Schaback, Robert

[["dc.bibliographiccitation.firstpage","67"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Advances in Computational Mathematics"],["dc.bibliographiccitation.lastpage","81"],["dc.bibliographiccitation.volume","34"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:00:35Z"],["dc.date.available","2018-11-07T09:00:35Z"],["dc.date.issued","2011"],["dc.description.abstract","The Wendland radial basis functions (Wendland, Adv Comput Math 4: 389-396, 1995) are piecewise polynomial compactly supported reproducing kernels in Hilbert spaces which are norm-equivalent to Sobolev spaces. But they only cover the Sobolev spaces H(d/2+ k+ 1/2)(R(d)), k is an element of N (1) and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half-integer k and even dimensions, reproducing integer-order Sobolev spaces in even dimensions, but they turn out to have two additional non-polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the \"dimension walk\" is applied. While the classical dimension walk proceeds in steps of two space dimensions taking single derivatives, the new one proceeds in steps of single dimensions and uses \"halved\" derivatives of fractional calculus."],["dc.identifier.doi","10.1007/s10444-009-9142-7"],["dc.identifier.isi","000285058200003"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/5986"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/24201"],["dc.notes.intern","Merged from goescholar"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","1019-7168"],["dc.relation.orgunit","Fakultät für Mathematik und Informatik"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.title","The missing Wendland functions"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","457"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","Advances in Computational Mathematics"],["dc.bibliographiccitation.lastpage","470"],["dc.bibliographiccitation.volume","31"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T11:22:19Z"],["dc.date.available","2018-11-07T11:22:19Z"],["dc.date.issued","2009"],["dc.description.abstract","We present a meshless technique which can be seen as an alternative to the method of fundamental solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artificial boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples."],["dc.identifier.doi","10.1007/s10444-008-9078-3"],["dc.identifier.isi","000271394800005"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?goescholar/3084"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/55969"],["dc.notes.intern","Merged from goescholar"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","1019-7168"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.title","Solving the Laplace equation by meshless collocation using harmonic kernels"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","155"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Advances in Computational Mathematics"],["dc.bibliographiccitation.lastpage","161"],["dc.bibliographiccitation.volume","32"],["dc.contributor.author","De Marchi, Stefano"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T08:45:58Z"],["dc.date.available","2018-11-07T08:45:58Z"],["dc.date.issued","2010"],["dc.description.abstract","It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L (2) or L (aaEuro parts per thousand) norms of interpolants can be bounded above by discrete a\"\"(2) and a\"\" (aaEuro parts per thousand) norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance."],["dc.description.sponsorship","CNR-DFG; GNCS-Indam"],["dc.identifier.doi","10.1007/s10444-008-9093-4"],["dc.identifier.isi","000273742600002"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/6785"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/20577"],["dc.notes.intern","Merged from goescholar"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","1572-9044"],["dc.relation.issn","1019-7168"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.title","Stability of kernel-based interpolation"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","629"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","Numerische Mathematik"],["dc.bibliographiccitation.lastpage","651"],["dc.bibliographiccitation.volume","114"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T08:46:27Z"],["dc.date.available","2018-11-07T08:46:27Z"],["dc.date.issued","2010"],["dc.description.abstract","A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong form without changing the theory. Non-stationary kernel-based trial and test spaces are shown to fit into the framework, disregarding the operator equation. As a special case, unsymmetric meshless kernel-based methods solving weakly posed problems with distributional data are treated in some detail. This provides a foundation of certain variations of the \"Meshless Local Petrov-Galerkin\" technique of S.N. Atluri and collaborators."],["dc.identifier.doi","10.1007/s00211-009-0265-z"],["dc.identifier.isi","000274045300004"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?goescholar/4024"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/20694"],["dc.notes.intern","Merged from goescholar"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","0029-599X"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.title","Unsymmetric meshless methods for operator equations"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","601"],["dc.bibliographiccitation.journal","European Journal of Applied Mathematics"],["dc.bibliographiccitation.lastpage","629"],["dc.bibliographiccitation.volume","24"],["dc.contributor.author","Scheuerer, Michael"],["dc.contributor.author","Schaback, Robert"],["dc.contributor.author","Schlather, Martin"],["dc.date.accessioned","2018-11-07T09:22:12Z"],["dc.date.available","2018-11-07T09:22:12Z"],["dc.date.issued","2013"],["dc.description.abstract","Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this paper we present the two approaches and discuss their modelling assumptions, notions of optimality and different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection and discuss the scope of these methods with a data example from the computer experiments literature."],["dc.identifier.doi","10.1017/S0956792513000016"],["dc.identifier.isi","000321206200005"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/11574"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/29284"],["dc.notes.intern","Merged from goescholar"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Cambridge Univ Press"],["dc.relation.issn","1469-4425"],["dc.relation.issn","0956-7925"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.title","Interpolation of spatial data - A stochastic or a deterministic problem?"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]