An increasing rank Riemannian method for generalized Lyapunov equations
Authors
Zhenwei Huang and Wen Huang
Abstract
In this paper, we consider finding a low-rank approximation to the solution of a large-scale generalized Lyapunov matrix equation in the form of $A X M + M X A = C$, where $A$ and $M$ are symmetric positive definite matrices. An algorithm called an Increasing Rank Riemannian method for generalized Lyapunov equation (IRRLyap) is proposed by merging the increasing rank technique and Riemannian optimization techniques on the quotient manifold $\mathbb{R}_*^{n \times p} / \mathcal{O}_p$. To efficiently solve the optimization problem on $\mathbb{R}_*^{n \times p} / \mathcal{O}_p$, a line-search-based Riemannian inexact Newton method is developed with its global convergence and local superlinear convergence rate guaranteed. Moreover, we investigate the influence of the existing three Riemannian metrics on $\mathbb{R}_*^{n \times p} / \mathcal{O}_p$ and derive new preconditioners which takes $M \neq I$ into consideration. Numerical experiments show that IRRLyap with one of the Riemannian metrics is most efficient and robust in general and is preferable compared to the tested state-of-the-art methods when the lowest rank solution is desired.
Key words
Generalized Lyapunov equations; Riemannian optimization; Low-rank approximation; Riemannian truncated Newton's method; Increasing rank method;
Status
Submitted
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