Ill-Posed Problems and Methods for Their Numerical Solution

Abstract

In the present chapter, the basic conceptions of the theory of ill-posed problems and numerical methods for their solving under different a priori information are described. Hadamard’s definition of well-posedness and examples of ill-posed problems are given. Tikhonov’s definition of a regularizing algorithm and classification of mathematical problems are described. The main properties of ill-posed problems are discussed. As an example of a priori information application for constructing regularizing algorithms an operator equation in Hilbert spaces is considered. If it is known that the exact solution belongs to a compact set then the quasisolution method can be used. An error of an approximate solution can be calculated also. If it is known that there is an a priori information concerning sourcewise representability of an exact solution with a completely continuous operator then the method of extending compacts can be applied. There exists a possibility to calculate an a posteriori error of an approximate solution. If strong a priori constraints are not available then the variational approach based on minimization of the Tikhonov functional with a choice of a regularization parameter, e.g., according to the generalized discrepancy principle is recommended. It is formulated by an equivalence of the generalized discrepancy principle and the generalized discrepancy method resulting in a possibility of the generalized discrepancy principle modification for solving incompatible ill-posed problems. Possible approaches for solving nonlinear ill-posed problems and iterative methods are described briefly.