Quasi-Solution Method and Global Minimization of the Residual Functional in Conditionally Well-Posed Inverse Problems

Abstract

A class of conditionally well-posed problems characterized by a Hölder conditional stability estimate on a convex compact set in a Hilbert space is considered. The operator of the direct problem and the right-hand side of the equation are given with errors, and the derivatives of the exact and perturbed operators are not assumed to be close to each other. The convexity and single-extremality of the residual functional of the quasi-solution method are examined. For this functional, each of its stationary points on the set of conditional well-posedness that lies not too far from the sought solution of the original inverse problem is shown to belong to a small neighborhood of the solution. The diameter of this neighborhood is estimated in terms of the errors in the input data. It is shown that this neighborhood is an attractor of the iterations of the gradient projection method, and the convergence rate of the iterations to the attractor is estimated. The necessity of the used conditional stability estimate for the existence of iterative processes with the indicated properties is established.