In this talk, I will present two kinds of numerical methods for mathematical models in biological pattern formation problems. The first method is the weighted essentially non-oscillatory (WENO) method for solving the nonlinear chemotaxis models. Chemotaxis is the phenomenon in which cells or organisms direct their movements according to certain gradients of chemicals in their environment. Chemotaxis plays an important role in many biological processes, such as bacterial aggregation, early vascular network formation, among others. While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this work, we combined two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. The second method is the Krylov implicit integration factor (IIF) method for nonlinear reaction–diffusion and advection-reaction-diffusion equations in pattern formations. Integration factor methods are a class of ‘‘exactly linear part’’ time discretization methods. Efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. Here, I will present our results in solving this problem by applying the Krylov subspace approximations to the matrix exponential operator. We applied this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction–diffusion equations. Then we extended the Krylov IIF method to solve advection-reaction-diffusion PDEs and achieved high order accuracy. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the methods.

Newly developed weak Galerkin finite element methods will be introduced for solving partial differential equations. Like discontinuous Galerkin methods, weak Galerkin finite element methods allow to use discontinuous functions in finite element procedure which makes weak Galerkin methods highly flexible. Unlike DG methods, weak Galerkin finite element methods enforce weak continuity of variables through well defined discrete differential operators. Therefore, weak Galerkin methods are parameter independent and absolutely stable. Error analysis and numerical experiments are presented.

Whether it is a bottle of soda, the tires of a car or the human body: Objects in everyday life can be described by only a few parameters, like temperature, pressure or volume. But how is this possible, if each of these systems are complex assemblies of atoms and molecules giving rise to a vast number of  coordinates, on the order of 10^23?

Ergodic theory is the mathematical attempt to provide an answer to this fundamental question. In this talk we will tackle the problem based on a prominent example - the Einstein model for a solid. It will be shown that this problem is reduced to studying rotations on a circle, for which we will prove a version of the ergodic theorem.

In this talk, we shall consider the near-invisibility cloaking in acoustic scattering by non-singular transformation media. A general lossy layer is included into our construction. We are especially interested in the cloaking of active/radiating objects. Our results on the one hand show how to cloak active contents more efficiently, and on the other hand indicate how to choose the lossy layer optimally.