My major research areas are: Multivariate Splines and Their Applications. Multivariate splines are piecewise polynomial functions defined on a collection of triangles in 2D or tetrahedrons in 3D and spherical triangles in the spherical setting. These functions have many nice properties such as efficiency in computation and approximation. They are very similar to univariate spline functions. I have implemented them in MATLAB and can use them for many applications: scattered data fitting, numerical solution of linear and nonlinear PDE, e.g., fluid flows, image enhancements, surface design, and data forecasting, ... Details can be found in my publications.
My other research interests are: 1) Wavelets, in particular, wavelets and wavelet frames in the multivariate setting. For example, I have used multivariate box splines to construct compactly supported biorthogonal wavelets of any regularity, compactly supported tight wavelet frames of any regularity and apply them for image processing. 2) Sparse solutions and Matrix Completion, in particular, the sparse solution of underdetermined linear system as well as matrix completion. 3) Generalized Barycentric coordinates (GBC) and Polygonal Splines. See my paper on GBC and polygonal splines. One of major results in the paper is the convergence of numerical solution of Poisson equation under pentagonal refinement. The rate of convergence is faster than that of standard finite element method based on uniform refinement of triangulation. Details can be found in the manuscript.
My current research interests are 1) GBC(generalized barycentric coordinates) for polygonal splines; 2) Matrix and Data Tensor Completion; 3) Numerical Solution of Monge-Ampere equations; 4) Subspace Clustering and Dictionary Learning; and etc..