The Wigner branching random walk: Efficient implementation and performance evaluation

TL;DR

This paper introduces two efficient strategies for implementing the Wigner branching random walk using signed particles, demonstrating improved performance and variance reduction through theoretical analysis and numerical experiments.

Contribution

It proposes novel signed-particle implementation methods for the Wigner branching random walk, including a probabilistic interpretation and a bootstrap filter, with comprehensive performance evaluation.

Findings

Both strategies are feasible and reduce variance.

The second approach improves stability and efficiency.

Thorough comparison with existing methods is provided.

Abstract

To implement the Wigner branching random walk, the particle carrying a signed weight, either $-1$ or $+1$, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from $-1$ to $+1$. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adjust the skewness of particle weights. Performance evaluations on the Gaussian barrier scattering (2D) and a Helium-like system (4D) demonstrate the feasibility of both strategies and the variance reduction property of the second approach. We provide an improvement of the first signed-particle implementation that partially alleviates the restriction on the time step and perform a thorough theoretical and numerical comparison among all the existing signed-particle implementations. Details on implementing the importance sampling according to the quasi-probability density and an efficient resampling or particle reduction are also provided.