VR-PIC: An entropic variance-reduction method for particle-in-cell solutions of the Vlasov-Poisson equation

TL;DR

This paper introduces VR-PIC, a variance reduction technique for particle-in-cell simulations of the Vlasov-Poisson equation, which improves computational efficiency while maintaining accuracy through a maximum cross-entropy correction.

Contribution

It extends an entropic variance reduction framework to PIC methods, proposing a bias correction that preserves conservation laws with minimal code modifications.

Findings

Significant speed-up in low signal regimes.

Maintains conservation laws with bias correction.

Effective in test cases like Sod's shock tube and Landau damping.

Abstract

We extend the recently developed entropic and conservative variance reduction framework [M. Sadr, N. G. Hadjiconstantinou, A variance-reduced direct Monte Carlo simulation method for solving the Boltzmann equation over a wide range of rarefaction, Journal of Computational Physics 472 (2023) 111677.] to the particle-in-cell (PIC) method of solving Vlasov-Poisson equation. We show that a zeroth-order approximation that freezes the importance weights during the velocity-space kick is stable at the expense of introducing bias. Then, we propose a correction for the weight distribution using maximum cross-entropy formulation to ensure conservation laws while minimizing the introduced bias. In several test cases including Sod's shock tube and Landau damping we show that the proposed method maintains the substantial speed-up of variance reduction method compared to the PIC simulations in the low signal regime with minimal changes to the simulation code.