A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations

TL;DR

This paper introduces a recursive polynomial chaos method for efficiently simulating stochastic differential equations over long times by dynamically updating basis functions to reduce computational complexity.

Contribution

The paper presents a novel recursive polynomial chaos evolution technique that maintains low-dimensional representations without sampling, improving long-term stochastic simulations.

Findings

Method accurately captures complex dynamics.

Achieves high accuracy with low computational cost.

Proven convergence in Wasserstein-1 distance.

Abstract

Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.