Dynamical Tensor Train Approximation for Kinetic Equations

TL;DR

This paper introduces a dynamical low-rank tensor train method for efficiently solving high-dimensional kinetic equations by exploiting low-rank structures in velocity space, reducing computational costs.

Contribution

It develops a novel tensor train-based dynamical low-rank method with a sweeping update procedure, improving efficiency over traditional approaches.

Findings

Reduces memory usage significantly compared to standard methods.

Achieves accurate solutions with relatively small TT-ranks.

Demonstrates effectiveness on several kinetic equations.

Abstract

The numerical solution of kinetic equations is challenging due to the high dimensionality of the underlying phase space. In this paper, we develop a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format. The key idea is to discretize the three-dimensional velocity variable using tensor trains while treating the spatial variable as a parameter, thereby exploiting the low-rank structure of the distribution function in velocity space. In contrast to the standard step-and-truncate approach, this method updates the tensor cores through a sweeping procedure, allowing the use of relatively small TT-ranks and leading to substantial reductions in memory usage and computational cost. We demonstrate the effectiveness of the proposed approach on several representative kinetic equations.