Preconditioned Low-Rank Riemannian Optimization for Symmetric Positive Definite Linear Matrix Equations
Ivan Bioli, Daniel Kreßner, Leonardo Robol · SIAM Journal on Scientific Computing · 2025
Abstract. This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form [Formula: see text], as they arise from discretized partial differential equations and control problems. One often finds that [Formula: see text] admits good low-rank approximations, in particular when the right-hand-side matrix [Formula: see text] has low rank. For [Formula: see text] terms, the solution of such equations is well studied, and effective low-rank solvers have been proposed, including alternating direction implicit (ADI) methods for Lyapunov and Sylvester equations. For [Formula: see text], several existing methods try to approach [Formula: see text] through combining a classical iterative method, such as the conjugate gradient (CG) method, with low-rank truncation. In this work, we consider a more direct approach that approximates [Formula: see text] on manifolds of fixed-rank matrices through Riemannian CG. One particular challenge is the incorporation of effective preconditioners into such a first-order Riemannian optimization method. We propose several novel preconditioning strategies, including a change of metric in the ambient space, preconditioning the Riemannian gradient, and a variant of ADI on the tangent space. Combined with a strategy for adapting the rank of the approximation, the resulting method is demonstrated to be competitive for a number of examples representative for typical applications. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/IvanBioli/riemannian-spdmatrixeq and in the supplementary materials ( riemannian-spdmatrixeq.zip [40.2MB]). [Formula: see text]