Some mathematical considerations in dealing with the inverse problem

Abstract

Many problems of mathematical physics can be formulated in terms of the operator equationAx = y, whereAis an integro-differential operator. GivenAandx, the solution foryis usually straightforward. However, the inverse problem which consists of the solution forxwhen givenAandyis much more difficult. The following questions relative to the inverse problem are explored. 1) Does specification of the operatorAdetermine the set\{y\}for which a solutionxis possible? 2) Does the inverse problem always have a unique solution? 3) Do small perturbations of the forcing functionyalways 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.