Adaptive sampling accelerates the hybrid deviational particle simulations

TL;DR

This paper tackles the costly sampling step in the hybrid deviational-particle (HDP) method for the Vlasov–Poisson–Landau system by introducing HDP‑AS, an adaptive sampling strategy. HDP‑AS first builds an adaptive, piecewise constant approximation of the high-dimensional source term $\Delta m_k(\mathbf{v}) = \Delta t\left(Q(f_p,m) - Q(f_n,m)\right)$ using a point distribution from TA collisions and a recursive binary partition guided by the mixture discrepancy $D^{\text{mix}}$, then samples deviational particles directly from the approximation without rejection. Replacing the NP-hard star discrepancy with $D^{\text{mix}}$ enables efficient adaptivity, yielding roughly a tenfold speedup over HDP while preserving accuracy across linear and nonlinear Landau damping, two-stream instability, bump-on-tail, and Rosenbluth test problems. The HDP‑AS framework thus significantly reduces the sampling bottleneck in HDP and can be applied to other high-dimensional sampling tasks in kinetic simulations.

Abstract

To avoid ineffective collisions between the equilibrium states, the hybrid method with deviational particles (HDP) has been proposed to integrate the Vlasov-Poisson-Landau system, while leaving a new issue in sampling deviational particles from the high-dimensional source term. In this paper, we present an adaptive sampling (AS) strategy that first adaptively reconstructs a piecewise constant approximation of the source term based on sequential clustering via discrepancy estimation, and then samples deviational particles directly from the resulting adaptive piecewise constant function without rejection. The mixture discrepancy, which can be easily calculated thanks to its explicit analytical expression, is employed as a measure of uniformity instead of the star discrepancy the calculation of which is NP-hard. The resulting method, dubbed the HDP-AS method, samples deviational particles through adaptive sampling instead of the acceptance-rejection method in the original HDP method. In the Landau damping, two stream instability, bump on tail and Rosenbluth's test problems, the HDP-AS method runs approximately ten times faster than the HDP method while keeping the same accuracy.

Figures (12)

• Figure 1: The wall time consumed by sampling new deviational particles, simulating collisions via TA and others for HDP and HDP-AS in the linear Landau damping corresponding to Figure \ref{['0.01dist']}. We find that sampling new deviational particles takes up only 22% (89.91s/403.26s) of the total wall time in HDP-AS, whereas nearly 90% (4120.41s/ 4629.91s) of the total wall time in HDP.
• Figure 2: One-dimensional schematic diagram: The HDP-AS method samples new particles from an adaptive piecewise constant approximation without rejection while the HDP method employs the acceptance-rejection sampling Yan2015Yan2016.
• Figure 3: Linear Landau damping: Snapshots of $p_{2d}(x, v_1, t_{fin})$ produced by HDP and HDP-AS with $\alpha = 0.01$, $N_{\text{eff }} = 5\times 10^{-6}$, $N_{\text{eff }}^F = 1\times 10^{-5}$ and $t_{fin} = 5$.
• Figure 4: Linear Landau damping with $\alpha = 0.01$, $N_{\text{eff }} = 5\times 10^{-6}$, $N_{\text{eff }}^F = 1\times 10^{-5}$ and $t_{fin} = 5$: Histogram of the acceptance rates in HDP for all grid points $x_k$ and time $t$. Over 73% of the acceptance rates less than 0.01, and over 95% of them less than 0.1. Low acceptance rates lead to low efficiency in the acceptance-rejection sampling within the HDP method.
• Figure 5: Linear Landau damping with $\alpha = 0.01$, $N_{\text{eff }} = 5\times 10^{-6}$, $N_{\text{eff }}^F = 1\times 10^{-5}$ and $t_{fin} = 5$: The projection of adaptive piecewise constant approximation $\overline{ \Delta m_k}(\bm{v})$ in $v_1v_2$-plane with $k = 50$ at different instants. In accordance with the changes of the distribution $f(x, \bm{v}, t)$ (see Figure \ref{['0.01dist']} for the 2-D projections in $v_1x$-plane), the projection exhibits significant changes in the region $(v_1, v_2)\in \left[-2,2\right]\times \left[-2,2\right]$, and thus we depict the snapshots only for $v_2 \in \left[-2,2\right]$.

Theorems & Definitions (1)

• Remark 1