Bauer, Frank

[["dc.bibliographiccitation.firstpage","837"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Inverse Problems"],["dc.bibliographiccitation.lastpage","858"],["dc.bibliographiccitation.volume","23"],["dc.contributor.author","Bauer, Frank"],["dc.date.accessioned","2018-11-07T11:03:33Z"],["dc.date.available","2018-11-07T11:03:33Z"],["dc.date.issued","2007"],["dc.description.abstract","Using a Bayesian-type approach to inverse problems many phenomena occurring in practice can be explained in a consistent way. In particular we can prove a discrete version of the quasi-optimality criterion for choosing the regularization parameter by simply imposing minor additional a priori assumptions about the solution and the measurement noise."],["dc.identifier.doi","10.1088/0266-5611/23/2/021"],["dc.identifier.isi","000245945100021"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/51646"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Iop Publishing Ltd"],["dc.relation.issn","0266-5611"],["dc.title","Some considerations concerning regularization and parameter choice algorithms"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","1437"],["dc.bibliographiccitation.issue","12"],["dc.bibliographiccitation.journal","Mathematical Methods in the Applied Sciences"],["dc.bibliographiccitation.lastpage","1451"],["dc.bibliographiccitation.volume","30"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Kannengiesser, Stephan"],["dc.date.accessioned","2018-11-07T11:00:18Z"],["dc.date.available","2018-11-07T11:00:18Z"],["dc.date.issued","2007"],["dc.description.abstract","Magnetic resonance imaging with parallel data acquisition requires algorithms for reconstructing the patient's image from a small number of measured k-space lines. In contrast to well-known algorithms like SENSE and GRAPPA and its flavours we consider the problem as a non-linear inverse problem. Fast computation algorithms for the necessary Frechet derivative and reconstruction algorithms are given. Copyright (C) 2007 John Wiley & Sons, Ltd."],["dc.identifier.doi","10.1002/mma.848"],["dc.identifier.isi","000248153200005"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/50890"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","John Wiley & Sons Ltd"],["dc.relation.issn","0170-4214"],["dc.title","An alternative approach to the image reconstruction for parallel data acquisition in MRI"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","45"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Applicable Analysis"],["dc.bibliographiccitation.lastpage","57"],["dc.bibliographiccitation.volume","87"],["dc.contributor.author","Bauer, Frank"],["dc.date.accessioned","2018-11-07T11:19:24Z"],["dc.date.available","2018-11-07T11:19:24Z"],["dc.date.issued","2008"],["dc.description.abstract","In the field of gravity determination, a special kind of boundary value problem respectively ill-posed satellite problem occurs; the data and hence side condition of our PDE are oblique second-order derivatives of the gravitational potential. For this computationally demanding problem with millions of data points, one classically just uses the derivatives to radial direction because they commute with the Laplace operator Delta. We will investigate in what extent this approach can also be used for other derivatives. We classify all first and purely second-order operators D which fulfill Delta Dv=0 if Delta v=0. This allows us to solve the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of D, i.e. integration."],["dc.identifier.doi","10.1080/00036810701603029"],["dc.identifier.isi","000253172800004"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/55267"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Taylor & Francis Ltd"],["dc.relation.issn","0003-6811"],["dc.title","Split operators for oblique boundary value problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","15"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Proceedings in Applied Mathematics and Mechanics"],["dc.bibliographiccitation.lastpage","18"],["dc.bibliographiccitation.volume","5"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Hohage, Thorsten"],["dc.date.accessioned","2017-09-07T11:52:31Z"],["dc.date.available","2017-09-07T11:52:31Z"],["dc.date.issued","2005"],["dc.description.abstract","Regularized Newton methods are one of the most popular approaches for the solution of inverse problems in differential equations. Since these problems are usually ill‐posed, an appropriate stopping rule is an essential ingredient of such methods. In this paper we suggest an a‐posteriori stopping rule of Lepskij‐type which is appropriate for data perturbed by random noise. The numerical results for this look promising."],["dc.identifier.doi","10.1002/pamm.200510005"],["dc.identifier.gro","3146341"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4112"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.title","A Lepskij's stopping rule for Newton-type methods with random noise"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","1975"],["dc.bibliographiccitation.issue","6"],["dc.bibliographiccitation.journal","Inverse Problems"],["dc.bibliographiccitation.lastpage","1991"],["dc.bibliographiccitation.volume","21"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Hohage, Thorsten"],["dc.date.accessioned","2017-09-07T11:52:34Z"],["dc.date.available","2017-09-07T11:52:34Z"],["dc.date.issued","2005"],["dc.description.abstract","We investigate an a posteriori stopping rule of Lepskij-type for a class of regularized Newton methods and show that it leads to order optimal convergence rates for Hölder and logarithmic source conditions without a priori knowledge of the smoothness of the solution. Numerical experiments show that this stopping rule yields results at least as good as, and in some situations significantly better than, Morozov's discrepancy principle."],["dc.identifier.doi","10.1088/0266-5611/21/6/011"],["dc.identifier.gro","3146340"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4110"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.title","A Lepskij-type stopping rule for regularized Newton methods"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","137"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Journal of Inverse and Ill-posed Problems"],["dc.bibliographiccitation.lastpage","148"],["dc.bibliographiccitation.volume","15"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Munk, Axel"],["dc.date.accessioned","2017-09-07T11:53:25Z"],["dc.date.available","2017-09-07T11:53:25Z"],["dc.date.issued","2007"],["dc.description.abstract","We present a strategy for choosing the regularization parameter (Lepskij-type balancing principle) for ill-posed problems in metric spaces with deterministic or stochastic noise. Additionally we improve the strategy in comparison to the previously used version for Hilbert spaces in some ways."],["dc.identifier.doi","10.1515/jiip.2007.007"],["dc.identifier.gro","3145070"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/2765"],["dc.language.iso","en"],["dc.notes.intern","Crossref Import"],["dc.notes.status","final"],["dc.relation.issn","0928-0219"],["dc.title","Optimal regularization for ill-posed problems in metric spaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","303"],["dc.bibliographiccitation.journal","European Journal of Applied Mathematics"],["dc.bibliographiccitation.lastpage","317"],["dc.bibliographiccitation.volume","16"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Pereverzev, S."],["dc.date.accessioned","2018-11-07T10:34:16Z"],["dc.date.available","2018-11-07T10:34:16Z"],["dc.date.issued","2005"],["dc.description.abstract","The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required the knowledge of smoothness of the solution or the overall error behaviour. We will show that we can guarantee an asymptotically almost optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behaviour. An application to the determination of the gravitational field out of satellite data will be shown."],["dc.identifier.doi","10.1017/S0956792505006236"],["dc.identifier.isi","000232994900001"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/44823"],["dc.language.iso","en"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.relation.issn","0956-7925"],["dc.title","Regularization without preliminary knowledge of smoothness and error behaviour"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","52"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Journal of Complexity"],["dc.bibliographiccitation.lastpage","72"],["dc.bibliographiccitation.volume","23"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Pereverzev, Sergei"],["dc.contributor.author","Rosasco, Lorenzo"],["dc.date.accessioned","2018-11-07T11:05:22Z"],["dc.date.available","2018-11-07T11:05:22Z"],["dc.date.issued","2007"],["dc.description.abstract","In this paper we discuss a relation between Learning Theory and Regularization of linear ill-posed inverse problems. It is well known that Tikhonov regularization can be profitably used in the context of supervised learning, where it usually goes under the name of regularized least-squares algorithm. Moreover, the gradient descent algorithm was studied recently, which is an analog of Landweber regularization scheme. In this paper we show that a notion of regularization defined according to what is usually done for ill-posed inverse problems allows to derive learning algorithms which are consistent and provide a fast convergence rate. It turns out that for priors expressed in term of variable Hilbert scales in reproducing kernel Hilbert spaces our results for Tikhonov regularization match those in Smale and Zhou [Learning theory estimates via integral operators and their approximations, submitted for publication, retrievable at (http://www.tti-c.org/smale.html), 2005] and improve the results for Landweber iterations obtained in Yao et al. [On early stopping in gradient descent learning, Constructive Approximation (2005), submitted for publication]. The remarkable fact is that our analysis shows that the same properties are shared by a large class of learning algorithms which are essentially all the linear regularization schemes. The concept of operator monotone functions turns out to be an important tool for the analysis. (c) 2006 Elsevier Inc. All rights reserved."],["dc.identifier.doi","10.1016/j.jco.2006.07.001"],["dc.identifier.isi","000245344800003"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/52056"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Academic Press Inc Elsevier Science"],["dc.relation.issn","0885-064X"],["dc.title","On regularization algorithms in learning theory"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","39"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Journal of Geodesy"],["dc.bibliographiccitation.lastpage","51"],["dc.bibliographiccitation.volume","81"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Mathe, Peter"],["dc.contributor.author","Pereverzev, Sergei"],["dc.date.accessioned","2018-11-07T11:06:52Z"],["dc.date.available","2018-11-07T11:06:52Z"],["dc.date.issued","2007"],["dc.description.abstract","In many geoscientific applications, one needs to recover the quantities of interest from indirect observations blurred by colored noise. Such quantities often correspond to the values of bounded linear functionals acting on the solution of some observation equation. For example, various quantities are derived from harmonic coefficients of the Earth's gravity potential. Each such coefficient is the value of the corresponding linear functional. The goal of this paper is to discuss new means to use information about the noise covariance structure, which allows order-optimal estimation of the functionals of interest and does not involve a covariance operator directly in the estimation process. It is done on the basis of a balancing principle for the choice of the regularization parameter, which is new in geoscientific applications. A number of tests demonstrate its applicability. In particular, we could find appropriate regularization parameters by knowing a small part of the gravitational field on the Earth's surface with high precision and reconstructing the rest globally by downward continuation from satellite data."],["dc.identifier.doi","10.1007/s00190-006-0049-5"],["dc.identifier.isi","000243047400004"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/52418"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","0949-7714"],["dc.title","Local solutions to inverse problems in geodesy"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","331"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Inverse Problems"],["dc.bibliographiccitation.lastpage","342"],["dc.bibliographiccitation.volume","23"],["dc.contributor.author","Bauer, Frank"],["dc.contributor.author","Ivanyshyn, Olha"],["dc.date.accessioned","2018-11-07T11:05:21Z"],["dc.date.available","2018-11-07T11:05:21Z"],["dc.date.issued","2007"],["dc.description.abstract","We will consider the situation of ill-posed problems with regularization methods which demand the choice of interdependent regularization parameters. We show that one can adapt the Lepskij-type balancing principle to this situation."],["dc.identifier.doi","10.1088/0266-5611/23/1/018"],["dc.identifier.isi","000244177100018"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/52052"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Iop Publishing Ltd"],["dc.relation.issn","0266-5611"],["dc.title","Optimal regularization with two interdependent regularization parameters"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]