Variance Reduction in the Fokker-Planck Particle Method for Rarefied Gases using Quasi-Random Numbers

TL;DR

This paper introduces a variance reduction technique for the Fokker-Planck particle method in rarefied-gas simulations by integrating Array-RQMC, leading to faster convergence and more accurate results.

Contribution

The study combines the FP method with Array-RQMC to effectively reduce variance and improve convergence in rarefied-gas simulations, a novel application of quasi-random sampling.

Findings

Enhanced convergence rates over pseudo-random sampling.

Smaller estimator errors with larger particle numbers.

Effective variance reduction in both homogeneous and inhomogeneous problems.

Abstract

The Fokker-Planck (FP) particle method accelerates rarefied-gas simulations by replacing the binary collisions of the commonly used Direct Simulation Monte Carlo (DSMC) method with a drift=diffusion process. Like all particle methods, the FP method is inherently stochastic, which leads to statistical fluctuations in macroscopic quantities and necessitates large particle numbers for accurate results. In this work, we investigate the use of quasi-random numbers, which sample distributions more evenly and thereby reduce the variance. To preserve the low-discrepancy structure across time steps, we employ the Array Randomized Quasi-Monte Carlo (Array-RQMC) technique. We combine the FP method with Array-RQMC and compare it in homogeneous and inhomogeneous problems with other commonly used variance-reduction techniques. The proposed FP-Array-RQMC approach achieves improved convergence rates compared with pseudo-random sampling and yields smaller estimator errors for sufficiently large particle numbers.