Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

TL;DR

This paper introduces implicit, rank-adaptive low-rank time-stepping methods for matrix differential equations, improving stability and convergence by merging spaces and adaptively controlling residuals, with proven error estimates and robust benchmarks.

Contribution

It proposes a novel modification to the BUG integrator for DLRA, enhancing stability and adaptivity in low-rank matrix differential equation solvers.

Findings

Stable and convergent schemes demonstrated on anisotropic diffusion and rotation problems.

Adaptive strategy effectively controls residuals and improves robustness.

Proven local truncation error estimates for the proposed methods.

Abstract

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac-Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using a base implicit solve for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. We prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.