High-dimensional approximation with kernel-based multilevel methods on sparse grids
Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique. While this has so far predominantly been done with either polynomials or splines, we suggest to employ a multilevel kernel-based a...
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| Published in: | Numerische Mathematik Vol. 154; no. 3-4; pp. 485 - 519 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2023
Springer Nature B.V |
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| ISSN: | 0029-599X, 0945-3245 |
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| Abstract | Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique. While this has so far predominantly been done with either polynomials or splines, we suggest to employ a multilevel kernel-based approximation scheme. In contrast to those schemes built upon polynomials and splines, this new method is capable of combining arbitrary low-dimensional domains instead of just intervals and arbitrarily distributed points in these low-dimensional domains. We introduce the method and analyse its convergence in the so-called isotropic and anisotropic cases. |
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| AbstractList | Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique. While this has so far predominantly been done with either polynomials or splines, we suggest to employ a multilevel kernel-based approximation scheme. In contrast to those schemes built upon polynomials and splines, this new method is capable of combining arbitrary low-dimensional domains instead of just intervals and arbitrarily distributed points in these low-dimensional domains. We introduce the method and analyse its convergence in the so-called isotropic and anisotropic cases. |
| Author | Wendland, Holger Kempf, Rüdiger |
| Author_xml | – sequence: 1 givenname: Rüdiger surname: Kempf fullname: Kempf, Rüdiger organization: Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth – sequence: 2 givenname: Holger surname: Wendland fullname: Wendland, Holger email: holger.wendland@uni-bayreuth.de organization: Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth |
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| Cites_doi | 10.1006/jcom.1995.1001 10.1090/conm/180/01952 10.1137/140966368 10.1016/j.jat.2016.02.006 10.1023/A:1018977404843 10.1007/s10444-013-9334-z 10.1007/s00607-009-0027-x 10.1002/9781118032725 10.1016/j.laa.2006.07.014 10.1016/S0168-9274(01)00030-7 10.1137/20M1325885 10.1016/j.jco.2018.02.003 10.1007/978-3-642-28027-6 10.1007/978-3-319-40528-5_7 10.1137/120898383 10.1137/120890715 10.1007/s11075-017-0340-y 10.1007/s00211-010-0313-8 10.1016/j.acha.2005.07.005 10.1137/1.9781611970128 10.1007/s00211-016-0845-7 10.1007/978-3-319-40528-5_9 10.21914/anziamj.v48i0.70 10.1142/6437 10.1093/imanum/dry080 10.1007/s00211-011-0428-6 10.1137/070680540 10.1063/1.4821348 10.1093/imanum/drs004 10.1137/060663660 10.1137/090774550 10.1016/0377-0427(96)00035-0 10.1093/imanum/drw006 10.1137/110859610 |
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| Title | High-dimensional approximation with kernel-based multilevel methods on sparse grids |
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