Adaptive Density Tracking by Quadrature for Stochastic Differential Equations

TL;DR

This paper introduces an adaptive density tracking method using quadrature for solving Fokker-Planck equations associated with stochastic differential equations, improving efficiency over traditional tensorized approaches.

Contribution

It extends existing DTQ methods by incorporating adaptive, unstructured meshes and provides a scalable procedure for higher-dimensional problems.

Findings

Adaptive DTQ is significantly more efficient than tensorized methods.

The method is demonstrated on two-dimensional examples.

Procedures are extendable to higher dimensions.

Abstract

Density tracking by quadrature (DTQ) is a numerical procedure for computing solutions to Fokker-Planck equations that describe probability densities for stochastic differential equations (SDEs). In this paper, we extend upon existing tensorized DTQ procedures by utilizing a flexible quadrature rule that allows for unstructured, adaptive meshes. We propose and describe the procedure for $N$-dimensions, and demonstrate that the resulting adaptive procedure is significantly more efficient than a tensorized approach. Although we consider two-dimensional examples, all our computational procedures are extendable to higher dimensional problems.