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Numerical methods for solving inverse problems of mathematical physics. (English) Zbl 1136.65105

Inverse and Ill-Posed Problems Series. Berlin: de Gruyter (ISBN 978-3-11-019666-5/hbk). xiv, 438 p. (2007).
Applied problems often require solving boundary value problems for partial differential equations. Elaboration of approximate solution methods for such problems rests on the development and examination of numerical methods for boundary value problems formulated for basic (fundamental, model) mathematical physics equations. If one considers second-order equation, then such equations are elliptic, parabolic and hyperbolic equations.
The solution of a boundary value problem is to be found from the equation and from some additional conditions. For time-independent equations have to be specified boundary conditions, and for time-dependent equation, in addition, initial conditions. Such classical problems are treated in all tutorials on mathematical physics equations and partial differential equations.
The above boundary value problems belong to the class of direct mathematical physics problems. A typical inverse problem is the problem in which it is required to find equation coefficients from some additional information about the solution; in the latter case the problem is called a coefficient inverse problem. In boundary inverse problems, to be reconstructed are unknown boundary conditions, and so on.
Inverse mathematical physics problems often belong to the class of classically ill-posed problems. First of all, ill-posed problems here are a consequence of lacking continuous dependence of the solution on input data. In this case, one has to narrow the class of admissible solutions and, to determine a stable solution, apply some special regularizing procedures.
Numerical solutions of direct mathematical physics problems are presently a well-studied matter. In solving multi-dimensional boundary value problems, difference methods and finite element methods are widely used. At present, tutorials and monographs on numerical solution methods for inverse problems are few in number. The latter was the primary motivation behind writing the present book.
By no means being a comprehensive guide, this book treats some particular inverse problems for time-dependent and time-independent equations often encountered in mathematical physics. Rather a complete and closed consideration of basic difficulties in approximate solution of inverse problems is given. A minimum mathematical apparatus is used, related with some basic properties of operators in finite-dimensional spaces.
A predominant contribution to the scope of problems dealt with in the theory and solution practice of inverse mathematical physics problems was made by Russian mathematicians.
I think it is useful to familiarize potential readers with the brief contents of this book:
1. Inverse mathematical physics problems.
a) Boundary value problems. b) Well-posed problems for partial differential equations. c) Ill-posed problems. d) Classification of inverse mathematical physics problems. e) Exercises.
2. Boundary value problems for ordinary differential equations.
a) Finite-difference problems. b) Convergence of difference schemes. c) Solution of the difference problems. d) Program realization and computational examples. e) Exercises.
3. Boundary value problems for elliptic equations.
a) The difference elliptic problem. b) Approximate-solution inaccuracy. c) Iteration solution methods for difference problems. d) Program realization and computational examples. e) Exercises.
4. Boundary value problems for parabolic equations.
a) Difference schemes. b) Stability of two-layer difference schemes. c) Three-layer operation-difference schemes. d) Consideration of difference schemes for a model problem. e) Program realization and computational examples. f) Exercises.
5. Ill-posed problems.
a) Solution methods for ill-posed problems. b) Tikhonov regularization method. c) The rate of convergence in the regularization method. d) Choice of regularization parameter. e) Iterative solution methods for ill-posed problems. f) Program realization and computational examples. g) Exercises
6. Right-hand side identification.
a) Right-hand side reconstruction from known solution: stationary problems. b) Right-hand side identification in the case of parabolic equation. c) Reconstruction of time-dependent right-hand side. d) Identification of time-independent right-hand side: parabolic equations. e) Right-hand side reconstruction from boundary data: elliptic equation. f) Exercises.
7. Evolutionary inverse problems.
a) Non-local perturbation of initial conditions. b) Regularized difference schemes. c) Iterative solution of retrospective problems. d) Second-order evolution equation. e) Continuation of non-stationary fields from point observation data. f) Exercises.
8. Other problems.
a) Continuation over spatial variable in boundary value inverse problems. b) Non-local distribution of boundary conditions. c) Identification of the boundary condition in two-dimensional problems. d) Coefficient inverse problem for the nonlinear parabolic equation. e) Coefficient inverse problem for elliptic equation. f) Exercises.

MSC:

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
34A55 Inverse problems involving ordinary differential equations
35R30 Inverse problems for PDEs
65L09 Numerical solution of inverse problems involving ordinary differential equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65L12 Finite difference and finite volume methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
35Qxx Partial differential equations of mathematical physics and other areas of application
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