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Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n ² ) to O(n). For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint ($\nabla \cdot \B = 0$) of its magnetic field $\B$. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements \cite{Li2005}, and another approach of the divergence cleaning technique given by Dedner et al. \cite{Dedner2002}. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

In this paper, a structure-preserving algorithm is developed for the computation of a semi-stabilizing solution of a Generalized Algebraic Riccati Equation (GARE). The semi-stabilizing solution of GAREs has been used to characterize the solvability of the (J, J ′ )-spectral factorization problem in control theory for general rational matrices which may have poles and zeros on the extended imaginary axis. The main difficulty in solving such a GARE lies in the fact that its associated Hamiltonian/skew-Hamiltonian pencil has eigenvalues on the extended imaginary axis. Consequently, it is not clear which eigenspace of the associated Hamiltonian/skew-Hamiltonian pencil can characterize the desired semi-stabilizing solution. That is, it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute. Hence, the well-known generalized eigenspace approach for the classical algebraic Riccati equations cannot be employed directly. The proposed algorithm consists of a structure-preserving doubling algorithm (SDA) and a postprocessing procedure to determine the desired eigenvectors and principal vectors corresponding to the purely imaginary and infinite eigenvalues. Under mild assumptions, linear convergence of rate 1/2 for the SDA is proved. Numerical experiments illustrate that the proposed algorithm performs efficiently and reliably.