On the clustering of stationary points of Tikhonov’s functional for conditionally well-posed inverse problems

Abstract

Abstract In a Hilbert space, we consider a class of conditionally well-posed inverse problems for which the Hölder type estimate of conditional stability on a bounded closed and convex subset holds. We investigate a finite-dimensional version of Tikhonov’s scheme in which the discretized Tikhonov’s functional is minimized over the finite-dimensional section of the set of conditional stability. For this optimization problem, we prove that each its stationary point that is located not too far from the desired solution of the original inverse problem in reality belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of discretization errors and error level in input data are also given.