An efficient stochastic particle method for high-dimensional nonlinear PDEs

TL;DR

This work tackles the curse of dimensionality in nonlinear PDEs by introducing a stochastic particle method (SPM) derived from the weak formulation of the Lawson-Euler time discretization. SPM represents the solution with real-valued weighted particles X_{t_m} = (1/N) sum_i w_i δ_{x_i} and employs adaptive particle motion, resampling, and reweighting, together with a virtual uniform grid (VUG) for efficient nonlinear term evaluation. The method demonstrates strong adaptivity and scalability to moderately high dimensions, achieving competitive accuracy on 6-D Allen-Cahn and 7-D HJB problems, with clear trade-offs between memory, time, and accuracy. Overall, SPM provides a principled, non-learning-based alternative to high-dimensional PDE solvers that leverages adaptive sampling and weak-form reconstruction to mitigate CoD in practice.

Abstract

Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

Figures (11)

• Figure 1: VUG: Only blue grids, containing the particles, are stored, and the corresponding grid coordinates for all particles are presented on the right.
• Figure 2: Example \ref{['exm:vug']}: The relative $L^2$ error $\mathcal{E}_2[p]$ against the side length $h$ for $d=6$ and $N = 4\times 10^7$.
• Figure 3: Strategy A (preliminary strategy) v.s. Strategy B (improved strategy): The relocating technique (see Eq. \ref{['x relocating']}) adopted by Strategy B may reduce the variance in long-time simulations. Here we set $N=1.6\times 10^7$, $\tau=0.1$ and $h=0.1$.
• Figure 4: The 6-D Allen-Cahn equation with diffusion coefficient $c=1$: Filled contour plots of $M(x_1, x_2, 2)$ defined in Eq. \ref{['P&M']} for (a) the reference solution given in Eq. \ref{['allencahnexact']} and (b) the numerical solution produced by SPM with $N=4\times 10^8$. The agreement between them is evident. SPM shows a highly adaptive characteristic since the particles with positive and negative weights concentrate in the area with positive and negative values of the solution, respectively. This is clearly demonstrated in (c) where we have randomly chosen $10^3$ particles from all samples at the final instant $T=2$ and projected their locations onto the $x_1x_2$-plane.
• Figure 5: The 6-D Allen-Cahn equation: Filled contour plots of $M(x_1, x_2, 2)$ for (a) $c=0.1$ and (b) $c=1.5$. As the diffusion coefficient $c$ increases, the support of the solution expands and the memory usage correspondingly increases as shown in (c). Here we set $N=4\times 10^8$.

Theorems & Definitions (6)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Example 3.1