In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. We invented numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow. We also showed broad applications using discrete Ricci flow in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.

In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of $O(h^{k+2})$, when piecewise polynomials of degree $k$ are used. We then prove that the highest superconvergence rate of the DG solution itself is $O(h^{k+\frac 32})$ as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.

In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

The problem of compressive-sensing (CS) L2-L1-TV reconstruc- tion of magnetic resonance (MR) scans from undersampled k-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regu- larization parameters are given, the solution for an MR reconstruction model is xed and is less eective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the pro- posed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via exible parameter slection.

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher-order) finite elements. This method can achieve high-order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright