Stochastic dynamical low-rank approximation method

TL;DR

This paper introduces a stochastic dynamical low-rank approximation method for high-dimensional stochastic differential equations, extending existing techniques to measure spaces and providing error analysis and numerical validation.

Contribution

It extends dynamical low-rank approximation to measure spaces and derives stochastic low-rank dynamics for SDEs and quantum master equations.

Findings

Effective in solving high-dimensional SDEs

Applicable to stochastic Burgers' equation

Validated through numerical examples

Abstract

In this paper, we extend the dynamical low-rank approximation method to the space of finite signed measures. Under this framework, we derive stochastic low-rank dynamics for stochastic differential equations (SDEs) coming from classical stochastic dynamics or unraveling of Lindblad quantum master equations. We justify the proposed method by error analysis and also numerical examples for applications in solving high-dimensional SDE, stochastic Burgers' equation, and high-dimensional Lindblad equation.