Reconstruction and Collocation of a Class of Non-periodic Functions by Sampling Along Tent-Transformed Rank-1 Lattices

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• 作者：Gowri Suryanarayana ; Dirk Nuyens…
• 关键词：Quasi ; Monte Carlo methods ; Cosine series ; Function approximation ; Hyperbolic crosses ; Rank ; 1 lattice rules ; Spectral methods ; Component ; by ; component construction ; 65D30 ; 65D32 ; 65C05 ; 65M70 ; 65T40 ; 65T50
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：22
• 期：1
• 页码：187-214
• 全文大小：1,634 KB
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• 作者单位：Gowri Suryanarayana (1)
  Dirk Nuyens (1)
  Ronald Cools (1)
  
  1. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

Spectral collocation and reconstruction methods have been widely studied for periodic functions using Fourier expansions. We investigate the use of cosine series for the approximation and collocation of multivariate non-periodic functions with frequency support mainly determined by hyperbolic crosses. We seek methods that work for an arbitrary number of dimensions. We show that applying the tent-transformation on rank-1 lattice points renders them suitable to be collocation/sampling points for the approximation of non-periodic functions with perfect numerical stability. Moreover, we show that the approximation degree—in the sense of approximating inner products of basis functions up to a certain degree exactly—of the tent-transformed lattice point set with respect to cosine series, is the same as the approximation degree of the original lattice point set with respect to Fourier series, although the error can still be reduced in the case of cosine series. A component-by-component algorithm is studied to construct such a point set. We are then able to reconstruct a non-periodic function from its samples and approximate the solutions to certain PDEs subject to Neumann and Dirichlet boundary conditions. Finally, we present some numerical results. Keywords Quasi-Monte Carlo methods Cosine series Function approximation Hyperbolic crosses Rank-1 lattice rules Spectral methods Component-by-component construction