CUR for Implicit Time Integration of Random Partial Differential Equations on Low-Rank Matrix Manifolds
Mohammad Hossein Naderi, Sara Akhavan, Hessam Babaee
TL;DR
The paper addresses the computational challenge of implicit time integration for nonlinear matrix differential equations (MDEs) arising from parametric PDEs by combining dynamical low-rank approximation (DLRA) with a CUR-based residual minimization strategy. It introduces an implicit TDB-CUR framework that solves nonlinear Newton steps only on a small set of strategically chosen columns and rows, using DEIM sampling to select rows/columns and a low-rank correction basis to handle row dependencies. The key contributions include a detailed CUR residual collocation formulation, efficient column/row solvers, and extensions to multistep and diagonally implicit Runge-Kutta schemes, plus oversampling and rank adaptivity to control error. Demonstrations on stochastic advection–diffusion, Burgers’, and Gray-Scott PDEs show that the method achieves high accuracy with substantial speedups over full-order integrations, enabling scalable simulations of nonlinear random PDEs on low-rank manifolds.
Abstract
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.