Methods for Solving Incorrectly Posed Problems - V. A. Morozov

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Résumé :
Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f ? F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following defini? tion of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F (sol vabi li ty condition); (2) The equality AU = AU for any u ,u ? DA implies the I 2 l 2 equality u = u (uniqueness condition); l 2 (3) The inverse operator A-I is continuous on F (stability condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any ill-posed (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation.

Sommaire:
1. The Regularization Method.- Section 1. The Basic Problem for Linear Operators.- Section 2. The Approximation of the Solution of the Basic Problem.- Section 3. The Euler Variation Inequality. Estimation of Accuracy.- Section 4. Stability of Regularized Solutions.- Section 5. Approximation of the Admissible Set. Choice of the Basis.- 2. Criteria for Selection of Regularization Parameter.- Section 6. Some Properties of Regularized Solutions.- Section 7. Methods for Choosing the Parameter: Case of Exact Information.- Section 8. The Residual Method and the Method of Quasi-solutions: Case of Exact Information.- Section 9. Properties of the Auxiliary Functions.- Section 10. Criteria for the Choice of a Parameter: Case of Inexact Data.- 3. Regular Methods for Solving Linear and Nonlinear Ill-Posed Problems.- Section 11. Regularity of Approximation Methods.- Section 12. The Theory of Accuracy of Regular Methods.- Section 13. The Computation of the Estimation Function.- Section 14. Examples of Regular Methods.- Section 15. The Principle of Residual Optimality for Approximate Solutions of Equations with Nonlinear Operators.- Section 16. The Regularization Method for Nonlinear Equations.- 4. The Problem of Computation and the General Theory of Splines.- Section 17. The Problem of Computation and the Parameter Identification Problem.- Section 18. Properties of Smoothing Families of Operators.- Section 19. The Optimality of Smoothing Algorithms.- Section 20. The Differentiation Problem and Algorithms of Approximation of the Experimental Data.- Section 21.The Theory of Splines and the Problem of Stable Computation of Values of an Unbounded Operator.- Section 22. Approximate Solution of Operator Equations Using Splines.- Section 23. Recovering the Solution of the Basic Problem From Approximate Values of the Functiona1s.- 5. Regular Methods for Special Cases of the Basic Problem. Algorithms for Choosing the Regularization Parameter.- Section 24. Pseudosolutions.- Section 25. Optimal Regularization.- Section 26. Numerical Algorithms for Regularization Parameters.- Section 27. Heuristic Methods for Choosing a Parameter.- Section 28. The Investigation of Adequacy of Mathematical Models.