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This work is concerned with the efficient optimization method for solving a large class of optimal mass transport problems. An inexact primal-dual algorithm is presented from the time discretization of a proper dynamical system, and by using the tool of Lyapunov function, the global (super-)linear convergence rate is established for function residual and feasibility violation. The proposed algorithm contains an inner problem that possesses strong semismoothness property and motivates the use of the semismooth Newton iteration. By exploring the hidden structure of the problem itself, the linear system arising from the Newton iteration is transferred equivalently into a graph Laplacian system, for which a robust algebraic multigrid method is proposed and also analyzed via the famous Xu--Zikatanov identity. Finally, numerical experiments are provided to validate the efficiency of our method.

This paper provides a self-contained ordinary differential equation solver approach for separable convex optimization problems. A novel primal-dual dynamical system with built-in time rescaling factors is introduced, and the exponential decay of a tailored Lyapunov function is established. Then several time discretizations of the continuous model are considered and analyzed via a unified discrete Lyapunov function. Moreover, two families of accelerated proximal alternating direction methods of multipliers are obtained, and nonergodic optimal mixed-type convergence rates shall be proved for the primal objective residual, the feasibility violation and the Lagrangian gap. Finally, numerical experiments are provided to validate the practical performances.

In this paper, we investigate fast algorithms in the small fraction order regime to approximate the Caputo derivative $^C_0D_t^\alpha u(t)$ when $\alpha$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme proposed in [26], and we show that the latter is superior when $\alpha$ is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as $\alpha$ gets small. Hence, only minimal cost for the fast evaluation is required in the small $\alpha$ regime, which matches prevailing protocols in engineering practice. We also present improved stability and error analysis of FIDR for solving linear fractional diffusion equations, which achieves clear dependence of the error bound on the fraction order $\alpha$. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when $\alpha$ is small., Comment: 42 pages, 5 figures, submitted to Journal of Scientific Computing

This paper presents a neural network based method Multi-Task Affect Net(MTANet) submitted to the Affective Behavior Analysis in-the-Wild Challenge in FG2020. This method is a multi-task network and based on SE-ResNet modules. By utilizing multi-task learning, this network can estimate and recognize three quantified affective models: valence and arousal, action units, and seven basic emotions simultaneously. MTANet achieve Concordance Correlation Coefficient(CCC) rates of 0.28 and 0.34 for valence and arousal, F1-score of 0.427 and 0.32 for AUs detection and categorical emotion classification., Comment: submitted to ABAW challenge in FG2020