Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than Petrov-Galerkin ones. This is mainly due to the different nature of the projected problems stemming from these two families of methods. While a Galerkin approach leads to the solution of a low-dimensional matrix equation per iteration, a matrix least-squares problem needs to be solved per iteration in a Petrov-Galerkin setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing minimization properties of Petrov-Galerkin schemes. In this paper we introduce a framework that allows for modifying the Galerkin approach by low-rank, additive corrections to the projected matrix equation problem with the two-fold goal of attaining monotonic convergence rates similar to those of Petrov-Galerkin schemes while maintaining essentially the same computational cost of the original Galerkin method. We analyze the well-posedness of our framework and determine possible scenarios where we expect the residual norm attained by two low-rank-modified variants to behave similarly to the one computed by a Petrov-Galerkin technique. A panel of diverse numerical examples shows the behavior and potential of our new approach.

翻译：在所有求解大规模李雅普诺夫矩阵方程的投影方法中，加勒金方法仍比彼得罗夫-加勒金方法更受欢迎。这主要源于这两类方法所生成的投影问题具有不同特性。加勒金方法每次迭代只需求解低维矩阵方程，而彼得罗夫-加勒金方法每次迭代则需解决矩阵最小二乘问题。尽管彼得罗夫-加勒金格式具有吸引人的最小化性质，但这些最小二乘问题的显著计算成本使研究者倾向于加勒金方法。本文提出了一种框架，通过低秩加性修正对加勒金方法生成的投影矩阵方程进行改进，实现双重目标：在保持原始加勒金方法基本相同的计算成本的同时，获得类似彼得罗夫-加勒金格式的单调收敛速率。我们分析了该框架的适定性，并确定了两种低秩修正变体残差范数与彼得罗夫-加勒金技术计算结果相似的预期情景。一系列多样化的数值算例展示了新方法的表现与潜力。