Schaback, Robert

[["dc.bibliographiccitation.firstpage","665"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","PAMM"],["dc.bibliographiccitation.lastpage","666"],["dc.bibliographiccitation.volume","15"],["dc.contributor.author","Peter, Thomas"],["dc.contributor.author","Plonka-Hoch, Gerlind"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2017-09-07T11:53:59Z"],["dc.date.available","2017-09-07T11:53:59Z"],["dc.date.issued","2015"],["dc.description.abstract","he problem of recovering translates and corresponding amplitudes of sparse sums of Gaussians out of sampling values as well as reconstructing sparse sums of exponentials are nonlinear inverse problems that can be solved for example by Prony's method. Here, we want to demonstrate a new extension to multivariate input data."],["dc.identifier.doi","10.1002/pamm.201510322"],["dc.identifier.gro","3146462"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4239"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.relation.issn","1617-7061"],["dc.title","Prony's Method for Multivariate Signals"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]

[["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","65"],["dc.bibliographiccitation.journal","Journal of Approximation Theory"],["dc.bibliographiccitation.lastpage","83"],["dc.bibliographiccitation.volume","171"],["dc.contributor.author","Zwicknagl, Barbara"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:23:17Z"],["dc.date.available","2018-11-07T09:23:17Z"],["dc.date.issued","2013"],["dc.description.abstract","The univariate Taylor formula without remainder allows to reproduce a function completely from certain derivative values. Thus one can look for Hilbert spaces in which the Taylor formula acts as a reproduction formula. It turns out that there are many Hilbert spaces which allow this, and they should be called Taylor spaces. They have certain reproducing kernels which are either polynomials or power series with nonnegative coefficients. Consequently, Taylor spaces can be spanned by translates of various classical special functions such as exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions. Since the theory of kernel-based interpolation and approximation is well-established, this leads to a variety of results. In particular, interpolation by shifted exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions provides exponentially convergent approximations to analytic functions, generalizing the classical Bernstein theorem for polynomial approximation to analytic functions. Finally, we prove sampling inequalities in Taylor spaces that allow to derive similar convergence rates for non-interpolatory approximations. (C) 2013 Elsevier Inc. All rights reserved."],["dc.identifier.doi","10.1016/j.jat.2013.03.006"],["dc.identifier.isi","000320833700003"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/29542"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Academic Press Inc Elsevier Science"],["dc.relation.issn","1096-0430"],["dc.relation.issn","0021-9045"],["dc.title","Interpolation and approximation in Taylor spaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["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.journal","IMA Journal of Numerical Analysis"],["dc.contributor.author","Davydov, Oleg"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2020-12-10T18:19:17Z"],["dc.date.available","2020-12-10T18:19:17Z"],["dc.date.issued","2017"],["dc.identifier.doi","10.1093/imanum/drx076"],["dc.identifier.eissn","1464-3642"],["dc.identifier.issn","0272-4979"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/75192"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-354"],["dc.title","Optimal stencils in Sobolev spaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","645"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Journal of Approximation Theory"],["dc.bibliographiccitation.lastpage","655"],["dc.bibliographiccitation.volume","161"],["dc.contributor.author","Mueller, Stefan"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T11:21:32Z"],["dc.date.available","2018-11-07T11:21:32Z"],["dc.date.issued","2009"],["dc.description.abstract","It is well known that representations of kernel-based approximants in terms of the standard basis of translated kernels are notoriously unstable. To come up with a more useful basis, we adopt the strategy known from Newton's interpolation formula, using generalized divided differences and a recursively Computable set of basis functions vanishing at increasingly many data points. The resulting basis turns out to be orthogonal in the Hilbert space in which the kernel is reproducing, and under certain assumptions it is complete and allows convergent expansions of functions into series of interpolants. Some numerical examples show that the Newton basis is much more stable than the standard basis of kernel translates. (C) 2008 Elsevier Inc. All rights reserved."],["dc.description.sponsorship","Deutsche Forschungsgemeinschaft [1023]"],["dc.identifier.doi","10.1016/j.jat.2008.10.014"],["dc.identifier.isi","000272562300013"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/55795"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Academic Press Inc Elsevier Science"],["dc.relation.issn","0021-9045"],["dc.title","A Newton basis for Kernel spaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","113"],["dc.bibliographiccitation.journal","Applied Mathematics and Computation"],["dc.bibliographiccitation.lastpage","123"],["dc.bibliographiccitation.volume","307"],["dc.contributor.author","Lenarduzzi, Licia"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T10:21:51Z"],["dc.date.available","2018-11-07T10:21:51Z"],["dc.date.issued","2017"],["dc.description.abstract","One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in those subdomains, and these sub-approximations can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes an algorithm that first calculates sub-approximations on non-overlapping subdomains, then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain. Consequently, there will be no Gibbs phenomenon along the boundaries of the subdomains. The method detects faults and gradient faults with good accuracy. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample. (C) 2017 Elsevier Inc. All rights reserved."],["dc.identifier.doi","10.1016/j.amc.2017.02.043"],["dc.identifier.isi","000399590900010"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/42172"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","PUB_WoS_Import"],["dc.publisher","Elsevier Science Inc"],["dc.relation.issn","1873-5649"],["dc.relation.issn","0096-3003"],["dc.title","Kernel-based adaptive approximation of functions with discontinuities"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","77"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","CALCOLO"],["dc.bibliographiccitation.lastpage","87"],["dc.bibliographiccitation.volume","41"],["dc.contributor.author","Bozzini, M."],["dc.contributor.author","Lenarduzzi, L."],["dc.contributor.author","Rossini, M."],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T10:47:30Z"],["dc.date.available","2018-11-07T10:47:30Z"],["dc.date.issued","2004"],["dc.description.abstract","Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications, the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results."],["dc.identifier.doi","10.1007/s10092-004-0085-6"],["dc.identifier.isi","000231278800002"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/47977"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Springer"],["dc.relation.issn","0008-0624"],["dc.title","Interpolation by basis functions of different scales and shapes"],["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","5224"],["dc.bibliographiccitation.issue","10"],["dc.bibliographiccitation.journal","Applied Mathematics and Computation"],["dc.bibliographiccitation.lastpage","5232"],["dc.bibliographiccitation.volume","219"],["dc.contributor.author","Dereli, Yilmaz"],["dc.contributor.author","Schaback, Robert"],["dc.date.accessioned","2018-11-07T09:29:10Z"],["dc.date.available","2018-11-07T09:29:10Z"],["dc.date.issued","2013"],["dc.description.abstract","The Equal Width equation governs nonlinear wave phenomena like waves in shallow water. Here, it is solved numerically by the Method of Lines using a somewhat unusual setup. There is no linearization of the nonlinear terms, no error in handling the starting approximation, and there are boundary conditions only at infinity. To achieve a space discretization of high accuracy with only few trial functions, meshless translates of radial kernels are used. In the numerical examples, the motion of solitary waves, the interaction of two and three solitary waves, the generation of wave undulation, the Maxwell initial condition, and the clash of two colliding solitary waves are simulated. Our numerical results compare favourably with results of earlier papers using other techniques. (C) 2012 Elsevier Inc. All rights reserved."],["dc.description.sponsorship","Scientific and Technological Research Council of Turkey (TUBITAK)"],["dc.identifier.doi","10.1016/j.amc.2012.10.086"],["dc.identifier.isi","000313825900028"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/30954"],["dc.notes.status","zu prüfen"],["dc.notes.submitter","Najko"],["dc.publisher","Elsevier Science Inc"],["dc.relation.issn","0096-3003"],["dc.title","The meshless kernel-based method of lines for solving the equal width equation"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.status","published"],["dspace.entity.type","Publication"]]