Transformed rank-1 lattices for high-dimensional approximation

Robert  Nasdala; Daniel  Potts

doi:doi:10.1553/etna_vol53s239

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Transformed rank-1 lattices for high-dimensional approximation

Robert Nasdala,

Daniel Potts

ETNA - Electronic Transactions on Numerical Analysis, pp. 239-282, 2020/02/04

doi: 10.1553/etna_vol53s239

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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.

Keywords: approximation on unbounded domains, change of variables, sparse multivariate trigonometric polynomials, lattice rule, multiple rank-1 lattice, fast Fourier transform

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Published Online: 2020/02/04 12:37:14

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Transformed rank-1 lattices for high-dimensional approximation

Abstract