Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs

TL;DR

This paper introduces a multilevel sparse tensor approximation method using the SALS algorithm for high-dimensional parametric PDEs, significantly reducing computational costs and demonstrating efficiency through theoretical analysis and numerical experiments.

Contribution

It develops a multilevel approach combining sparse tensor approximations with Galerkin discretization, achieving level-independent computational complexity.

Findings

Work overhead is independent of discretization level.

Significant computational savings demonstrated in high-dimensional tests.

The method is adaptable to various tensor formats and model classes.

Abstract

In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS) algorithm is employed to construct adaptive tensor train (TT) approximations of quantities of interest (QoI). By combining this tensor-based approach with a multilevel Galerkin discretization strategy, the solution's regularity can be exploited to significantly reduce computational costs by level-adapted sample sizes. A rigorous theoretical analysis is derived, demonstrating that the work overhead for the proposed multilevel method remains independent of the discretization level, which stands in stark contrast to the exponential growth observed in single-level approaches. The presented analysis is quite general and not constrained to the sparse TT format but uses a generic framework that can be extended to other model classes. Numerical experiments validate the predicted efficiency gains in high-dimensional settings.