时间相关扩散的无网格蒙特卡洛方法
Grid-Free Monte Carlo for Time-Dependent Diffusion
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- Dartmouth College（达特茅斯学院）
- NVIDIA（英伟达）
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中文总结 AI 辅助
本文提出无网格蒙特卡洛方法，推广WoS和WoSt至瞬态扩散问题，通过时间预算和退出时间采样直接估计任意时刻解，消除网格划分与时间步进偏差。
中文摘要 AI 辅助
许多科学应用需要模拟扩散系统随时间的演化，而不仅仅是其最终的稳态。虽然传统上对复杂几何体上的偏微分方程（PDEs）进行稳态分析已受到昂贵的体积网格划分的阻碍，瞬态分析还进一步需要顺序时间步进和仔细的步长选择。诸如球面行走（WoS）和星面行走（WoSt）等无网格蒙特卡洛求解器避免了这一网格划分瓶颈，但主要局限于稳态问题。我们将WoS（针对纯Dirichlet问题）和WoSt（针对混合Dirichlet-Neumann问题）推广到具有初始条件以及时间相关源项和边界数据的热方程。我们为每次随机游走配备有限的时间预算，并在每个空间步骤采样一个退出时间。如果退出时间超过剩余预算，游走则采样一个内部点并评估初始条件；否则，它以缩减的预算继续，累积源项和边界贡献。我们的主要技术贡献是一套核采样和方差缩减技术，包括一个低偏差、免制表的退出时间采样器和高效的拒绝采样器。与基于网格的瞬态求解器不同，我们的方法无需体积网格划分或顺序时间推进，即可直接估计任意所需时刻的解。它还保留了WoS和WoSt的并行、渐进和输出敏感评估特性，同时完全消除了时间步长选择和时域离散化偏差。最后，我们展示了如何通过共享游走来实现对多个目标时间的高效估计。
英文摘要
Many scientific applications require modeling how diffusive systems evolve over time, not merely their eventual steady states. While conventional steady-state analysis of partial differential equations (PDEs) on complex geometries is already hindered by costly volumetric meshing, transient analysis further requires sequential time stepping and careful step size selection. Grid-free Monte Carlo solvers such as walk on spheres (WoS) and walk on stars (WoSt) avoid this meshing bottleneck but remain largely limited to steady-state problems. We generalize WoS, for pure Dirichlet problems, and WoSt, for mixed Dirichlet--Neumann problems, to heat equations with initial conditions and time-dependent source and boundary data. We equip each random walk with a finite time budget and sample an exit time at every spatial step. If the exit time exceeds the remaining budget, the walk samples an interior point and evaluates the initial condition; otherwise, it continues with a reduced budget, accumulating source and boundary contributions. Our main technical contribution is a suite of kernel sampling and variance reduction techniques, including a low-bias, tabulation-free exit time sampler and efficient rejection samplers. Unlike grid-based transient solvers, our method directly estimates the solution at any requested time without volumetric meshing or sequential time marching. It also retains the parallel, progressive, and output-sensitive evaluation of WoS and WoSt while eliminating time step selection and temporal discretization bias entirely. Finally, we show how sharing walks enables efficient estimates at multiple target times.
