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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Bibliographic Details
Published in:Numerische Mathematik Vol. 154; no. 3-4; pp. 485 - 519
Main Authors: Kempf, Rüdiger, Wendland, Holger
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023
Springer Nature B.V
Subjects:
Approximation
Kernels
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Polynomials
Simulation
Spline functions
Theoretical
65D15
65D12
46E22
41A63
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary: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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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-023-01363-x