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

TL;DR

It is shown that a zeroth-order approximation that freezes the importance weights during the velocity-space kick is stable at the expense of introducing bias and a correction for the weight distribution using maximum cross-entropy formulation to ensure conservation laws while minimizing the introduced bias is proposed.

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.

Figures (8)

• Figure 1: Normalized maximum particle weight $\langle \lVert W(t) \rVert_\infty / \lVert W(0) \rVert_\infty \rangle$ as a function of time $t$ for increasing perturbation amplitudes $\alpha$ for VR-PIC without MxE (top) and with MxE (middle), as well as the average number of iterations needed for the MxE algorithm \ref{['alg:ME_optimization']} to converge. Solid lines represent the mean over 100 ensembles with the respective shaded areas indicating the standard deviation ($\pm 1\sigma$).
• Figure 2: Normalized number density ($n/n_0$) and temperature ($T/T_0$) profiles for the Sod's shock tube problem at different times for initial perturbation of $\alpha=0.01$ (top) and $0.2$ (bottom). The black markers represent the reference solution obtained using the PIC simulation averaged over $3 \times 10^6$ and $2 \times 10^5$ ensembles, respectively. The VR-PIC solution with (red) and without maximum cross-entropy correction (blue) are obtained using 100 and 2000 ensembles, respectively.
• Figure 3: Relative statistical error in the density profile as a function of the initial signal strength $\alpha$. The PIC simulation error (black) increases quadratically as the signal $\alpha$ reduces, while the uncertainty of the VR-PIC solution (red) remains constant and orders-of-magnitudes smaller than the one for PIC. The data points were averaged over 100 ensembles and compared to a reference solution obtained with $5\times 10^4$ ensembles.
• Figure 4: Evolution of the normalized charge density $\rho/\rho_0$ for the Landau Damping test case computed using PIC (top) and VR-PIC (middle) method with $10^6$ particles as well as PIC with $10^8$ as the reference (bottom), respectively, at $t/\Delta t=50, 100, 150$ and $200$ time steps.
• Figure 5: Evolution of the normalized temperature $T/T_0$ for the Landau Damping test case computed using PIC (top) and VR-PIC (middle) method with $10^6$ particles as well as PIC with $10^8$ as the reference (bottom), respectively, at $t/\Delta t=50, 100, 150$ and $200$ time steps.

Theorems & Definitions (11)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4