对称正定矩阵方程的预条件化低秩黎曼优化

本文关注大规模对称正定矩阵方程的数值求解，方程形式为 $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$，其源自离散化的偏微分方程和控制问题。 often find 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.