In this paper, we propose a Riemannian Acceleration with Preconditioning (RAP) for symmetric eigenvalue problems, which is one of the most important geodesically convex optimization problem on Riemannian manifold, and obtain the acceleration. Firstly, the preconditioning for symmetric eigenvalue problems from the Riemannian manifold viewpoint is discussed. In order to obtain the local geodesic convexity, we develop the leading angle to measure the quality of the preconditioner for symmetric eigenvalue problems. A new Riemannian acceleration, called Locally Optimal Riemannian Accelerated Gradient (LORAG) method, is proposed to overcome the local geodesic convexity for symmetric eigenvalue problems. With similar techniques for RAGD and analysis of local convex optimization in Euclidean space, we analyze the convergence of LORAG. Incorporating the local geodesic convexity of symmetric eigenvalue problems under preconditioning with the LORAG, we propose the Riemannian Acceleration with Preconditioning (RAP) and prove its acceleration. Additionally, when the Schwarz preconditioner, especially the overlapping or non-overlapping domain decomposition method, is applied for elliptic eigenvalue problems, we also obtain the rate of convergence as $1-C\kappa^{-1/2}$, where $C$ is a constant independent of the mesh sizes and the eigenvalue gap, $\kappa=\kappa_{\nu}\lambda_{2}/(\lambda_{2}-\lambda_{1})$, $\kappa_{\nu}$ is the parameter from the stable decomposition, $\lambda_{1}$ and $\lambda_{2}$ are the smallest two eigenvalues of the elliptic operator. Numerical results show the power of Riemannian acceleration and preconditioning.

翻译：本文针对对称特征值问题——黎曼流形上最重要的测地凸优化问题之一——提出了一种带预条件的黎曼加速方法（RAP），并实现了加速效果。首先，从黎曼流形视角讨论了对称特征值问题的预条件处理。为获取局部测地凸性，我们引入领先角（leading angle）来度量对称特征值问题预条件器的质量。提出了一种新的黎曼加速方法——局部最优黎曼加速梯度法（LORAG），以克服对称特征值问题的局部测地凸性。借鉴RAGD的类似技巧及欧氏空间中局部凸优化的分析方法，我们分析了LORAG的收敛性。将预条件下对称特征值问题的局部测地凸性与LORAG相结合，提出了带预条件的黎曼加速方法（RAP）并证明了其加速效果。此外，当椭圆特征值问题采用Schwarz预条件器（特别是重叠或非重叠区域分解法）时，我们得到了收敛速率$1-C\kappa^{-1/2}$，其中$C$是与网格尺寸和特征值间隙无关的常数，$\kappa=\kappa_{\nu}\lambda_{2}/(\lambda_{2}-\lambda_{1})$，$\kappa_{\nu}$为稳定分解参数，$\lambda_{1}$和$\lambda_{2}$是椭圆算子的最小两个特征值。数值结果展示了黎曼加速与预条件的强大性能。