AI 中文总结
针对 blob 方法的 O(N²) 计算瓶颈，该研究提出随机多速率方法，证明其收敛速率理论上尖锐且优于随机批处理方法，可高效模拟多种非线性扩散方程。
AI 中文摘要
线性和非线性扩散方程出现在一系列具有数学研究价值的现象中，包括慢扩散、快扩散、沙堆动力学、高度约束输运、二维 Navier-Stokes 方程以及概率测度采样动力学。近年来，blob 方法作为数值模拟这类偏微分方程（PDE）的方法受到广泛关注。为解决 blob 方法的 O(N²) 计算瓶颈，我们考虑时空的随机离散化。我们将随机批处理方法与一种新方法（我们称之为随机多速率方法）进行比较。这两种方法都基于优化文献中的经典随机方法，与随机梯度下降和随机坐标下降有诸多相似之处。我们发现，对于线性和非线性扩散方程，在计算复杂度与精度的权衡中，随机多速率方法的性能最佳。一方面，我们证明随机多速率方法以 O(kΔt) 的速率收敛到对应的常微分方程（ODE）系统，与前向欧拉法匹配，并通过实例表明该速率在理论上是尖锐的。另一方面，我们观察到将随机多速率方法应用于扩散 PDE 的 blob 方法所对应的 ODE 时，其收敛速率甚至更优。最后，由于该方法能够模拟包括高度约束输运和沙堆动力学在内的多种非线性扩散方程，它成功捕捉到了少数数值方法可处理的 PDE 的关键特征。
英文摘要
Linear and nonlinear diffusion equations arise in a range of phenomena of mathematical interest, including slow and fast diffusion, sandpile dynamics, height-constrained transport, the two-dimensional Navier-Stokes equation, and dynamics for sampling probability measures. In recent years, blob methods have attracted significant interest as an approach for numerically simulating these types of PDEs. To address the $O(N^2)$ computational bottleneck of blob methods, we consider stochastic discretizations of space and time. We compare the random batch method with a new approach, which we call the random multirate method. Both of these methods build on classical stochastic methods in the optimization literature, with many similarities to stochastic gradient descent and random coordinate descent. We find that, for linear and nonlinear diffusion equations, in the tradeoff between computational complexity and accuracy, the random multirate method has the best performance. On one hand, we prove that the random multirate method converges to the underlying ODE system at a rate of $O(k Δt)$, matching forward Euler, and show by example that this rate is theoretically sharp. On the other hand, we observe even better rates of convergence for the random multirate method when applied to ODEs arising from blob methods for diffusive PDEs. Finally, due to its ability to simulate a wide range of nonlinear diffusion equations, including height-constrained transport and sandpile dynamics, our method succeeds in capturing key features of PDEs for which few numerical approaches exist.
