Author	Morozov, V. A. author
Title	Methods for Solving Incorrectly Posed Problems [electronic resource] / by V. A. Morozov
Imprint	New York, NY : Springer New York, 1984
Connect to	http://dx.doi.org/10.1007/978-1-4612-5280-1
Descript	257p. online resource

Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f โ{130}ฌ 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 definiยญ 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 โ{130}ฌ 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

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