The Impact of Multiplication Operators on the Ill-Posedness of Inverse Problems

In this work we deal with the degree of ill-posednessof linear operators in Hilbert spaces, where theoperator may be decomposed into a compact linearintegral operator with a well-known decay rate ofsingular values and a multiplication operator. This case occurs, for example, for nonlinear operatorequations. Then the local degree of ill-posedness isinvestigated via the Fr¿¿chet derivative, providingthe situation described above.If the multiplier function has got zeroes, thedetermination of the local degree of ill-posedness isnot trivial. We are going to investigate thissituation, provide analytical tools as well as theirlimitations. By using several numerical approachesfor computing the singular values we find that thedegree of ill-posedness does not change through thosemultiplication operators. We provide a conjecture,verified by several numerical studies, how thoseoperators influence the singular values. Finally, we analyze the influence of thesemultiplication operators on Tikhonov regularizationand corresponding convergence rates. In this contextwe also provide a short summary on the relationshipbetween nonlinear problems and their linearizations.

Dr. Melina Freitag (MSc, Dipl.-Math, PhD), Studium der Mathematikan der TU Chemnitz mit Schwerpunkt Inverse Probleme undNumerische Mathematik, Master- und PhD Studium an der Universit¿¿tBath (England) mit Spezialisierung in Angewandter Mathematik undNumerischer Linearer Algebra.