Constructing efficient spatial discretizations of spans of multivariate Chebyshev polynomials

TL;DR

This paper introduces an efficient method for constructing spatial discretizations of multivariate Chebyshev polynomial spans, enabling unique reconstruction with fewer samples using structured rank-1 lattices and fast Fourier transforms.

Contribution

The paper presents a novel structured sampling approach combining rank-1 lattices and Fourier transforms for efficient polynomial reconstruction in high dimensions.

Findings

Numerical tests confirm the theoretical efficiency and practicality of the proposed method.

The approach handles spans with over 50 variables effectively.

Improvements significantly enhance the method's applicability.

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.