A cross algorithm for implicit time integration of random partial differential equations on low-rank matrix manifolds

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), where the columns of the MDE can be solved independently. Cost-effectiveness is achieved by solving the MDE at the minimum number of entries required for a rank- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> approximation. We present a novel cross low-rank approximation that requires solving the parametric PDE at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> strategically selected parameters and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 multi-step 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.