Tensor approximation of the self-diffusion matrix of tagged particle processes

TL;DR

This paper introduces a tensor-based numerical method for approximating the self-diffusion matrix of tagged particle processes, reducing statistical noise compared to traditional long-time average methods.

Contribution

It develops a novel tensor approximation approach combined with variance reduction for efficient and less noisy computation of the self-diffusion matrix.

Findings

Tensor method reduces statistical noise significantly.

Iterative low-rank approximation efficiently computes the matrix.

Method outperforms classical approaches in accuracy and stability.

Abstract

The objective of this paper is to investigate a new numerical method for the approximation of the self-diffusion matrix of a tagged particle process defined on a grid. While standard numerical methods make use of long-time averages of empirical means of deviations of some stochastic processes, and are thus subject to statistical noise, we propose here a tensor method in order to compute an approximation of the solution of a high-dimensional quadratic optimization problem, which enables to obtain a numerical approximation of the self-diffusion matrix. The tensor method we use here relies on an iterative scheme which builds low-rank approximations of the quantity of interest and on a carefully tuned variance reduction method so as to evaluate the various terms arising in the functional to minimize. In particular, we numerically observe here that it is much less subject to statistical noise than classical approaches.