Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC

Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz
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Abstract

This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise.
通过 McKean-Vlasov 动力学向前传播低差异:从 QMC 到 MLQMC
这项研究开发了一种粒子系统,用于近似麦金-弗拉索夫随机微分方程(SDE)。该方法的新颖之处在于,在构建具有耦合噪声和初始条件的粒子系统时,非难涉及低差异序列。数值研究证明,与标准粒子系统相比,本文提出的新方法使平均场极限的弱逼近和强逼近的收敛率分别提高了一倍。这些收敛率在均场常微分方程的简化设置中得到了证明,即对于具有群结构(如 Rank-1 格点)的低差异序列,涉及星差异的适当边界。这一结构提供了一个反向多层次准蒙特卡罗估计器。渐近误差分析表明,所提出的方法优于基于具有独立初始条件和噪声的经典粒子系统的方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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