Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than minimal-residual 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 minimal-residual setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing properties of minimal-residual 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 minimal-residual 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 minimal-residual 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Lyapunov矩阵方程的投影方法中，Galerkin方法仍比最小残差法更受欢迎。这主要是由于这两类方法产生的投影问题性质不同：Galerkin方法每步迭代需要求解低维矩阵方程，而最小残差方法每步需处理矩阵最小二乘问题。尽管最小残差方案具有吸引人的特性，但求解这些最小二乘问题的显著计算成本促使研究者更倾向于Galerkin方法。本文提出一种框架，允许通过对投影矩阵方程问题进行低秩加性修正来改进Galerkin方法，旨在达成双重目标：在保持原始Galerkin方法基本相同的计算成本的同时，实现与最小残差方案类似的单调收敛速率。我们分析了该框架的适定性，并确定了两种低秩修正变体预期能达到与最小残差技术相似残差范数表现的可能情景。一系列多样化的数值算例展示了新方法的行为特性与应用潜力。