Stochastic particle method with birth-death dynamics

TL;DR

This paper introduces an active birth-death particle dynamics to improve the efficiency of stochastic particle methods for solving high-dimensional nonlinear PDEs, achieving better accuracy with less frequent resampling.

Contribution

The paper proposes a novel birth-death dynamics mechanism within stochastic particle methods, providing rigorous error analysis and demonstrating improved efficiency over existing methods.

Findings

SPM-birth-death achieves higher efficiency than SPM.

The method attains first-order convergence in time and space.

Numerical experiments show smaller errors at similar computational costs.

Abstract

In order to numerically solve high-dimensional nonlinear PDEs and alleviate the curse of dimensionality, a stochastic particle method (SPM) has been proposed to capture the relevant feature of the solution through the adaptive evolution of particles [J. Comput. Phys. 527 (2025) 113818]. In this paper, we introduce an active birth-death dynamics of particles to improve the efficiency of SPM. The resulting method, dubbed SPM-birth-death, sample new particles according to the nonlinear term and execute the annihilation strategy when the number of particles exceeds a given threshold. A rigorous error estimation for SPM-birth-death is established, elucidating the first-order convergence in time and space, as well as half-order accuracy in the initial sample size with explicit variance estimates. We also extend the analysis framework to SPM and provide theoretical justification for the existing numerical convergence study. Our theoretical results reveal that the introduced active birth-death dynamics of particles results into less frequent resampling and SPM-birth-death is thus able to achieve higher efficiency than SPM. Validating benchmarks are provided. In particular, preliminary numerical experiments on the Allen-Cahn equation demonstrate that SPM-birth-death can achieve smaller errors at the same computational cost compared with the original SPM.