In this talk, I will present two deblurring methods, one exploits the spatial interactions in images, i.e. the self-similarity; and the other explicitly takes into account the sparse characteristics of natural images and does not entail solving a numerically ill-conditioned backward-diffusion.

In particular, the self-similarity is defined by a weight function, which induces two types of regularization in a nonlocal fashion. Furthermore, we get superior results using preprocessed data as input for the weighted functionals.

The second part of the talk is based on the observation that the sparse coefficients that encode a given image with respect to an over-complete basis are the same that encode a blurred version of the image with respect to a modified basis. An explicit generative model is used to compute a sparse representation of the blurred image, and the coefficients of which are used to combine elements of the original basis to yield a restored image.

A layered tissue model has a thin, epithelial layer supported underneath by a thick, stromal layer. Each layer has its own optical properties which, in turn, provide useful medical diagnostic information. We present an asymptotic analysis of the boundary value problem for the vector radiative transport equation that governs a polarized beam incident on layered tissues. In doing so, we are able to propose novel methods to study the optical properties of epithelial tissue layers which have applications in the early diagnosis of cancer.

This work involves collaborations with Miguel Moscoso, Shelley Rohde, and Julia Clark.

Consider the global optimization problem of minimizing a polynomial function subject to polynomial equalities and/or inequalities. Jacobian SDP Relaxation is the first method that can solve this problem globally and exactly by using semidefinite programming. This solves an open problem in the field of polynomial optimization. Its basic idea is to use the minors of Jacobian matrix of the given polynomials, add new redundant polynomial equations about the minors to the constraints, and then apply the hierarchy of Lasserre's semidefinite programming relaxations. The main result is that this new semidefinite programming relaxation will be exact for a sufficiently high (but finite) order, that is, the global minimum of the polynomial optimization can be computed by solving a semidefinite programming problem.

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.