Approximation in the extended functional tensor train format

TL;DR

This paper introduces the extended functional tensor train (EFTT) format, a new method for efficiently compressing multivariate functions on tensor domains using adaptive low-rank approximation and Chebyshev interpolation, significantly reducing function evaluations.

Contribution

The paper presents the EFTT format and a novel adaptive compression algorithm that outperforms existing methods in reducing storage and evaluation costs.

Findings

Achieves up to 96% reduction in function evaluations

Maintains accuracy while reducing storage requirements

Demonstrates superior adaptivity over previous methods

Abstract

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].