High-Order Implicit Low-Rank Method with Spectral Deferred Correction for Matrix Differential Equations

In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. Low rank techniques exploit the fact that the solution to a differential equation can be approximated by a low-rank matrix, therefore, by only storing and operating on the low-rank factors, we can save computational cost and storage. The dynamic low-rank approximation (DLRA) is a well-known technique to realize such computational advantages for time-dependent problems. In [1], a numerical method is proposed to correct the modeling error of the basis update and the Galerkin (BUG) method, which is a computational approach for DLRA. This method (merge-BUG/mBUG method) has been demonstrated to compute numerically convergent solutions for general advection-diffusion problems. However, similar to the original BUG method, it is only first-order accurate in time. On the other hand, SDC is a general framework to construct high-order time discretizations from low-order time discretizations. In this paper, we explore using SDC to elevate the convergence order of the mBUG method. In SDC, we start by computing a first-order solution by mBUG, and then perform successive updates by computing low-rank solutions to the Picard integral equation. Rather than a straightforward application of SDC with mBUG, we propose two aspects to improve computational efficiency. The first is to reduce the intermediate numerical rank by detailed analysis of dependence of truncation parameter on the correction levels. It turns out that truncation tolerances depends on the correction levels, and larger tolerance can be used in the initial and early correction levels, which will reduce the numerical rank of the intermediate solutions. The second aspect is a careful choice of subspaces in the successive correction to avoid inverting large linear systems (from the K- and L-steps in BUG). We prove that the resulting scheme is high-order accurate for the Lipschitz continuous and bounded dynamical system. We further consider numerical rank control in our framework by comparing two low-rank truncation strategies: the hard truncation strategy by truncated singular value decomposition and the soft truncation strategy by soft thresholding. We demonstrate numerically that soft thresholding offers better rank control in particular for higher-order schemes for weakly (or non-)dissipative problems. Numerical results on benchmark tests are provided validating the performance of the method.