Title: The Impact of Multiplication Operators on the Ill-Posedness of Inverse Problems
Publisher: VDM Verlag Dr. Mueller Aktiengesellschaft & Co. KG
Publication Date: 2008
Language: English
ISBN 10: 3639069137
ISBN 13: 9783639069136
Binding: Soft cover
Condition: New

About this title

Synopsis

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.

About the Author

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.

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