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

TL;DR

This work addresses solving linear matrix differential equations $\frac{d}{dt}X(t)=F(X(t),t)$ with a low-rank structure, where $F(X,t)=\sum_j A_j X B_j^T + G(t)$. It introduces two implicit, rank-adaptive schemes, Merge and Merge-adapt, that merge the explicit-step truncation spaces with BUG-prediction spaces to neutralize the tangent-projection (modeling) error and to ensure robust convergence even for cross-term operators. The authors prove stability and derive local truncation-error estimates, and they demonstrate through solid-body rotation and anisotropic-diffusion benchmarks that the methods achieve comparable accuracy to implicit Euler while often reducing rank growth and computation time. The adaptive approach selectively computes BUG spaces based on a residual check, offering efficiency gains for moderately stiff problems and paving the way for higher-order and tensor-extended formulations.

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.

Figures (4)

• Figure 1: The solution to the solid body rotation problem computed using the Merge method (left), the a highly accurate reference solution (middle) and the BUG method (right). The BUG method does not see the rotation (it is outside the tangent space) and remains stationary.
• Figure 2: Displayed is the truncated rank for the Merge, Merge-adapt and Implicit Euler method for the problem with solid body rotation problem. Here "Fail for MA" is the indicator where the BUG space is needed for the Merge-adapt method. For the Implicit Euler method we constantly use the threshold $\Delta t^2$ when computing the rank via the truncated SVD. All the computations are done with $m_1 = m_2 = 799$ and $n_{T} = 320.$
• Figure 3: Displayed is the truncated rank for the Merge, Merge-adapt and Implicit Euler method for the problem with rotation and anisotropic diffusion. Here "Fail for MA" is the indicator where the BUG space is needed for the Merge-adapt method. For the Implicit Euler method we constantly use the threshold $\Delta t^2$ when computing the rank via the truncated SVD. All the computations are done with $m_1 = m_2 = 799$ and $n_{T} = 320.$
• Figure 4: Displayed is the truncated rank for the Merge, Merge-adapt and Implicit Euler method for the problem with anisotropic diffusion initial data $\sin( \pi x_1) \sin(\pi x_2),$ (left) and $\sin(2 \pi x_1) \sin(2\pi x_2)$ (right) and the solid body rotation problem (right). Here "Fail for MA" is the indicator where the BUG space is needed for the Merge-adapt method. For the Implicit Euler method we constantly use the threshold $\Delta t^2$ when computing the rank via the truncated SVD. All the computations are done with $m_1 = m_2 = 799$ and $n_{T} = 320.$

Theorems & Definitions (6)

• Theorem 3.1
• proof
• Theorem 3.2: Stability for semi-bounded operator
• proof
• Lemma 3.3: Local truncation error with cheap prediction space
• proof