A low-rank approach to the computation of path integrals

TL;DR

This paper introduces a low-rank convolution algorithm for efficiently computing path integrals related to reaction-diffusion equations with general potentials, reducing computational complexity and memory usage.

Contribution

It proposes a novel low-rank approximation method for Hankel matrices to accelerate the computation of path integrals in reaction-diffusion problems.

Findings

Algorithm achieves $ ext{O}(nr M ext{log} M + nr^2 M)$ complexity.

Requires $ ext{O}(M r)$ memory, significantly less than traditional methods.

Applicable to higher-order diffusion processes.

Abstract

We present a method for solving the reaction-diffusion equation with general potential in free space. It is based on the approximation of the Feynman-Kac formula by a sequence of convolutions on sequentially diminishing grids. For computation of the convolutions we propose a fast algorithm based on the low-rank approximation of the Hankel matrices. The algorithm has complexity of $\mathcal{O}(nr M \log M + nr^2 M)$ flops and requires $\mathcal{O}(M r)$ floating-point numbers in memory, where $n$ is the dimension of the integral, $r \ll n$, and $M$ is the mesh size in one dimension. The presented technique can be generalized to the higher-order diffusion processes.