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

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.

Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problems

We obtain rate of convergence estimates for approximations delivered by Tikhonov’s scheme of solving ill-posed nonconvex optimization problems in a Hilbert space. The problems under investigation involve minimization of twice Frechet differentiable functionals given with errors on a closed convex set having a nonempty interior and smooth boundary. Assuming that the desired solution satisfies an appropriate source condition which includes the second derivative of the cost functional and depends on the geometry of constraints near the solution, we establish accuracy estimates in terms of the error level. Both the a priori and a posteriori parameter choice rules are analyzed.