Interpolatory Dynamical Low-Rank Approximation: Theoretical Foundations and Algorithms
Benjamin Carrel, Daniel Kressner, Hei Yin Lam, Bart Vandereycken
TL;DR
The paper develops DLRA-DEIM, a continuous-time framework that replaces orthogonal tangent projections in dynamical low-rank approximation with DEIM-based oblique projections to address nonlinear evaluation costs. It establishes existence, exactness, and error bounds for the resulting differential inclusion, with an explicit polytope characterization in the QDEIM case, and introduces PRK-DEIM (and PERK-DEIM for exponential integrators) to maintain high-order accuracy at reduced cost. The methodology extends to low-order tensor differential equations in Tucker format, with Tucker PRK-DEIM and tensor-specific DEIM index selection, backed by theoretical error bounds and numerical experiments on nonlinear Schrödinger and Allen–Cahn equations. The results demonstrate substantial computational savings with minimal accuracy loss, broadening the applicability of DLRA to large-scale nonlinear and tensor-structured dynamics.
Abstract
Dynamical low-rank approximation (DLRA) is a widely used paradigm for solving large-scale matrix differential equations, as they arise, for example, from the discretization of time-dependent partial differential equations on tensorized domains. Through orthogonally projecting the dynamics onto the tangent space of a low-dimensional manifold, DLRA achieves a significant reduction of the storage required to represent the solution. However, the need for evaluating the velocity field can make it challenging to attain a corresponding reduction of computational cost in the presence of nonlinearities. In this work, we address this challenge by replacing orthogonal tangent space projections with oblique, data-sparse projections selected by a discrete empirical interpolation method (DEIM). At the continuous-time level, this leads to DLRA-DEIM, a well-posed differential inclusion (in the Filippov sense) that captures the discontinuities induced by changes in the indices selected by DEIM. We establish an existence result, exactness property and error bound for DLRA-DEIM that match existing results for DLRA. For the particular case of QDEIM, a popular variant of DEIM, we provide an explicit convex-polytope characterization of the differential inclusion. Building on DLRA-DEIM, we propose a new class of projected integrators, called PRK-DEIM, that combines explicit Runge--Kutta methods with DEIM-based projections. We analyze the convergence order of PRK-DEIM and show that it matches the accuracy of previously proposed projected Runge-Kutta methods, while being significantly cheaper. Extensions to exponential Runge--Kutta methods and low-order tensor differential equations demonstrate the versatility of our framework.