Weidling, Frederic

[["dc.bibliographiccitation.firstpage","598"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","SIAM Journal on Numerical Analysis"],["dc.bibliographiccitation.lastpage","620"],["dc.bibliographiccitation.volume","55"],["dc.contributor.author","Hohage, Thorsten"],["dc.contributor.author","Weidling, Frederic"],["dc.date.accessioned","2020-12-10T18:37:19Z"],["dc.date.available","2020-12-10T18:37:19Z"],["dc.date.issued","2017"],["dc.description.abstract","We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of illposedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates of spectral regularization methods, which are slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. In many cases variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems."],["dc.identifier.doi","10.1137/16M1067445"],["dc.identifier.eissn","1095-7170"],["dc.identifier.gro","3146380"],["dc.identifier.isi","000401780500007"],["dc.identifier.issn","0036-1429"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/76914"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-354"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","PUB_WoS_Import"],["dc.publisher","Siam Publications"],["dc.relation.issn","1095-7170"],["dc.relation.issn","0036-1429"],["dc.title","Characterizations of Variational Source Conditions, Converse Results, and Maxisets of Spectral Regularization Methods"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.artnumber","075006"],["dc.bibliographiccitation.firstpage","1"],["dc.bibliographiccitation.issue","7"],["dc.bibliographiccitation.journal","Inverse Problems"],["dc.bibliographiccitation.lastpage","14"],["dc.bibliographiccitation.volume","31"],["dc.contributor.author","Hohage, Thorsten"],["dc.contributor.author","Weidling, Frederic"],["dc.date.accessioned","2017-09-07T11:52:57Z"],["dc.date.available","2017-09-07T11:52:57Z"],["dc.date.issued","2015"],["dc.description.abstract","This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory."],["dc.identifier.doi","10.1088/0266-5611/31/7/075006"],["dc.identifier.gro","3146381"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4151"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.title","Verification of a variational source condition for acoustic inverse medium scattering problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.contributor.author","Weidling, Frederic"],["dc.date.accessioned","2022-04-05T08:59:19Z"],["dc.date.available","2022-04-05T08:59:19Z"],["dc.date.issued","2019"],["dc.description.abstract","Inverse Probleme befassen sich mit der Schätzung eines Parameters aus indirekten Messungen. Die Messung ist üblicherweise fehlerbehaftet während die Identifizierung des Parameters aus der Messung schlecht gestellt (d.h. unstetig) ist. Daher kommen zur Schätzung des Parameters Regularisierungsverfahren zum Einsatz, die diese Faktoren einbeziehen, um eine stabile Rekonstruktion zu erhalten. Um jedoch eine Abschätzung des Rekonstruktionsfehlers zu erhalten werden Zusatzannahmen an den zu Grunde liegenden Parameter benötigt. Diese werden üblicherweise in Form von sogenannten Variationellen Quellbedingungen formuliert. In dieser Doktorarbeit wird eine Strategie zur Verifizierung dieser Bedingungen entwickelt, die auf der Glattheit des Parameters und der Schlechtgestelltheit des Problems beruht; dabei wird ausgenutzt, dass letztere Ähnlichkeiten zu Stabilitätsabschätzungen aufweist. Anschließend wird diese Strategie eingesetzt um Variationelle Quellbedingungen für Probleme in der Parameteridentifikation, inversen Streutheorie sowie elektrischen Impedanztomographie nachzuweisen."],["dc.description.abstract","In inverse problems one wants to find some parameter of interest which is not directly observable by indirect measurement. These measurements are usually noisy while the mapping of measurement to parameter is typically illposed (that is unstable). Therefore one applies regularization techniques that balance these two factors to find a stable approximation of the sought for parameter. However, in order to bound the reconstruction error, one needs additional information on the true parameter, which is nowadays typically formulated in terms of variational source conditions. In this thesis, we develop a general strategy to verify these conditions based on smoothness of the true parameter and the illposedness of the problem; the latter will be characterized by exploiting structural similarities to stability estimates. Following this, we apply our strategy to verify variational source conditions for parameter identification problems, inverse scattering and electrical impedance tomography."],["dc.format.extent","222"],["dc.identifier.doi","10.17875/gup2019-1165"],["dc.identifier.isbn","978-3-86395-411-6"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?univerlag-isbn-978-3-86395-411-6"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/106352"],["dc.identifier.urn","urn:nbn:de:gbv:7-isbn-978-3-86395-411-6-0"],["dc.language.iso","en"],["dc.notes.intern","Import GROB-550"],["dc.publisher","Universitätsverlag Göttingen"],["dc.publisher.place","Göttingen"],["dc.rights.uri","http://creativecommons.org/licenses/by-sa/4.0/deed.de"],["dc.title","Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs"],["dc.type","thesis"],["dc.type.subtype","dissertation"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","203"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Inverse Problems and Imaging"],["dc.bibliographiccitation.lastpage","220"],["dc.bibliographiccitation.volume","11"],["dc.contributor.author","Weidling, Frederic"],["dc.contributor.author","Hohage, Thorsten"],["dc.date.accessioned","2017-09-07T11:52:58Z"],["dc.date.available","2017-09-07T11:52:58Z"],["dc.date.issued","2017"],["dc.description.abstract","This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature."],["dc.identifier.doi","10.3934/ipi.2017010"],["dc.identifier.gro","3146398"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4168"],["dc.language.iso","en"],["dc.notes.intern","Not valid abstract: This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a \\\\.fxed frequency. The main results are two variational source conditions for near and far \\\\.feld data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature."],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.title","Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

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