---
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.'
address:
- 'Department of Mathematics, Duke University, Box 90320, Durham, NC 27708 USA'
- 'Departments of Mathematics, Physics, and Chemistry, Duke University, Box 90320, Durham, NC 27708 USA'
author:
- Yu Cao
- Jianfeng Lu
bibliography:
- 'ref.bib'
title: 'Stochastic dynamical low-rank approximation method'
---

Dynamical low-rank approximation, stochastic differential equation, [Lindblad]{} equation, model reduction.

Introduction
============

Many problems in computational physics are challenging to solve due to *curse of dimensionality*, such as high dimensional master equations and many-body quantum dynamics. In attempt to resolve the difficulty, many ideas have been proposed: model reduction method [@antoulas_survey_2001; @Chatterjee_00_pod], Monte Carlo method [@Gisin92; @Dalibard92; @Tully], [*etc*.]{} In many situations, several methods of dimension reduction need to be combined together. For instance, after applying Monte Carlo method to some deterministic dynamics by simulating a 