On stable finite dimensional approximation of conditionally well-posed inverse problems

Abstract

We consider nonlinear conditionally well-posed inverse problems with Hölder-type stability estimates on closed, convex, and bounded subsets of a Hilbert space. A finite dimensional version of Ivanov's quasisolution method is investigated. The method involves minimization of the discrepancy functional over the section of the set of conditional well-posedness by a finite dimensional subspace. For this multiextremal minimization problem, we prove that if its stationary point is located not too far from the desired solution of the original inverse problem, then the mentioned point necessarily belongs to a small neighborhood of the solution. The diameter of the neighborhood is estimated in terms of an error level in input data and properties of approximating finite dimensional subspaces. The results are used in a convergence analysis of the gradient projection method, as applied to the finite dimensional subproblem of Ivanov's method.