A Posteriori Analysis of Inverse Problems for Elliptic Differential Equations

Abstract This article is concerned with a posteriori error estimation methods in the context of inverse problems, which are typically incorrect in Hadamard’s sense. The main question discussed is what quantitative information can be reliably obtained based on the results of computations. A posteriori estimates of the functional type derived for the corresponding class of direct (well-posed) boundary value problems provide the basis for addressing this issue. They generate computable quantities that estimate the distance between a function (approximate solution to the inverse problem) and the exact solution of a “close” well-posed initial boundary value problem associated with the same differential equation. The solution of this problem exactly satisfies one part of the imposed conditions and the others are satisfied approximately with the accuracy ϵ, which is fully determined. The estimates are not connected with a particular numerical method and can be used for approximations generated by known methods based on regularization or iteration procedures. Also, they suggest natural cost functionals that can be used in direct minimization procedures aimed at finding a solution of an inverse problem.