Interpolatory Dynamical Low-Rank Approximation: Theoretical Foundations and Algorithms

TL;DR

This paper introduces DLRA-DEIM, a novel approach combining dynamical low-rank approximation with empirical interpolation, providing a computationally efficient method with strong theoretical guarantees for large-scale matrix differential equations.

Contribution

It develops DLRA-DEIM, a new framework replacing orthogonal projections with data-sparse DEIM projections, and introduces PRK-DEIM, a class of efficient integrators with proven convergence properties.

Findings

DLRA-DEIM is well-posed and captures discontinuities effectively.

PRK-DEIM achieves the same accuracy as existing methods at lower computational cost.

Extensions to exponential Runge--Kutta methods demonstrate framework versatility.

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.