Dynamical low-rank approximation allows for solving large-scale matrix differential equations (MDEs) with significantly fewer degrees of freedom and has been applied to a growing number of applications. However, most existing techniques rely on explicit time integration schemes. In this work, we introduce a cost-effective Newton's method for the implicit time integration of stiff, nonlinear MDEs on low-rank matrix manifolds. Our methodology is focused on MDEs resulting from the discretization of random partial differential equations (PDEs). Cost-effectiveness is achieved by solving the MDE at the minimum number of entries required for a rank-$r$ approximation. We present a novel CUR low-rank approximation that requires solving the parametric PDE at $r$ strategically selected parameters and $\mathcal{O}(r)$ grid points using Newton's method. The selected random samples and grid points adaptively vary over time and are chosen using the discrete empirical interpolation method or similar techniques. The proposed methodology is developed for high-order implicit multistep and Runge-Kutta schemes and incorporates rank adaptivity, allowing for dynamic rank adjustment over time to control error. Several analytical and PDE examples, including the stochastic Burgers' and Gray-Scott equations, demonstrate the accuracy and efficiency of the presented methodology.

翻译：动态低秩逼近能够以显著减少的自由度数求解大规模矩阵微分方程，并已在越来越多的应用中得到使用。然而，现有技术大多依赖于显式时间积分格式。本文针对刚性非线性矩阵微分方程在低秩矩阵流形上的隐式时间积分，提出了一种计算高效的牛顿法。我们的方法主要关注由随机偏微分方程离散化得到的矩阵微分方程。计算高效性通过以秩-$r$逼近所需的最少条目数求解矩阵微分方程来实现。我们提出了一种新颖的CUR低秩逼近方法，该方法要求使用牛顿法在$r$个策略性选择的参数和$\mathcal{O}(r)$个网格点上求解参数化偏微分方程。所选的随机样本和网格点随时间自适应变化，并通过离散经验插值法或类似技术进行选取。所提出的方法适用于高阶隐式多步法和龙格-库塔格式，并包含秩自适应性，允许随时间动态调整秩以控制误差。包括随机Burgers方程和Gray-Scott方程在内的若干解析与偏微分方程算例验证了所提方法的精确性与效率。