The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn– Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method.

Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.

LetMbe a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1, 2, . . . , n, is defined as the k-th ele- mentary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a con- formal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.

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In the research of computer vision and machine perception, 3D objects are usually represented by 2-manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct iso-contours, bisectors, and Voronoi diagrams of point sites onM, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams onMexhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesicmetric- based Voronoi diagrams onM, this paper studies the analytic structure of iso-contours, bisectors, and Voronoi diagrams onM. After a necessary preprocessing of model M, practical algorithms are proposed for quickly obtaining full information about iso-contours, bisectors, and Voronoi diagrams on M. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications—surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis—are presented that show the potential power of the proposed algorithms in pattern analysis.