Robert Nasdala,
Daniel Potts
S. 239 - 282
doi:
10.1553/etna_vol53s239
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol53s239
Abstract:
This paper describes an extension of Fourier approximation methods for multivariate functions defined on thetorus $\\mathbb{T}^d$ to functions in a weighted Hilbert space $L_{2}(\\mathbb{R}^d, \\omega)$via a multivariate change of variables $\\psi:\\left(-\\frac{1}{2},\\frac{1}{2}\\right)^d\\to\\mathbb{R}^d$.We establish sufficient conditions for $\\psi$ and $\\omega$ such that the composition of a function in such a weighted Hilbert space with $\\psi$ yields a function in theSobolev space $H_{\\rm mix}^{m}(\\mathbb{T}^d)$ of functions on the torus with mixed smoothness of natural order $m \\in \\mathbb{N}_{0}$.In this approach we adapt algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus $\\mathbb{T}^d$based on single and multiple reconstructing rank-$1$ lattices.Since in applications it may be difficult to choose a related function space, we make use of dimension incremental construction methods for sparse frequency sets.Various numerical tests confirm the obtained theoretical results for the transformed methods.
approximation on unbounded domains, change of variables, sparse multivariate trigonometric polynomials, lattice rule, multiple rank-1 lattice, fast Fourier transform
Published Online:
2020/02/04 12:37:14
Object Identifier:
0xc1aa5576 0x003b464b
Rights:All rights reserved.For questions regarding copyright and copies please contact us by email.
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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