type

(8.183) is being set. These input data for the inverse problem are perturbed with some

random function. Conditions are modeled in which an exact ﬁrst-kind boundary con-

dition (see (8.166)) is set, whereas the additional, second-kind boundary conditions

(see (8.167)) are set with some inaccuracy. In these conditions, for instance, at the

near-boundary nodes closest to the boundary 

we have:

y(h

, x

) = φ

((h

, x

)) = φ((h

, x

)) + 2h

δ(σ(h

, x

) − 1/2),

where

φ((h

, x

)) = ϕ(h

, x

) − h

ψ(h

, x

and σ(h

, x

) is a random function.

The data obtained in the reconstruction of the coefﬁcient with δ = 0.02 are shown

in Figure 8.28. The solution of the direct problem obtained with this value of the

428 Chapter 8 Other problems

coefﬁcient is shown in Figure 8.29. The effect due to the inaccuracy level can be

ﬁgured out considering Figures 8.30 and 8.31 (twice decreased and twice increased

inaccuracy, respectively).

Figure 8.27 Solution of the direct problem

Figure 8.28 Reconstruction of the coefﬁcient with δ = 0.02

Section 8.5 Coefﬁcient inverse problem for elliptic equation 429

Figure 8.29 Contour lines for the inverse-problem solution obtained with δ = 0.02

Figure 8.30 Inverse-problem solution obtained with δ = 0.01

430 Chapter 8 Other problems

Figure 8.31 Inverse-problem solution obtained with δ = 0.04

8.6 Exercises

Exercise 8.1 Consider the matter of using the quasi-inversion method in continuation

over the spatial variable in solving the boundary value inverse problem in the rectangle

 ={x | x = (x

, x

), 0 < x

< l

,α= 1, 2}

for the two-dimensional parabolic equation

∂u

∂t

−



β=1

∂

∂x

= 0, x ∈ , 0 < t < T ,

u(0, x

, t) = ϕ(x

, t),

∂u

∂x

(0, x

, t) = 0,

u(x

, 0, t) = 0, u(x

, l

, t) = 0, 0 < t < T ,

u(x, t) = 0, x ∈ .

Exercise 8.2 Examine the convergence of the difference scheme (8.43) in the approx-

imate solution of the Cauchy problem (8.12), (8.13), (8.25).

Exercise 8.3 Based on the program PROBLEM15, write a program that realizes the

quasi-inversion method in the variant in which the equation

∂

∂x

−

∂v

∂t

+ α

∂

∂t

= 0, 0 < x < l, 0 < t < T

is to be solved. Perform a comparative analysis of the variants of the quasi-inversion

method as applied to the approximate solution of the model problems.

Section 8.6 Exercises 431

Exercise 8.4 In the Tikhonov regularization method as applied to approximate solu-

tion of the boundary value inverse problem for one-dimensional parabolic equation, to

be minimized is the functional

(v) =



(u(0, t) − ϕ

(t))

dt + α



(t) dt

with the constraints

∂u

∂t

∂

∂x



k(x)

∂u

∂x



, 0 < x < l, 0 < t ≤ T,

k(x)

∂u

∂x

(0, t) = 0, 0 < t ≤ T,

u(l, t) = v(t), 0 < t ≤ T,

u(x, 0) = 0, 0 ≤ x ≤ l.

Obtain the optimality conditions for this optimal control problem (Euler equation).

Exercise 8.5 In the uniform norm, examine stability of the difference scheme (8.79)–

(8.82) as applied for the solution of the problem with a non-local boundary condition.

Exercise 8.6 In the program PROBLEM16, provide a scheme for mesh smoothing of

input data for obtaining a smooth solution of the inverse problem. Perform numerical

experiments showing how the smoothing parameter affects the approximate-solution

accuracy in model inverse problems.

Exercise 8.7 Construct a gradient iteration method for the approximate solution of the

boundary value inverse problem

∂u

∂t

+ b(x)

∂u

∂x

∂

∂x



k(x)

∂u

∂x



, 0 < x < l, 0 < t ≤ T, (8.195)

k(x)

∂u

∂x

(0, t) = 0, 0 < t ≤ T, (8.196)

u(0, t) = ϕ(t), 0 < t ≤ T , (8.197)

u(x, 0) = 0, 0 ≤ x ≤ l, (8.198)

by reﬁning the boundary condition (function v(t)) on the right boundary:

u(l, t) = v(t), 0 < t ≤ T.

Exercise 8.8 Construct an additive difference scheme (splitting scheme) over spatial

variables for the realization of the method with non-locally perturbed boundary condi-

tions in solving the boundary value inverse problem in the rectangle :

∂u

∂t

−



β=1

∂

∂x



k(x)

∂u

∂x



= 0, x ∈ , 0 < t < T ,

432 Chapter 8 Other problems

∂u

∂x

(0, x

, t) = 0, u(0, x

, t) + αu(l

, x

, t) = ϕ(x

, t),

u(x

, 0, t) = 0, u(x

, l

, t) = 0, 0 < t < T ,

u(x, t) = 0, x ∈ .

Exercise 8.9 The program PROBLEM17 implements an algorithm for iterative reﬁne-

ment of a second-kind boundary condition. Modify this program to realize an algo-

rithm for iterative reﬁnement of the third-kind boundary condition

∂u

∂n

(x, t) + σ u(x, t) = μ(x

, t), x ∈ γ.

Perform computational experiments to investigate the effect due to the numerical pa-

rameter σ ≥ 0.

Exercise 8.10 Suppose that in the problem

∂u

∂t

−

∂

∂x



k(x, t)

∂u

∂x



= 0, 0 < x < l, 0 < t ≤ T,

u(0, t) = 0, u(l, t) = g(t), 0 < t ≤ T ,

u(x, 0) = 0, 0 ≤ x ≤ l

the coefﬁcient k(x, t) is a piecewise constant function, but the position of the interface

between the medium is unknown. In the representation

k(x, t) =



, 0 < x <γ(t),

,γ(t)<x < l

the constants k

, β = 1, 2 are given, and the function γ(t) is to be determined from

additional observations of the solution performed at the internal points z

∈ , m =

1, 2,...,M:

u(z

, t) ≈ ϕ

(t), 0 < t ≤ T, m = 1, 2,...,M.

Examine the possibility of constructing gradient iterative methods for the approximate

solution of this coefﬁcient inverse problem.

Exercise 8.11 Examine the possibility of successive identiﬁcation (local regulariza-

tion) of the nonlinear coefﬁcient k(u) in the approximate solution of the inverse prob-

lem (8.132)–(8.136) using the parametric identiﬁcation (8.147) in the class of piece-

wise linear functions.

Exercise 8.12 Modify the program PROBLEM18 using the gradient iterative method

for determining the piecewise constant coefﬁcient k(u) instead of the golden-section

method applied to minimize the discrepancy functional on an individual subinterval.

Compare the computational efﬁciencies of these approaches as exempliﬁed by the so-

lutions of model problems.

Section 8.6 Exercises 433

Exercise 8.13 Consider the matter of solution unicity for the coefﬁcient inverse prob-

lem on ﬁnding the function pair {u(x), k(x

)} from the conditions



β=1

∂

∂x



k(x

)

∂u

∂x



= 0, x ∈ ,

u(x) = ϕ(x), x ∈ ∂,

∂u

∂n

(x) = ψ(x), x ∈ ∂.

Exercise 8.14 Consider on the difference level the coefﬁcient inverse problem formu-

lated in the previous exercise. Obtain the gradient of the discrepancy functional under

the assumption that the ﬁrst-kind boundary conditions are given exactly and that the

second-kind boundary conditions are given approximately.

Exercise 8.15 Using the program PROBLEM19, perform numerical experiments on

the determination of the lowest coefﬁcient c(x

) in equation (8.165) with additional

information available not on the whole boundary ∂, but only on some individual

segments of the boundary, namely, on the sides 

, β = 1, 2, 3, 4.

Bibliography

Alifanov, O. M. (1995). Inverse Heat Transfer Problems. Springer-Verlag Telos.

Alifanov, O. M., Artiukhin, E. A., and Rumiantsev, S. V. (1995). Extreme Methods for Solv-

ing Ill-Posed Problems With Applications to Inverse Heat Transfer Problems. Begell

House Publishers.

Bakushinskii, A. B., and Goncharskii, A. V. (1989). Iterative Solution Methods for Ill-Posed

Problems. Nauka, Moscow.

Bakushinskii, A. B., and Goncharskii, A. V. (1989). Ill-Posed Problems: Numerical Meth-

ods. Moscow State Univ. Publ., Moscow.

Denisov, A. M. (1999). Elements of the Theory of Inverse Problems (Inverse and Ill-Posed

Problems). Brill Academic Publishers.

Ivanov, V. K., Vasin, V. V., and Tanana, V. P. (2002). Theory of Linear Ill-Posed Problems

and Its Applications. Brill Academic Publishers.

Isakov, V. (1997). Inverse Problems for Partial Differential Equations. Springer-Verlag.

Lavrent’ev, M. M., Reznitskaya, K. G., and Yakhno, V. G. (1982). One-Dimensional Ill-

Posed Problems in Mathematical Physics. Nauka, Novosibirsk.

Lavrent’ev, M. M., Romanov, V. G., and Shishatskii, S. P. (1980). Ill-Posed Problems in

Mathematical Physics. Nauka, Moscow.

Lattes, R., and Lions, J.-L. (1969). Method of Quasireversibility. Elsevier.

Lions, J.-L. (1972). Optimal Control of Systems Governed by Partial Differential Equations.

Mir, Moscow.

Marchuk, G. I. (1989). Computational-Mathematics Methods. Nauka, Moscow.

Morozov, V. A. (1987). Regularization Methods for Non-Stationary Problems. Moscow State

Univ. Publ., Moscow.

Osipov, Yu. S., Vasil’ev, F. P., and Potapov, M. M. (1999). Basics of the Dynamic Regulariza-

tion Method. Moscow State Univ. Publ., Moscow.

Prilepko, A. I., Orlovsky, D. G., and Vasin, I. A. (2000). Methods for Solving Inverse Prob-

lems in Mathematical Physics. Marcel Dekker.

Romanov, V. G. (2002). Investigation Methods for Inverse Problems. Brill Academic Pub-

lishers.

Romanov, V. G., and Kabanikhin, S. I. (1991). Inverse Problems in Geoelectricity. Nauka,

Moscow.

Samarskii, A. A. (2001). Theory of Difference Schemes. Marcel Dekker.

Samarskii, A. A., and Andreev, V. B. (1976). Difference Methods for Elliptic Equations.

Nauka, Moscow.

Samarskii, A. A., and Gulin, A. V. (1973). Stability of Difference Schemes. Nauka, Moscow.

Samarskii, A. A., and Nikolaev, E. (2001). Numerical Methods for Grid Equations, Volume

II. Birkh

auser.