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

TL;DR

This paper addresses variance in stochastic FP particle simulations for rarefied gases by integrating Array-RQMC, which preserves low-discrepancy sampling across time steps through Morton-ordered particle updates and Sobol' sequences. The FP method, using a linear drift-diffusion model with OU dynamics, is implemented with quasi-random inputs to reduce estimator variance without increasing particle count. Results across homogeneous and inhomogeneous test cases show that FP-Array-RQMC significantly improves convergence rates and reduces estimator errors for large $N$, with mean quantities benefiting most and second-order moments receiving substantial gains in homogeneous settings; in inhomogeneous cases the gains persist but are more modest due to transport and boundary effects. Overall, FP-Array-RQMC enhances efficiency of FP simulations and demonstrates the viability of quasi-random variance reduction in spatially discretized kinetic models, paving the way for extensions to higher-order FP models and polyatomic gases.

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.

Figures (6)

• Figure 1: Convergence of MC and QMC estimators for the first four moments of a uniform random variable. The plot shows the relative RMSE of the estimated moments versus the sample size $N$ on a log-log scale. Reference slopes proportional to $N^{-1/2}$ and $N^{-3/2}$ are added for comparison. QMC consistently outperforms MC.
• Figure 2: Convergence of the average RMSE for a relaxation with constant coefficients. The four panels show the errors for the mean velocity component $y$, the translational energy, and the $\sigma_{xy}$ and $\sigma_{yz}$ components of the stress tensor. The figure compares pseudo-random sampling, normalized pseudo-random sampling, antithetical sampling, control variate, shuffled quasi-random sampling, and quasi-random sampling with the Array-RQMC technique.
• Figure 3: Convergence of the average RMSE for the homogeneous McKean--Vlasov relaxation test case. The four panels show the errors for the mean velocity component $y$, the translational energy, and the $\sigma_{xy}$ and $\sigma_{yz}$ components of the stress tensor. The figure compares pseudo-random sampling, normalized pseudo-random sampling, antithetical sampling, shuffled quasi-random sampling, and quasi-random sampling with the Array-RQMC technique.
• Figure 4: Schematic of the inhomogeneous test configurations. A monoatomic argon gas is initially at rest with $T_0=300\,\mathrm{K}$ between two parallel plates. The left plate is fixed at $T_\text{left}=300\,\mathrm{K}$ and $v_\text{left}=0$. The Couette-flow is obtained by moving the right plate with $v_\text{right}=100\,\mathrm{m\,s^{-1}}$ at constant temperature, while the heat-flux is obtained by heating the right plate to $T_\text{right}=400\,\mathrm{K}$ while keeping it stationary. Dotted lines indicate the spatial cell discretization.
• Figure 5: Convergence of the average RMSE for the inhomogeneous Couette flow test case. The four panels show the errors for the mean velocity component $y$, the translational energy, and the $\sigma_{xy}$ and $\sigma_{yz}$ components of the stress tensor. The figure compares pseudo-random sampling, normalized pseudo-random sampling, antithetical sampling, shuffled quasi-random sampling, and quasi-random sampling with the Array-RQMC technique.