A novel variable-separation method based on sparse representation for stochastic partial differential equations

TL;DR

This paper introduces a new variable-separation method for stochastic PDEs that reduces computational complexity and enhances efficiency through sparse representation and hierarchical tensor approximation techniques.

Contribution

The paper develops a novel variable-separation approach with no iteration at each step, combined with ILARS and HSLRTA for efficient high-dimensional stochastic PDE solutions.

Findings

NVS achieves systematic solution representation with less computation.

ILARS improves efficiency in sparse regularization tasks.

HSLRTA effectively approximates high-dimensional stochastic solutions.

Abstract

In this paper, we propose a novel variable-separation (NVS) method for generic multivariate functions. The idea of NVS is extended to to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs). Compared with many widely used variation-separation methods, NVS shares their merits but has less computation complexity and better efficiency. NVS can be used to get the separated representation of the solution for SPDE in a systematic enrichment manner. No iteration is performed at each enrichment step. This is a significant improvement compared with proper generalized decomposition. Because the stochastic functions of the separated representations obtained by NVS depend on the previous terms, this impacts on the computation efficiency and brings great challenge for numerical simulation for the problems in high stochastic dimensional spaces. In order to overcome the difficulty, we propose an improved least angle regression algorithm (ILARS) and a hierarchical sparse low rank tensor approximation (HSLRTA) method based on sparse regularization. For ILARS, we explicitly give the selection of the optimal regularization parameters at each step based on least angle regression algorithm (LARS) for lasso problems such that ILARS is much more efficient. HSLRTA hierarchically decomposes a high dimensional problem into some low dimensional problems and brings an accurate approximation for the solution to SPDEs in high dimensional stochastic spaces using limited computer resource. A few numerical examples are presented to illustrate the efficacy of the proposed methods.