Functional Tucker approximation using Chebyshev interpolation

TL;DR

This paper introduces Chebfun3F, a new algorithm that efficiently approximates trivariate functions using Chebyshev interpolation combined with Tucker decomposition, significantly reducing function evaluations compared to previous methods.

Contribution

The paper presents Chebfun3F, a novel approach that uses univariate fibers for Tucker decomposition, improving efficiency over the existing Chebfun3 algorithm.

Findings

Reduces function evaluations by up to 98%.

Achieves significant computational cost savings.

Provides accurate low-rank function approximations.

Abstract

This work is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function approximations that can be computed and stored very efficiently. The existing Chebfun3 algorithm [Hashemi and Trefethen, SIAM J. Sci. Comput., 39 (2017)]uses a similar format but the construction of the approximation proceeds indirectly, via a so called slice-Tucker decomposition. As a consequence, Chebfun3 sometimes uses unnecessarily many function evaluations and does not fully benefit from the potential of the Tucker decomposition to reduce, sometimes dramatically, the computational cost. We propose a novel algorithm Chebfun3F that utilizes univariate fibers instead of bivariate slices to construct the Tucker decomposition. Chebfun3F reduces the cost for the approximation in terms of the number of function evaluations for nearly all functions considered, typically by 75%, and sometimes by over 98%.