随机微分方程动态低秩逼近的数值方法——第二部分：随机离散化
Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization
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中文总结 AI 辅助
本文分析随机微分方程动态低秩逼近算法的随机离散化，采用蒙特卡洛方法，给出椭圆扩散下DLR投影分裂的收敛率，并讨论一般扩散情形及正则化影响。
中文摘要 AI 辅助
在本文（第二部分）中，我们从随机离散化的角度分析了第一部分（arXiv:2601.21428）中引入的随机微分方程（SDE）动态低秩逼近（DLRA）的数值算法。具体而言，我们采用包含 $M$ 个样本的蒙特卡洛方法来逼近随机空间及所有相关感兴趣的量。因此，这些算法产生带噪声的相互作用粒子系统，其误差分析并非标准。假设初始条件具有高阶矩和亚高斯尾部，在椭圆扩散情形下，我们为第一部分中提出的DLR投影分裂（DLR Projector Splitting）算法提供了收敛性结果。当完全离散化的Gramian矩阵在整个时间演化过程中保持满秩时，对于较大的 $M$，收敛速率接近通常的蒙特卡洛速率，即 $O(\frac{1}{\sqrt{M}})$。如果在每个时间步对该矩阵进行正则化处理，则对于较大的 $M$，速率接近 $O(\frac{1}{\sqrt[3]{M}})$。另一方面，如果仅在Gramian矩阵的最小奇异值小于给定正阈值时才进行正则化，则随机收敛速率介于上述两个结果之间。此外，对于一般扩散情形，我们提供了一种易于实现的收敛算法。关于DLR Euler-Maruyama和DLR投影分裂用于Euler-Maruyama（EM）的情形，这些结果的推导并非直接，本文也简要讨论了为何这是一项具有挑战性的任务。数值模拟将完善这一分析。
英文摘要
In this second article (Part II), we analyze the numerical algorithms for the Dynamical Low-Rank Approximation (DLRA) of Stochastic Differential Equations (SDEs) introduced in Part I arXiv:2601.21428 under the perspective of the stochastic discretization. Specifically, we employ a Monte Carlo method with $M$ samples to approximate the stochastic space and all the related quantities of interest. Consequently, these algorithms produce noisy interacting particle systems whose error analysis is not standard. Assuming high moments and subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the DLR Projector Splitting for SDEs presented in Part I. When the fully discretized Gramian is of full rank for all the time evolution, then one observes a convergence rate close to the usual Monte Carlo one, i.e. $O(\frac{1}{\sqrt{M}})$, for large $M$. If a regularization of this matrix is employed at each time-step, the rate is close to $O(\frac{1}{\sqrt[3]{M}})$ for large $M$. On the other hand, in case the regularization occurs only when the smallest singular value of the Gramian is smaller than a prescribed positive threshold, the stochastic convergence rate is located between the aforementioned results. Furthermore, in the case of general diffusion, we provide an easy-to-implement convergent algorithm. Concerning the DLR Euler-Maruyama and the DLR Projector Splitting for Euler-Maruyama (EM), these results are not straightforward to derive and a brief discussion on why this is a challenging task is provided, too. Numerical simulations will complete this analysis.
发表机构
- University of Manchester（曼彻斯特大学）
- École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
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