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.

翻译：本文针对时变矩阵微分方程，发展了隐式秩自适应格式。动态低秩逼近（DLRA）是一种基于狄拉克-弗伦克尔时变变分原理捕捉动态低秩结构的著名技术。近年来，因其广泛适用性而备受关注。我们的格式受DLRA非传统鲁棒积分器（即所谓的BUG积分器）秩自适应版本中三步过程的启发。首先，进行预测（基更新）步骤，计算下一时间层的近似列空间和行空间。其次，利用基于隐式求解小核心矩阵的基础过程调用伽辽金演化步骤。最后，根据预设误差阈值进行截断。由于DLRA将微分方程投影到低秩流形的切空间上演化，BUG积分器的误差估计包含难以通过网格细化控制的切向投影（建模）误差。这可能导致含交叉项方程的收敛性问题。为解决此问题，我们提出简单修改：在预测步骤中将显式步截断方法的行空间和列空间与BUG空间合并。此外，我们提出自适应策略：仅当通过显式步截断方法从预测空间获得的解的残差过大时，才计算BUG空间。我们在假设条件下证明了格式的稳定性并估计了局部截断误差。通过各向异性扩散、刚体旋转及两者组合等多个测试对格式进行基准测试，展示了鲁棒的收敛特性。