Preconditioned Low-Rank Riemannian Optimization for Symmetric Positive Definite Linear Matrix Equations

This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form A₁XB₁^ + A₂XB₂^ + + A_ X B_^ = 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 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 > 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.