Analytical and numerical studies on the influence of multiplication operators for the ill-posedness of inverse problems

Abstract

In this paper we deal with the degree of ill-posedness of linear operator equations Ax = y , x ∈ X , y ∈ Y , in the Hilbert space X = Y = L 2 (0, 1), where A = M о J is a compact operator that may be decomposed into the simple integration operator J with a well-known decay rate of singular values and a multiplication operator M determined by the multiplier function m . This case occurs for example for nonlinear operator equations F ( x ) = y with a forward operator F = N о J where N is a Nemytskii operator. Then the local degree of ill-posedness of the nonlinear equation at a point x 0 of the domain of F is investigated via the Fréchet derivative of F which has the form F ' ( x 0 ) = M о J . We show the restricted influence of such multiplication operators M mapping in L 2 (0,1). If the multiplier function m has got zeros, the determination of the degree of ill-posedness is not trivial. We are going to investigate this situation and provide analytical tools as well as their limitations. For power and exponential type multiplier functions with essential zeros we will show by using several numerical approaches that the unbounded inverse of the injective multiplication operator does not influence the local degree of ill-posedness. We provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of A = M о J . Finally we analyze the influence of those multiplication operators M on the possibilities of Tikhonov regularization and corresponding convergence rates. We investigate the role of approximate source conditions in the method of Tikhonov regularization for linear and nonlinear ill-posed operator equations. Based on the studies on approximate source conditions we indicate that only integrals of m and not the decay of multiplier functions near zero determines the convergence behavior of the regularized solution.