Numerical Methods for Solving Inverse Problems of Mathematical Physics / Edition 1

by A. A. Samarskii, Petr N. Vabishchevich

Overview

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.

Product Details

ISBN-13:	9783110196665
Publisher:	De Gruyter
Publication date:	12/14/2007
Series:	Inverse and Ill-Posed Problems Series , #52
Pages:	452
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

About the Author

Alexander A. Samarskii and Peter N. Vabishchevich, Russian Academy of Sciences, Moscow, Russia.

Table of Contents

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