Numerical Methods for Solving Inverse Problems of Mathematical Physics

by Samarskii, Alexander A.

• ISBN13: 9783110196665
• ISBN10: 3110196662
• Format: Hardcover
• Copyright: 2007-12-14
• Publisher: De Gruyter

Summary

The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in applied mathematics, computational mathematics, and mathematical modelling.

Table of Contents

Preface	p. v
Main definitions and notations	p. vii
Inverse mathematical physics problems	p. 1
Boundary value problems	p. 1
Stationary mathematical physics problems	p. 1
Nonstationary mathematical physics problems	p. 2
Well-posed problems for partial differential equations	p. 4
The notion of well-posedness	p. 4
Boundary value problem for the parabolic equation	p. 4
Boundary value problem for the elliptic equation	p. 8
Ill-posed problems	p. 9
Example of an ill-posed problem	p. 10
The notion of conditionally well-posed problems	p. 11
Condition for well-posedness of the inverted-time problem	p. 11
Classification of inverse mathematical physics problems	p. 13
Direct and inverse problems	p. 13
Coefficient inverse problems	p. 14
Boundary value inverse problems	p. 15
Evolutionary inverse problems	p. 16
Exercises	p. 16
Boundary value problems for ordinary differential equations	p. 19
Finite-difference problem	p. 19
Model differential problem	p. 19
Difference scheme	p. 20
Finite element method schemes	p. 23
Balance method	p. 25
Convergence of difference schemes	p. 26
Difference identities	p. 27
Properties of the operator A	p. 28
Accuracy of difference schemes	p. 30
Solution of the difference problem	p. 31
The sweep method	p. 32
Correctness of the sweep algorithm	p. 33
The Gauss method	p. 34
Program realization and computational examples	p. 35
Problem statement	p. 35
Difference schemes	p. 37
Program	p. 39
Computational experiments	p. 43
Exercises	p. 45
Boundary value problems for elliptic equations	p. 49
The difference elliptic problem	p. 49
Boundary value problems	p. 49
Difference problem	p. 50
Problems in irregular domains	p. 52
Approximate-solution inaccuracy	p. 54
Elliptic difference operators	p. 54
Convergence of difference solution	p. 56
Maximum principle	p. 57
Iteration solution methods for difference problems	p. 59
Direct solution methods for difference problems	p. 59
Iteration methods	p. 60
Examples of simplest iteration methods	p. 62
Variation-type iteration methods	p. 64
Iteration methods with diagonal reconditioner	p. 66
Alternate-triangular iteration methods	p. 67
Program realization and numerical examples	p. 70
Statement of the problem and the difference scheme	p. 70
A subroutine for solving difference equations	p. 71
Program	p. 79
Computational experiments	p. 83
Exercises	p. 85
Boundary value problems for parabolic equations	p. 90
Difference schemes	p. 90
Boundary value problems	p. 90
Approximation over space	p. 92
Approximation over time	p. 93
Stability of two-layer difference schemes	p. 95
Basic notions	p. 95
Stability with respect to initial data	p. 97
Stability with respect to right-hand side	p. 100
Three-layer operator-difference schemes	p. 102
Stability with respect to initial data	p. 102
Passage to an equivalent two-layer scheme	p. 104
[phi]-stability of three-layer schemes	p. 106
Estimates in simpler norms	p. 108
Stability with respect to right-hand side	p. 110
Consideration of difference schemes for a model problem	p. 110
Stability condition for a two-layer scheme	p. 111
Convergence of difference schemes	p. 112
Stability of weighted three-layer schemes	p. 113
Program realization and computation examples	p. 114
Problem statement	p. 114
Linearized difference schemes	p. 115
Program	p. 118
Computational experiments	p. 121
Exercises	p. 124
Solution methods for ill-posed problems	p. 127
Tikhonov regularization method	p. 127
Problem statement	p. 127
Variational method	p. 128
Convergence of the regularization method	p. 129
The rate of convergence in the regularization method	p. 131
Euler equation for the smoothing functional	p. 131
Classes of a priori constraints imposed on the solution	p. 132
Estimates of the rate of convergence	p. 133
Choice of regularization parameter	p. 134
The choice in the class of a priori constraints on the solution	p. 135
Discrepancy method	p. 136
Other methods for choosing the regularization parameter	p. 137
Iterative solution methods for ill-posed problems	p. 138
Specific features in the application of iteration methods	p. 138
Iterative solution of ill-posed problems	p. 139
Estimate of the convergence rate	p. 141
Generalizations	p. 143
Program implementation and computational experiments	p. 144
Continuation of a potential	p. 144
Integral equation	p. 146
Computational realization	p. 147
Program	p. 148
Computational experiments	p. 152
Exercises	p. 154
Right-hand side identification	p. 157
Right-hand side reconstruction from known solution: stationary problems	p. 157
Problem statement	p. 157
Difference algorithms	p. 158
Tikhonov regularization	p. 161
Other algorithms	p. 163
Computational and program realization	p. 164
Examples	p. 172
Right-hand side identification in the case of parabolic equation	p. 175
Model problem	p. 175
Global regularization	p. 176
Local regularization	p. 178
Iterative solution of the identification problem	p. 180
Computational experiments	p. 189
Reconstruction of the time-dependent right-hand side	p. 191
Inverse problem	p. 192
Boundary value problem for the loaded equation	p. 192
Difference scheme	p. 194
Non-local difference problem and program realization	p. 194
Computational experiments	p. 199
Identification of time-independent right-hand side: parabolic equations	p. 201
Statement of the problem	p. 201
Estimate of stability	p. 202
Difference problem	p. 204
Solution of the difference problem	p. 207
Computational experiments	p. 215
Right-hand side reconstruction from boundary data: elliptic equations	p. 218
Statement of the inverse problem	p. 218
Uniqueness of the inverse-problem solution	p. 219
Difference problem	p. 220
Solution of the difference problem	p. 224
Program	p. 226
Computational experiments	p. 234
Exercises	p. 237
Evolutionary inverse problems	p. 240
Non-local perturbation of initial conditions	p. 240
Problem statement	p. 240
General methods for solving ill-posed evolutionary problems	p. 241
Perturbed initial conditions	p. 243
Convergence of approximate solution to the exact solution	p. 246
Equivalence between the non-local problem and the optimal control problem	p. 250
Non-local difference problems	p. 252
Program realization	p. 256
Computational experiments	p. 260
Regularized difference schemes	p. 263
Regularization principle for difference schemes	p. 263
Inverted-time problem	p. 267
Generalized inverse method	p. 269
Regularized additive schemes	p. 277
Program	p. 281
Computational experiments	p. 288
Iterative solution of retrospective problems	p. 291
Statement of the problem	p. 291
Difference problem	p. 292
Iterative refinement of the initial condition	p. 292
Program	p. 295
Computational experiments	p. 302
Second-order evolution equation	p. 305
Model problem	p. 305
Equivalent first-order equation	p. 307
Perturbed initial conditions	p. 308
Perturbed equation	p. 311
Regularized difference schemes	p. 314
Program	p. 319
Computational experiments	p. 324
Continuation of non-stationary fields from point observation data	p. 326
Statement of the problem	p. 326
Variational problem	p. 327
Difference problem	p. 329
Numerical solution of the difference problem	p. 331
Program	p. 333
Computational experiments	p. 340
Exercises	p. 343
Other problems	p. 345
Continuation over spatial variable in boundary value inverse problems	p. 345
Statement of the problem	p. 346
Generalized inverse method	p. 347
Difference schemes for the generalized inverse method	p. 350
Program	p. 354
Examples	p. 359
Non-local distribution of boundary conditions	p. 362
Model problem	p. 362
Non-local boundary value problem	p. 362
Local regularization	p. 363
Difference non-local problem	p. 365
Program	p. 367
Computational experiments	p. 372
Identification of the boundary condition in two-dimensional problems	p. 374
Statement of the problem	p. 374
Iteration method	p. 376
Difference problem	p. 378
Iterative refinement of the boundary condition	p. 380
Program realization	p. 383
Computational experiments	p. 390
Coefficient inverse problem for the nonlinear parabolic equation	p. 394
Statement of the problem	p. 395
Functional optimization	p. 396
Parametric optimization	p. 399
Difference problem	p. 402
Program	p. 405
Computational experiments	p. 411
Coefficient inverse problem for elliptic equation	p. 414
Statement of the problem	p. 414
Solution uniqueness for the inverse problem	p. 415
Difference inverse problem	p. 417
Iterative solution of the inverse problem	p. 419
Program	p. 421
Computational experiments	p. 427
Exercises	p. 430
Bibliography	p. 435
Index	p. 437
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