Hohage, Thorsten

[["dc.bibliographiccitation.firstpage","2610"],["dc.bibliographiccitation.issue","6"],["dc.bibliographiccitation.journal","SIAM Journal on Numerical Analysis"],["dc.bibliographiccitation.lastpage","2636"],["dc.bibliographiccitation.volume","45"],["dc.contributor.author","Bissantz, N."],["dc.contributor.author","Hohage, T."],["dc.contributor.author","Munk, A."],["dc.contributor.author","Ruymgaart, F."],["dc.date.accessioned","2017-09-07T11:49:53Z"],["dc.date.available","2017-09-07T11:49:53Z"],["dc.date.issued","2007"],["dc.description.abstract","Previously, the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov-type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as v-methods and the Land-weber iteration. The latter estimators converge at the same rate as spectral cut-off but require only matrix-vector products. Our results are applied to various problems; in particular we obtain precise convergence rates for satellite gradiometry, L-2-boosting, and errors in variable problems."],["dc.identifier.doi","10.1137/060651884"],["dc.identifier.gro","3143567"],["dc.identifier.isi","000253017000015"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/1095"],["dc.language.iso","en"],["dc.notes.intern","WoS Import 2017-03-10"],["dc.notes.status","final"],["dc.notes.submitter","PUB_WoS_Import"],["dc.relation.eissn","1095-7170"],["dc.relation.issn","0036-1429"],["dc.title","Convergence rates of general regularization methods for statistical inverse problems and applications"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.subtype","original"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.artnumber","239025"],["dc.bibliographiccitation.firstpage","17"],["dc.bibliographiccitation.journal","International Journal of Mathematics and Mathematical Sciences"],["dc.bibliographiccitation.volume","2009"],["dc.contributor.author","Gilliam, D. S."],["dc.contributor.author","Hohage, Thorsten"],["dc.contributor.author","Ji, X."],["dc.contributor.author","Ruymgaart, F."],["dc.date.accessioned","2019-07-09T11:52:46Z"],["dc.date.available","2019-07-09T11:52:46Z"],["dc.date.issued","2009"],["dc.description.abstract","The main result in this paper is the determination of the Fréchet derivative of an analytic function of a bounded operator, tangentially to the space of all bounded operators. Some applied problems from statistics and numerical analysis are included as a motivation for this study. The perturbation operator (increment) is not of any special form and is not supposed to commute with the operator at which the derivative is evaluated. This generality is important for the applications. In the Hermitian case, moreover, some results on perturbation of an isolated eigenvalue, its eigenprojection, and its eigenvector if the eigenvalue is simple, are also included. Although these results are known in principle, they are not in general formulated in terms of arbitrary perturbations as required for the applications. Moreover, these results are presented as corollaries to the main theorem, so that this paper also provides a short, essentially self-contained review of these aspects of perturbation theory."],["dc.identifier.doi","10.1155/2009/239025"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/5909"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/60271"],["dc.language.iso","en"],["dc.notes.intern","Merged from goescholar"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.subject.ddc","510"],["dc.title","The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]