Some Mathematical Considerations in Dealing with the Inverse Problem

Many problems of mathematical physics can be formulated in terms of the operator equation Ax = y, where A is an integro-differential operator. Given A and x, the solution for y is usually straightforward. However, the inverse problem which consists of the solution for x when given A and y is much more difficult. The following questions relative to the inverse problem are explored. 1) Does specification of the operator A determine the set {y} for which a solution x is possible? 2) Does the inverse problem always have a unique solution? 3) Do small perturbations of the forcing function y always result in small perturbations of the solution? 4) What are some of the considerations that enter into the choice of a solution technique for a specific problem? The concept of an ill-posed problem versus that of a well-posed problem is discussed. Specifically, the manner by which an ill-posed problem may be regularized to a well-posed problem is presented. The concepts are illustrated by several examples.