Multivariate Splines My main interest lies in the theory and application of multivariate splines which I have studied for over twenty years. One of the reasons that multivariate splines have interested me for so long is that it is a fundamental tool for approximating any known or unknown functions. Typical applications are scattered data fitting and interpolations over scattered locations, numerical solutions of partial differential equations over domain of irregular shape, image enhancements, and data forecasting. Numerical Analysis, Scientific Computation, and Applied Mathematics in general need such a tool. In theory, I mainly study the approximation properties of multivariate splines, construction of locally supported spline functions, approximation properties of data fitting using multivariate splines, computational schemes using multivariate splines for applications. A typical paper that represents my work can be found in On the Approximation Power of Bivariate Splines. They have various applications in data fitting and hole filling in Computer Aided Geometric Design, numerical solution of PDE, e.g. Fluid Flow Simulation, and in geodetic application, e.g. geopotential reconstruction. Bivariate Splines can be used for image enhancements as seen in these numerical results. I have used bivariate splines for statistical applications, e.g. to forecast ozone concentration level at Atlanta based on spatial data around Atlanta. We are able to predict the level of ozone concentration very well. In particular, a usefulness of these multivariate splines can be seen in a Ph.D. dissertation at the aerospace engineering department of Delft University of Technology, Netherlands. Their usage for a global system of identification based on a NASA wind tunnel dataset and other data sets demonstrates their excellence, much better than other data fitting methods. See Visser's Dissertation for detail. For another example, trivariate splines have been useful for study of extending battery life by finding optimal parameters from spline interpolation of experimental data values.
"Spline Functions on Triangulations" Larry Schumaker and I wrote a monograph "Spline Functions on Triangulations" together which was published by Cambridge University Press in 2007.
Wavelets I have used multivariate box splines to construct various wavelet functions and studied their application for image processing. My interest is to construct various compactly supported biorthogonal wavelets, tight wavelet frames, orthonormal wavelets in Sobolev spaces, and pre-wavelets in the multivariate setting using multivariate box splines. The regularity of these wavelets are inherited from the box spline functions used. Tight wavelet frames based on bivariate box splines are implemented and used for image edge detection and denoising. In addition, orthonormal wavelets, orthonormal multi-wavelets were constructed.
PDE/Numerical PDE I have used bivariate splines to numerically solve 2D Navier-Stokes equations, 2D nonlinear biharmonic equations associated to a model for thin-films which are similar to Ginzburg-Laudau equations. I have used bivariate splines, finite element and finite difference methods to solve time dependent and steady state nonlinear PDE associated with ROF model for image denoising. In addition to the use of wavelets for image processing, I am also interested in the PDE approach for image denoising, mainly using bivariate splines to solve the nonlinear PDE associated with ROF model for image denoising and use the approach for image resizing, image imprinting, and image enhancements. Some numerical results can be found here
An application of multivariate splines for Fluid Flow Simulation won me a research medal from the University of Georgia in 2002. The representative paper is Bivariate Splines for Fluid Flows. In this paper, many standard fluid flows, e.g. cavity flows, backward step flows, flows around a circular object with various Reynolds numbers are simulated. In particular, a cavity flow over a triangular domain is simulated. Flows passing through a narrowed channel is simulated. Bivariate splines of various degrees were used in various simulations. The highest degree I used is 12. In fact, the degree of splines is an input variable and my matlab code is programmed for arbitrary degree $d\ge 1$, for arbitrary smoothness $r\ge 0$ and arbitrary triangulation $\triangle$.
Multivariate Spline Approximation Currently, I am using bivariate splines to approximate the functions in ill-posed problems, e.g., de-convolution, to approximate functions of bounded variation (BV), to approximate probability density functions(PDF), and to approximate measurable functions (Learning Theory). More results are coming up soon.
Compressed Sensing and Low-rank Matrix Reconstruction Currently I am interested in sparest solutions of undetermined linear systems and their applications in compressed sensing and low-rank matrix recovery. In my joint paper with Simon Foucart, we show how to use quasi norm lq, 0 < q ≤ 1 to find the sparsest solution. This paper has many citations since its publication. I have worked with W. T. Yin on unconstrained lq minimization for sparse vector recovery and matrix completion recently. Two joint papers have been accepted for publications: Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed lq Minimization and Augmented l1 and Nuclear Norm Models with a Globally Linearly Convergent Algorithm. In addition, I have several other preprints and reprints on this topic available on-line at my publication section of this page.