Preconditioned Low-Rank Riemannian Optimization for Symmetric Positive Definite Linear Matrix Equations
Ivan Bioli, Daniel Kressner, Leonardo Robol
TL;DR
This work addresses the numerical solution of large-scale multiterm symmetric positive definite matrix equations of the form $A_1 X B_1^\top + \cdots + A_\ell X B_\ell^\top = F$. It proposes a direct low-rank approach on the fixed-rank matrix manifold $\mathcal{M}_r$ using a preconditioned Riemannian Nonlinear Conjugate Gradient method, augmented by various preconditioning strategies and a tangential ADI (tangADI) scheme to approximate inverse actions efficiently. A rank-adaptive framework (RRAM) is developed to balance accuracy and rank growth, combining fixed-rank Riemannian optimization with strategic rank updates. Numerical tests on 2D PDE discretizations, stochastic Galerkin problems, and a modified Rail Lyapunov-type problem demonstrate that the proposed methods—particularly when paired with tangADI and rank adaptivity—achieve competitive wall-clock times and scalable performance compared to iterate-and-truncate approaches and truncated CG. The results highlight the practical impact of structure-exploiting, preconditioned Riemannian optimization for large multiterm matrix equations in PDEs and control applications.
Abstract
This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$, as they arise from discretized partial differential equations and control problems. One often finds that $X$ admits good low-rank approximations, in particular when the right-hand side matrix $F$ has low rank. For $\ell \le 2$ 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 $\ell > 2$, several existing methods try to approach $X$ 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 $X$ 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.