Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics
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x Contents
2.4.2 Difference schemes ....................... 37
2.4.3 Program ............................. 39
2.4.4 Computational experiments .................. 43
2.5 Exercises ................................ 45
3 Boundary value problems for elliptic equations 49
3.1 The difference elliptic problem ..................... 49
3.1.1 Boundary value problems ................... 49
3.1.2 Difference problem ....................... 50
3.1.3 Problems in irregular domains ................. 52
3.2 Approximate-solution inaccuracy ................... 54
3.2.1 Elliptic difference operators .................. 54
3.2.2 Convergence of difference solution .............. 56
3.2.3 Maximum principle ....................... 57
3.3 Iteration solution methods for difference problems .......... 59
3.3.1 Direct solution methods for difference problems ....... 59
3.3.2 Iteration methods ........................ 60
3.3.3 Examples of simplest iteration methods ............ 62
3.3.4 Variation-type iteration methods ................ 64
3.3.5 Iteration methods with diagonal reconditioner ......... 66
3.3.6 Alternate-triangular iteration methods ............. 67
3.4 Program realization and numerical examples ............. 70
3.4.1 Statement of the problem and the difference scheme ..... 70
3.4.2 A subroutine for solving difference equations ......... 71
3.4.3 Program ............................. 79
3.4.4 Computational experiments .................. 83
3.5 Exercises ................................ 85
4 Boundary value problems for parabolic equations 90
4.1 Difference schemes ........................... 90
4.1.1 Boundary value problems ................... 90
4.1.2 Approximation over space ................... 92
4.1.3 Approximation over time .................... 93
4.2 Stability of two-layer difference schemes ............... 95
4.2.1 Basic notions .......................... 95
4.2.2 Stability with respect to initial data .............. 97
4.2.3 Stability with respect to right-hand side ............ 100
4.3 Three-layer operator-difference schemes ................ 102
4.3.1 Stability with respect to initial data .............. 102
4.3.2 Passage to an equivalent two-layer scheme .......... 104
4.3.3 ρ-stability of three-layer schemes ............... 106
4.3.4 Estimates in simpler norms ................... 108
4.3.5 Stability with respect to right-hand side ............ 110
Contents xi
4.4 Consideration of difference schemes for a model problem ....... 110
4.4.1 Stability condition for a two-layer scheme ........... 111
4.4.2 Convergence of difference schemes .............. 112
4.4.3 Stability of weighted three-layer schemes ........... 113
4.5 Program realization and computation examples ............ 114
4.5.1 Problem statement ....................... 114
4.5.2 Linearized difference schemes ................. 115
4.5.3 Program ............................. 118
4.5.4 Computational experiments .................. 121
4.6 Exercises ................................ 124
5 Solution methods for ill-posed problems 127
5.1 Tikhonov regularization method .................... 127
5.1.1 Problem statement ....................... 127
5.1.2 Variational method ....................... 128
5.1.3 Convergence of the regularization method ........... 129
5.2 The rate of convergence in the regularization method ......... 131
5.2.1 Euler equation for the smoothing functional .......... 131
5.2.2 Classes of a priori constraints imposed on the solution .... 132
5.2.3 Estimates of the rate of convergence .............. 133
5.3 Choice of regularization parameter ................... 134
5.3.1 The choice in the class of a priori constraints on the solution . 135
5.3.2 Discrepancy method ...................... 136
5.3.3 Other methods for choosing the regularization parameter . . . 137
5.4 Iterative solution methods for ill-posed problems ........... 138
5.4.1 Specific features in the application of iteration methods .... 138
5.4.2 Iterative solution of ill-posed problems ............ 139
5.4.3 Estimate of the convergence rate ................ 141
5.4.4 Generalizations ......................... 143
5.5 Program implementation and computational experiments ....... 144
5.5.1 Continuation of a potential................... 144
5.5.2 Integral equation ........................ 146
5.5.3 Computational realization ................... 147
5.5.4 Program ............................. 148
5.5.5 Computational experiments .................. 152
5.6 Exercises ................................ 154
6 Right-hand side identification 157
6.1 Right-hand side reconstruction from known solution: stationary prob-
lems ................................... 157
6.1.1 Problem statement ....................... 157
6.1.2 Difference algorithms ..................... 158
6.1.3 Tikhonov regularization .................... 161
xii Contents
6.1.4 Other algorithms ........................ 163
6.1.5 Computational and program realization ............ 164
6.1.6 Examples ............................ 172
6.2 Right-hand side identification in the case of parabolic equation .... 175
6.2.1 Model problem ......................... 175
6.2.2 Global regularization ...................... 176
6.2.3 Local regularization ...................... 178
6.2.4 Iterative solution of the identification problem ......... 180
6.2.5 Computational experiments .................. 189
6.3 Reconstruction of the time-dependent right-hand side ......... 191
6.3.1 Inverse problem ......................... 192
6.3.2 Boundary value problem for the loaded equation ....... 192
6.3.3 Difference scheme ....................... 194
6.3.4 Non-local difference problem and program realization .... 194
6.3.5 Computational experiments .................. 199
6.4 Identification of time-independent right-hand side: parabolic equations 201
6.4.1 Statement of the problem .................... 201
6.4.2 Estimate of stability ...................... 202
6.4.3 Difference problem ....................... 204
6.4.4 Solution of the difference problem ............... 207
6.4.5 Computational experiments .................. 215
6.5 Right-hand side reconstruction from boundary data: elliptic equations 218
6.5.1 Statement of the inverse problem ................ 218
6.5.2 Uniqueness of the inverse-problem solution .......... 219
6.5.3 Difference problem ....................... 220
6.5.4 Solution of the difference problem ............... 224
6.5.5 Program ............................. 226
6.5.6 Computational experiments .................. 234
6.6 Exercises ................................ 237
7 Evolutionary inverse problems 240
7.1 Non-local perturbation of initial conditions .............. 240
7.1.1 Problem statement ....................... 240
7.1.2 General methods for solving ill-posed evolutionary problems . 241
7.1.3 Perturbed initial conditions ................... 243
7.1.4 Convergence of approximate solution to the exact solution . . 246
7.1.5 Equivalence between the non-local problem and the optimal
control problem ......................... 250
7.1.6 Non-local difference problems ................. 252
7.1.7 Program realization ....................... 256
7.1.8 Computational experiments .................. 260
7.2 Regularized difference schemes .................... 263
7.2.1 Regularization principle for difference schemes ........ 263
Contents xiii
7.2.2 Inverted-time problem ..................... 267
7.2.3 Generalized inverse method .................. 269
7.2.4 Regularized additive schemes ................. 277
7.2.5 Program ............................. 281
7.2.6 Computational experiments .................. 288
7.3 Iterative solution of retrospective problems .............. 291
7.3.1 Statement of the problem .................... 291
7.3.2 Difference problem ....................... 292
7.3.3 Iterative refinement of the initial condition ........... 292
7.3.4 Program ............................. 295
7.3.5 Computational experiments .................. 302
7.4 Second-order evolution equation .................... 305
7.4.1 Model problem ......................... 305
7.4.2 Equivalent first-order equation ................. 307
7.4.3 Perturbed initial conditions ................... 308
7.4.4 Perturbed equation ....................... 311
7.4.5 Regularized difference schemes ................ 314
7.4.6 Program ............................. 319
7.4.7 Computational experiments .................. 324
7.5 Continuation of non-stationary fields from point observation data . . . 326
7.5.1 Statement of the problem .................... 326
7.5.2 Variational problem ....................... 327
7.5.3 Difference problem ....................... 329
7.5.4 Numerical solution of the difference problem ......... 331
7.5.5 Program ............................. 333
7.5.6 Computational experiments .................. 340
7.6 Exercises ................................ 343
8 Other problems 345
8.1 Continuation over spatial variable in boundary value inverse problems 345
8.1.1 Statement of the problem .................... 346
8.1.2 Generalized inverse method .................. 347
8.1.3 Difference schemes for the generalized inverse method .... 350
8.1.4 Program ............................. 354
8.1.5 Examples ............................ 359
8.2 Non-local distribution of boundary conditions ............. 362
8.2.1 Model problem ......................... 362
8.2.2 Non-local boundary value problem .............. 362
8.2.3 Local regularization ...................... 363
8.2.4 Difference non-local problem ................. 365
8.2.5 Program ............................. 367
8.2.6 Computational experiments .................. 372
8.3 Identification of the boundary condition in two-dimensional problems 374
xiv Contents
8.3.1 Statement of the problem .................... 374
8.3.2 Iteration method ........................ 376
8.3.3 Difference problem ....................... 378
8.3.4 Iterative refinement of the boundary condition ......... 380
8.3.5 Program realization ....................... 383
8.3.6 Computational experiments .................. 390
8.4 Coefficient inverse problem for the nonlinear parabolic equation . . . 394
8.4.1 Statement of the problem .................... 395
8.4.2 Functional optimization .................... 396
8.4.3 Parametric optimization .................... 399
8.4.4 Difference problem ....................... 402
8.4.5 Program ............................. 405
8.4.6 Computational experiments .................. 411
8.5 Coefficient inverse problem for elliptic equation ............ 414
8.5.1 Statement of the problem .................... 414
8.5.2 Solution uniqueness for the inverse problem .......... 415
8.5.3 Difference inverse problem ................... 417
8.5.4 Iterative solution of the inverse problem ............ 419
8.5.5 Program ............................. 421
8.5.6 Computational experiments .................. 427
8.6 Exercises ................................ 430
Bibliography 435
Index 437
1 Inverse mathematical physics problems
We associate direct mathematical physics problems with classical boundary value
problems often encountered in mathematical physics. In a direct problem, it is required
to find a solution that satisfies some given partial differential equation and some initial
and boundary conditions. In inverse problems, the master equation and/or initial con-
ditions and/or boundary conditions are not fully specified but, instead, some additional
information is available. So separating out inverse mathematical physics problems,
we can speak of coefficient inverse problems (in which the equation is not specified
completely as some equation coefficients are unknown), boundary inverse problems
(in which boundary conditions are unknown), and evolutionary inverse problems (in
which initial conditions are unknown). Very often, inverse problems are problems
ill-posed in the classical sense. A typical feature here is the violated requirement of
solution continuity on input data. An ill-posed problem can be moved up into the class
of well-posed problems by narrowing the class of admissible solutions.
1.1 Boundary value problems
The core of applied mathematical models is made up by partial differential equations.
Here, the solution can be found from mathematical physics equations and some ad-
ditional relations. Such additional relations are, first of all, boundary and initial con-
ditions. In courses on mathematical physics equations, equations most important for
applications are second-order equations. Among such equations are elliptic, parabolic,
and hyperbolic equations.
1.1.1 Stationary mathematical physics problems
As an example, consider several two-dimensional boundary value problems. The so-
lution u(x), x = (x
1
, x
2
) is to be found in some bounded domain with a sufficiently
smooth boundary ∂. This solution is defined by the second-order elliptic equation
−
2
α=1
∂
∂x
α
k(x)
∂u
∂x
α
+ q(x)u = f (x), x ∈ . (1.1)
Normally, the constraints imposed on the equation coefficients look as
k(x) ≥ κ>0, q(x) ≥ 0, x ∈ .
A typical example of an elliptic equation (1.1) is given by the Poisson equation
−u ≡−
2
α=1
∂
2
u
∂x
2
α
= f (x), x ∈ , (1.2)
2 Chapter 1 Inverse mathematical physics problems
i. e., in (1.1) we have k(x) = 1 and q(x) = 0.
For equation (1.1), we consider first-kind boundary conditions
u(x) = μ(x), x ∈ ∂. (1.3)
On the boundary of or on a part of the boundary, second- or third-kind boundary
conditions can also be given. In the case of third-kind boundary conditions,wehave
k(x)
∂u
∂n
+ σ(x)u = μ(x), x ∈ ∂, (1.4)
where n is the external normal to .
Many key features of stationary mathematical physics problems for second-order
elliptic equations can be figured out considering simplest boundary value problems for
the second-order ordinary differential equation. A prototype of (1.1) is the equation
−
d
dx
k(x)
du
dx
+ q(x)u = f (x), 0 < x < l (1.5)
with variable coefficients
k(x) ≥ κ>0, q(x) ≥ 0.
For the unknown function u(x) to be uniquely determined, equation (1.5) must
be supplemented with two boundary conditions given at the end points of the seg-
ment [0, l]. Here, either the function u(x) (first-kind boundary condition), the flux
w(x) =−k(x)
du
dx
(x) (second-kind boundary condition), or a linear combination of
the above conditions (third-kind boundary condition) can be considered:
u(0) = μ
1
, u(l) = μ
2
, (1.6)
−k(0)
du
dx
(0) = μ
1
, k(l)
du
dx
(l) = μ
2
, (1.7)
−k(0)
du
dx
(0) + σ
1
u(0) = μ
1
, k(l)
du
dx
(l) + σ
2
u(l) = μ
2
. (1.8)
1.1.2 Nonstationary mathematical physics problems
A fundamental time-dependent equation in mathematical physics is the one-
dimensional second-order parabolic equation. In a rectangle
Q
T
= × [0, T ], ={x | 0 ≤ x ≤ l}, 0 ≤ t ≤ T
we consider the equation
∂u
∂t
=
∂
∂x
k(x)
∂u
∂x
+ f (x, t), 0 < x < l, 0 < t ≤ T. (1.9)
Section 1.1 Boundary value problems 3
This equation is supplemented (the first boundary value problem) with the boundary
conditions
u(0, t) = 0, u(l, t) = 0, 0 < t ≤ T (1.10)
and with the initial condition
u(x, 0) = u
0
(x), 0 ≤ x ≤ l. (1.11)
For simplicity, here we have restricted ourselves to homogeneous boundary conditions
and to the case in which the coefficient k depends on the spatial variable only and, in
addition, k(x) ≥ κ>0.
Instead of first-kind conditions (1.10), other boundary conditions can be considered.
For instance, in many applied problems one has to formulate third-type boundary con-
ditions:
−k(0)
du
dx
(0, t) + σ
1
(t)u(0, t) = μ
1
(t),
k(l)
du
dx
(l, t) + σ
2
(t)u(l, t) = μ
2
(t),
0 < t ≤ T. (1.12)
Among other nonstationary boundary value problems, the second-order hyperbolic
equation is to be considered. In the spatially one-dimensional case, we seek the solu-
tion of the following equation:
∂
2
u
∂t
2
=
∂
∂x
k(x)
∂u
∂x
+ f (x, t), 0 < x < l, 0 < t ≤ T. (1.13)
For the solution to be defined completely, in addition to (1.10) the following two
initial conditions are to be considered:
u(x, 0) = u
0
(x),
∂u
∂t
(0, t) = u
1
(x), 0 ≤ x ≤ l. (1.14)
Particular attention must be given to multi-dimensional nonstationary mathematical
physics problems. An example here is the two-dimensional parabolic equation. We
seek, in some domain , a function u(x, t) which satisfies the equation
∂u
∂t
=
2
α=1
∂
∂x
α
k(x)
∂u
∂x
α
− q(x, t)u + f (x, t),
x ∈ , 0 ≤ t ≤ T
(1.15)
and the conditions
u(x, t) = 0, x ∈ ∂, 0 < t ≤ T, (1.16)
u(x, 0) = u
0
(x), x ∈ . (1.17)
In a similar way, other nonstationary multidimensional boundary value problems for
partial differential equations can be formulated.
4 Chapter 1 Inverse mathematical physics problems
1.2 Well-posed problems for partial differential equations
Here, we introduce the notion of well-posed boundary value problem, related with
the existence of a unique solution that continuously depends on input data. Results
on stability of classical boundary value problems for partial differential equations are
presented.
1.2.1 The notion of well-posedness
Boundary and initial conditions are formulated to identify, among the whole set of
possible solutions of a partial differential equation, a desired solution. These additional
conditions must be not too numerous (solutions must exist), nor they must be few
in number (solutions must be not numerous). With this circumstance, the notion of
well-posed statement of a problem is related. Let us dwell first on the notion of well-
posedness of a problem according to J. Hadamard (well-posedness in the classical
sense).
A problem is well-posed if:
1) the solution of the problem does exist;
2) the solution is unique;
3) the solution continuously depends on input data.
It is the third condition for well-posedness that is of primary significance here. This
condition provides for smallness of the solution changes resulting from small input-
data variation. The input data are the equation coefficients, the right-hand side and
the boundary and initial conditions, taken from an experiment and always known to
some limited accuracy. In fact, solution stability with respect to small perturbations of
initial and boundary conditions, coefficients, and right-hand side justifies the problem
statement itself, as well as its cognitive essence, and makes the whole study valuable.
In consideration of boundary value problems for mathematical physics equations,
the existence, uniqueness and stability theorems taken as a whole provide a complete
study of well-posedness of a posed problem. Of course, conditions for well-posedness
must be rendered concrete considering each particular problem. The latter is related
with the fact that the solution of the problem and the input data are considered as
elements in a certain fully defined functional space. That is why a given problem can
be ill-posed with one choice of spaces and well-posed with another choice of spaces.
Hence, a statement that this or that problem is a well- (ill-)posed one is never global:
such statements must be supplemented with necessary amendments.
1.2.2 Boundary value problem for the parabolic equation
Some fundamental points in a consideration of well- or ill-posedness of a boundary
value mathematical physics problem can be illustrated with the example of a sim-
plest boundary value problem for the one-dimensional parabolic equation (1.9)–(1.11).
Section 1.2 Well-posed problems for partial differential equations 5
Here, we do not touch on points concerning the solution existence; instead, we restrict
ourselves to the uniqueness problem and to the property of continuous dependence of
solution on input data. We assume that problem (1.9)–(1.11) indeed has a classical
solution u(x, t) (for instance, this solution is doubly differentiable with respect to x
and continuously differentiable with respect to t).
We write problem (1.9)–(1.11) as a Cauchy problem for a first-order operator dif-
ferential equation. For functions given in the domain = (0, 1) and vanishing at
the boundary points of this domain (at the boundary ∂), we define a Hilbert space
H = L
2
() in which the scalar product is defined as
(v, w) =
v(x)w(x) dx.
For the norm in H we use
v=(v, v)
1/2
=
v
2
(x) dx
1/2
.
For functions satisfying the boundary conditions (1.10), we define the operator
Au =−
∂
∂x
k(x)
∂u
∂x
, 0 < x < l. (1.18)
With the notation introduced, equation (1.9) supplemented with conditions (1.10) at
the boundary ∂ can be written as an operator differential equation for the function
u(t) ∈ H:
du
dt
+ Au = f (t), 0 < t ≤ T. (1.19)
The initial condition (1.11) can be written as
u(0) = u
0
. (1.20)
The following main properties of the operator A defined by (1.18) are worthy of
note. The operator A is a self-adjoint operator non-negative in H :
A
∗
= A ≥ 0. (1.21)
The property of self-adjointness stems from the expression
(Av, w) =
l
0
Av(x)w(x) dx =
l
0
k(x)
∂v
∂x
∂w
∂x
dx = (v, A
∗
w),
derived with regard to the fact that the functions v(x) and w(x) vanish at the points
x ∈ ∂. For the functions u(x) = 0, x ∈ ∂,wehave
(Av, v) =
l
0
k(x)
∂v
∂x
2
dx ≥ 0