标题： 高阶隐式低秩方法与谱延迟校正用于矩阵微分方程 显示英文标题

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

摘要： 在本文中，我们开发了一种使用谱延迟校正（SDC）的低秩方法，以实现高阶时间精度来计算线性矩阵微分方程。 在[1]中，提出了一种低秩数值方法，以校正基函数更新和Galerkin（BUG）方法的建模误差，这是一种DLRA的计算方法。 该方法（合并BUG/mBUG方法）已被证明对于一般的对流-扩散问题是一阶收敛的。 在本文中，我们探讨使用SDC来提高mBUG方法的收敛阶数。 在SDC中，我们首先通过mBUG计算一阶解，然后通过计算Picard积分方程的低秩解进行连续更新。 与直接应用SDC结合mBUG的方法不同，我们提出了两个方面来提高计算效率。 第一个方面是通过详细分析截断参数对校正层次的依赖性，以降低中间数值秩。 第二个方面是在连续校正中仔细选择子空间，以避免求解大型线性系统（来自BUG中的K-和L-步骤）。 我们证明了所得到的方案对于利普希茨连续且有界的动力系统是高阶准确的。 我们在框架中通过比较两种低秩截断策略来考虑数值秩控制：通过截断奇异值分解的硬截断策略和通过软阈值处理的软截断策略。 我们通过数值实验表明，特别是在针对弱（或非）耗散问题的高阶方案中，软阈值处理提供了更好的秩控制。 显示英文摘要

摘要： 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. In [1], a low rank 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 be first order convergent for general advection-diffusion problems. 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. 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 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.