Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean-Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant speed-up (with respect to a standard Monte Carlo method) in the simulation of the Dean-Kawasaki equation. Specifically, we quantify how the speed-up factor increases as the average particle density increases, and show that sizeable speed-ups can be obtained even in regimes of low particle density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean-Kawasaki equation being highly singular.

翻译：涨落流体力学中的随机偏微分方程是描述多粒子系统涨落现象的强有力工具。本文针对该类型随机偏微分方程的核心代表——Dean-Kawasaki方程，提出并分析了一种多层蒙特卡洛方案。我们从解析角度证明并数值验证了该方案在Dean-Kawasaki方程模拟中相较于标准蒙特卡洛方法具有显著加速效果。具体而言，我们量化了加速因子随平均粒子密度增大而增加的特征，并证明即使在低粒子密度区域也能获得可观的速度提升。通过二维数值模拟验证了理论预测。本文结论完全基于分布律而非强空间范数进行表述：这一关键特性使得即使在Dean-Kawasaki方程高度奇异的情况下，仍能实现多层蒙特卡洛加速。