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.

翻译：本研究关注大规模对称正定矩阵方程 $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$ 的数值求解问题，此类方程常见于离散化偏微分方程及控制问题。当右侧矩阵 $F$ 具有低秩特性时，解 $X$ 通常存在良好的低秩近似。对于 $\ell \le 2$ 项的情形，此类方程求解已有深入研究，并提出了有效的低秩求解器，包括针对李雅普诺夫方程和西尔维斯特方程的交替方向隐式（ADI）方法。对于 $\ell > 2$ 的情形，现有方法多尝试通过结合经典迭代法（如共轭梯度法）与低秩截断来逼近 $X$。本文采用更直接的途径，通过黎曼共轭梯度法在固定秩矩阵流形上逼近 $X$。其中关键挑战在于如何将高效预处理策略融入此类一阶黎曼优化方法。我们提出了若干创新的预处理策略，包括环境空间中的度量变换、黎曼梯度预处理以及切空间上的ADI变体方法。结合近似秩自适应策略，数值实验表明所提方法在多个典型应用场景中具有显著竞争力。