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Constructing efficient spatial discretizations of spans of multivariate Chebyshev polynomials

Lutz Kämmerer

TL;DR

This work addresses constructing efficient spatial discretizations for spans of multivariate Chebyshev polynomials by embedding them into cosine-transformed multiple rank-1 lattices. The authors develop a framework that achieves three types of efficiency—construction, sampling, and reconstruction—by exploiting the connection between Chebyshev and trigonometric systems, notably via the mirror operator and the Not-In-Span-Of-Rest criterion, enabling FFT-based reconstruction. They prove probabilistic guarantees on the discretization quality and introduce practical improvements (greedy, index-set–aware, and halving strategies) that markedly reduce the number of sampling nodes while preserving accuracy. Numerical experiments up to high dimensions and across diverse index sets demonstrate favorable sampling sizes, conditioning, and robustness compared to single-rank-1 approaches, with strong potential for dimension-incremental polynomial approximation in applications like PDEs with random coefficients.

Abstract

For an arbitrary given span of high-dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each polynomial of this span. The approach presented here combines three different types of efficiency. First, the construction of the spatial discretization should be efficient with respect to the dimension of the span of the Chebyshev polynomials. Second, the number of sampling nodes within the constructed discretizations should be efficient, i.e., the oversampling factors should be reasonable. Third, there should be an efficient method for the unique reconstruction of a polynomial from given sampling values at the sampling nodes of the discretization. The first two mentioned types of efficiency are also present in constructions based on random sampling nodes, but the lack of structure here causes the inefficiency of the reconstruction method. Our approach uses a combination of cosine transformed rank-1 lattices whose structure allows for applications of univariate fast Fourier transforms for the reconstruction algorithm and is thus a priori efficiently realizable. Besides the theoretical estimates of numbers of sampling nodes and failure probabilities due to a random draw of the used lattices, we present several improvements of the basic design approach that significantly increases its practical applicability. Numerical tests, which discretize spans of multivariate Chebyshev polynomials depending on up to more than 50 spatial variables, corroborate the theoretical results and the significance of the improvements.

Constructing efficient spatial discretizations of spans of multivariate Chebyshev polynomials

TL;DR

This work addresses constructing efficient spatial discretizations for spans of multivariate Chebyshev polynomials by embedding them into cosine-transformed multiple rank-1 lattices. The authors develop a framework that achieves three types of efficiency—construction, sampling, and reconstruction—by exploiting the connection between Chebyshev and trigonometric systems, notably via the mirror operator and the Not-In-Span-Of-Rest criterion, enabling FFT-based reconstruction. They prove probabilistic guarantees on the discretization quality and introduce practical improvements (greedy, index-set–aware, and halving strategies) that markedly reduce the number of sampling nodes while preserving accuracy. Numerical experiments up to high dimensions and across diverse index sets demonstrate favorable sampling sizes, conditioning, and robustness compared to single-rank-1 approaches, with strong potential for dimension-incremental polynomial approximation in applications like PDEs with random coefficients.

Abstract

For an arbitrary given span of high-dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each polynomial of this span. The approach presented here combines three different types of efficiency. First, the construction of the spatial discretization should be efficient with respect to the dimension of the span of the Chebyshev polynomials. Second, the number of sampling nodes within the constructed discretizations should be efficient, i.e., the oversampling factors should be reasonable. Third, there should be an efficient method for the unique reconstruction of a polynomial from given sampling values at the sampling nodes of the discretization. The first two mentioned types of efficiency are also present in constructions based on random sampling nodes, but the lack of structure here causes the inefficiency of the reconstruction method. Our approach uses a combination of cosine transformed rank-1 lattices whose structure allows for applications of univariate fast Fourier transforms for the reconstruction algorithm and is thus a priori efficiently realizable. Besides the theoretical estimates of numbers of sampling nodes and failure probabilities due to a random draw of the used lattices, we present several improvements of the basic design approach that significantly increases its practical applicability. Numerical tests, which discretize spans of multivariate Chebyshev polynomials depending on up to more than 50 spatial variables, corroborate the theoretical results and the significance of the improvements.
Paper Structure (17 sections, 13 theorems, 57 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 13 theorems, 57 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Linear algebra, Fourier vs. Chebyshev, rank-1 Lattices
  3. Linear algebra
  4. Fourier vs. Chebyshev
  5. rank-1 Lattices
  6. Joining suitable sampling sets for discretization
  7. Theory on (cosine transformed) multiple rank-1 lattices
  8. Improvements for practical implementation
  9. Construction without determining index sets
  10. Construction with determined index sets
  11. Greedy improvement
  12. Improvement taking Section \ref{['sec:add_impro_general']} into account
  13. Further improvement
  14. Numerical tests
  15. Comparison to single rank-1 Chebyshev lattices
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let the matrix ${\boldsymbol{A}}=\left(a_{kj}\right)_{k=1,\dots,n; j=1,\dots,m}\in\mathbb{C}^{n\times m}$ be given. For a subset $K\subset\{1,\dots,n\}$ we denote where the elements in $K$ are assumed to be naturally ordered. If there exists a set $K\subset\{1,\dots,n\}$ such that then ${\boldsymbol{a}}_1\not\in\textnormal{span}\left\{{\boldsymbol{a}}_j\colon j\in\{2,\dots,m\}\right\}$ holds.

Figures (3)

  • Figure 6.1: Oversampling factors of spatial discretizations consisting of cosine transformed rank-1 lattices for $\textnormal{C}\Pi(\bar{H}_n^d)$, $n\in\{2,3,4,5\}$ fixed and dimensions $d$ up to $53$.
  • Figure 6.2: Oversampling factors of spatial discretizations consisting of cosine transformed rank-1 lattices for spans of Chebyshev polynomials with dyadic hyperbolic cross index sets $\bar{H}_n^6$ and the condition numbers of associated Chebyshev matrices.
  • Figure 6.3: Oversampling factors of spatial discretizations consisting of cosine transformed rank-1 lattices for spans of Chebyshev polynomials with random index sets in $\mathbb{N}_0^{25}$ where the number $d_s$ of active dimensions is fixed.

Theorems & Definitions (35)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 25 more