Parallel numerical tensor methods for high-dimensional PDEs

TL;DR

This paper introduces parallel tensor-based algorithms for efficiently solving high-dimensional PDEs, demonstrating their accuracy and computational advantages in complex multi-variable equations.

Contribution

The paper develops novel parallel algorithms combining tensor methods with ALS and hierarchical SVD for high-dimensional PDEs, including implicit and explicit schemes.

Findings

Algorithms accurately solve 6-variable advection equations.

Efficiently compute linearized Boltzmann equations.

Show improved performance over traditional methods.

Abstract

High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and Fokker-Planck equations. We develop new parallel algorithms to solve high-dimensional PDEs. The algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition. Both implicit and explicit integration schemes are presented and discussed. We demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the Boltzmann equation.