Speaker
Description
In this talk we focus on the numerical solution of large-scale symmetric positive definite matrix equations of the form $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$, which arise from discretized partial differential equations and control problems. These equations frequently admit low-rank approximations of the solution $X$, particularly when the right-hand side matrix $F$ has low rank. For cases where $\ell \leq 2$, effective low-rank solvers have been developed, 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 talk, we consider a more direct approach that approximates $X$ on manifolds of fixed-rank matrices through Riemannian nonlinear CG. A significant challenge is the integration of effective preconditioners into this first-order Riemannian optimization method. We propose 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. Along with a rank adaptation strategy, the proposed method demonstrates competitive performance on a range of representative examples.
References
[1] Ivan Bioli, Daniel Kressner, and Leonardo Robol. Preconditioned Low-Rank Riemannian Optimization for Symmetric Positive Definite Linear Matrix Equations. Aug. 29, 2024. arXiv:2408.16416[math.NA].
