In this work we deal with the degree of ill-posedness of linear operators in Hilbert spaces, where the operator may be decomposed into a compact linear integral operator with a well-known decay rate of singular values and a multiplication operator. This case occurs, for example, for nonlinear operator equations. Then the local degree of ill-posedness is investigated via the Frechet derivative, providing the situation described above. If the multiplier function has got zeroes, the determination of the local degree of ill-posedness is not trivial. We are going to investigate this situation, provide analytical tools as well as their limitations. By using several numerical approaches for computing the singular values we find that the degree of ill-posedness does not change through those multiplication operators. We provide a conjecture, verified by several numerical studies, how those operators influence the singular values. Finally, we analyze the influence of these multiplication operators on Tikhonov regularization and corresponding convergence rates. In this context we also provide a short summary on the relationship between nonlinear problems and their linearizations.
|Place of Publication||Germany|
|Number of pages||168|
|Publication status||Published - Aug 2008|