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We consider the problem of large‐scale finite‐sum minimization on Riemannian manifold. We develop a sub‐sampled adaptive trust region method on Riemannian manifolds. Based on inexact information, we adopt adaptive techniques to flexibly adjust the trust region radius in our method. We present the iteration complexity is Omax{εg−2εH−1,εH−3}$$ O\left(\max \left\{{\varepsilon}_g^{-2}{\varepsilon}_H^{-1},{\varepsilon}_H^{-3}\right\}\right) $$ when the algorithm attains an (εg,εH)$$ \left({\varepsilon}_g,{\varepsilon}_H\right) $$‐second‐order stationary point, which matches the result on trust region method. Numerical results for PCA on Grassmann manifold and low‐rank matrix completion are reported to demonstrate the effectiveness of the proposed Riemannian method. [ABSTRACT FROM AUTHOR]

Due to the non-local property of the fractional derivative, the analytic solutions of fractional differential equations are hardly available in most cases. Consequently, researches on the numerical solutions of fractional differential equations are of great importance. In this paper, we will propose a new numerical method to obtain the solutions of the Caputo-type fractional differential equations. The error analysis is given to prove the accuracy of our method when the order $ p $ of the fractional derivative satisfying $ p>1 $. The numerical experiments also show that both linear fractional equations and nonlinear fractional equations can be solved with this method. [ABSTRACT FROM AUTHOR]

This paper presents a differential equation including both random matrices and a Liu process. Then we demonstrate that the solution to this equation exists and is unique. Under the framework of chance theory, problems of optimal control and two‐person zero‐sum game subject to differential equations are considered. An equation of optimality is provided for solving a problem of optimal control. Then equilibrium equations are proposed to identify the saddle‐point of a two‐person zero‐sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including both random matrices and multiple Liu processes. Finally, we utilize the acquired theoretical results to analyze a portfolio selection game problem. [ABSTRACT FROM AUTHOR]