Schaback, Robert

[["dc.bibliographiccitation.firstpage","177"],["dc.bibliographiccitation.issue","2-3"],["dc.bibliographiccitation.journal","Applied Mathematics and Computation"],["dc.bibliographiccitation.lastpage","186"],["dc.bibliographiccitation.volume","119"],["dc.contributor.author","Hong, Yun-Chul"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:09:50Z"],["dc.date.available","2018-11-07T09:09:50Z"],["dc.date.issued","2001"],["dc.description.abstract","Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system was still missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort. (C) 2001 Elsevier Science Inc. All rights reserved."],["dc.identifier.doi","10.1016/S0096-3003(99)00255-6"],["dc.identifier.isi","000168075800005"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/26355"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Elsevier Science Inc"],["dc.relation.issn","0096-3003"],["dc.title","On unsymmetric collocation by radial basis functions"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","220"],["dc.bibliographiccitation.journal","Applied Mathematics and Computation"],["dc.bibliographiccitation.lastpage","226"],["dc.bibliographiccitation.volume","258"],["dc.contributor.author","Hong, Yun-Chul"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:58:07Z"],["dc.date.available","2018-11-07T09:58:07Z"],["dc.date.issued","2015"],["dc.description.abstract","We provide a class of positive definite kernels that allow to solve certain evolution equations of parabolic type for scattered initial data by kernel-based interpolation or approximation, avoiding time intergation completely. Some numerical illustrations are given. (C) 2015 Elsevier Inc. All rights reserved."],["dc.description.sponsorship","City University of Hong Kong, Hong Kong [ARG 9667078]"],["dc.identifier.doi","10.1016/j.amc.2014.12.140"],["dc.identifier.isi","000351668500024"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/37308"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Elsevier Science Inc"],["dc.relation.issn","1873-5649"],["dc.relation.issn","0096-3003"],["dc.title","Direct meshless kernel techniques for time-dependent equations"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","1"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Advances in Computational Mathematics"],["dc.bibliographiccitation.lastpage","19"],["dc.bibliographiccitation.volume","38"],["dc.contributor.author","Hong, Yun-Chul"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:30:32Z"],["dc.date.available","2018-11-07T09:30:32Z"],["dc.date.issued","2013"],["dc.description.abstract","This paper solves the Laplace equation Delta u = 0 on domains Omega aS,aEuro parts per thousand a\"e(3) by meshless collocation on scattered points of the boundary . Due to the use of new positive definite kernels K(x, y) which are harmonic in both arguments and have no singularities for x = y, one can directly interpolate on the boundary, and there is no artificial boundary needed as in the Method of Fundamental Solutions. In contrast to many other techniques, e.g. the Boundary Point Method or the Method of Fundamental Solutions, we provide a solid and comprehensive mathematical foundation which includes error bounds and works for general star-shaped domains. The convergence rates depend only on the smoothness of the domain and the boundary data. Some numerical examples are included."],["dc.identifier.doi","10.1007/s10444-011-9224-1"],["dc.identifier.isi","000314033600001"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/31327"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","1572-9044"],["dc.relation.issn","1019-7168"],["dc.title","Solving the 3D Laplace equation by meshless collocation via harmonic kernels"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","13"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Numerical Algorithms"],["dc.bibliographiccitation.lastpage","25"],["dc.bibliographiccitation.volume","32"],["dc.contributor.author","Hong, Yun-Chul"],["dc.contributor.author","Schaback, Robert"],["dc.contributor.author","Zhou, X."],["dc.date.accessioned","2018-11-07T10:42:46Z"],["dc.date.available","2018-11-07T10:42:46Z"],["dc.date.issued","2003"],["dc.description.abstract","The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and ill-conditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding \"sparse\" approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given."],["dc.identifier.doi","10.1023/A:1022253303343"],["dc.identifier.isi","000180783100002"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/46884"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Kluwer Academic Publ"],["dc.relation.issn","1017-1398"],["dc.title","An adaptive greedy algorithm for solving large RBF collocation problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","2057"],["dc.bibliographiccitation.issue","12"],["dc.bibliographiccitation.journal","Computers & Mathematics with Applications"],["dc.bibliographiccitation.lastpage","2067"],["dc.bibliographiccitation.volume","68"],["dc.contributor.author","Hong, Yun-Chul"],["dc.contributor.author","Schaback, Robert"],["dc.contributor.author","Zhong, M."],["dc.date.accessioned","2018-11-07T09:31:42Z"],["dc.date.available","2018-11-07T09:31:42Z"],["dc.date.issued","2014"],["dc.description.abstract","Using the heat equation as a simple example, we give a rigorous theoretical analysis of the Method of Lines, implemented as a meshless method based on spatial trial spaces spanned by translates of positive definite kernels. The technique can be generalized to other parabolic problems, and some numerical illustrations are given. (C) 2014 Elsevier Ltd. All rights reserved."],["dc.identifier.doi","10.1016/j.camwa.2014.09.015"],["dc.identifier.isi","000347758600031"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/31594"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.relation.issn","1873-7668"],["dc.relation.issn","0898-1221"],["dc.title","The meshless Kernel-based method of lines for parabolic equations"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dspace.entity.type","Publication"]]