Professor Ming-Jun Lai Mail: Department of Mathematics University of Georgia Athens, GA 30602-7403 Phone: (706) 542-2065 Fax: (706) 542-2573 Office: Rm 540, Boyd Grad. Studies Building E-mail: mjlai at math.uga.edu

Research Profiles:

• I am also interested in the PDE approach for image denoising. Mainly I use bivariate splines to solve the nonlinear PDE associated with image denoising, image impainting, and image enhancements. See webpage for details.
• I am currently interested in sparsest solutions of underdetermined linear systems and their applications in compressed sensing and other areas. In my paper joint with Simon Foucart, we show how to use quasi-norm \ell_q, 0<= q<1 to find the sparsest solution. I give a simple explanation of the sparse solution of underdetermined linear systems in a survey paper. More publications can be found at my publication webpage.
• I have studied the theory and applications of multivariate splines for twenty years. One of the reasons that I am so interested in this topic for so long is that multivariate splines are a fundamental tool for Numerical Analysis, Scientific Computation, and Applied Mathematics. I mainly study the approximation properties of multivariate splines, construction of locally supported spline functions, approximation properties of data fitting using multivariate splines, computation with multivariate splines and applications of multivariate splines. Some typical papers are On the Approximation Power of Bivariate Splines , C^r Clough-Tocher Macro-Elements , and Spherical Splines for Scattered Data Fitting . They have various applications in data fitting and hole filling in Computer Aided Geometric Design, Fluid Flow Simulation , and in Geodetic Applications. A Ph.D. at the aerospace engineering department of Delft University of Technology, Netherlands finds a successful usage of the multivariate splines for global system of identification based on an NASA wind tunnel dataset and other datasets. See Visser's dissertation. A lot of my papers on multivariate splines are cited.
• Larry Schumaker and I wrote a monograph on "Spline Functions on Triangulations" which is recently published by Cambridge University Press. Three book reviews are available, review1 , review2, and review3.
• An application of multivariate splines for Fluid Flow Simulation won me a research medal from the University of Georgia in 2002. The main representative paper is Bivariate Splines for Fluid Flows .
• I have also studied Wavelets Analysis and Its Applications for Image Processing for about ten years. My interest is to construct various compactly supported orthonormal wavelets , orthonormal multi-wavelets , biorthogonal wavelets , tight wavelet frames , pre-wavelets, Riesz basis, and pre-Riesz basis in the multivariate setting. I am also interested in wavelet applications in image processing .
• I organized a summer school on Multivariate Splines and Their Applications in May 11--22, 2009.

Teaching Profiles:

• I have taught many courses for undergraduate students and graduate students. For graduate students, I mainly teach Advances in Numerical Analysis including Numerical Linear Algebra, Numerical Approximation, Numerical Solution of PDE's, Wavelets Analysis, and Optimization
• Eleven Ph.D's were graduated under my supervision. They are Dr. Wenjie He, Dr. Xiangming Xu, Dr. Gerard Awanou, Dr. V. Baramidze and Dr. K. Nam Dr. Wenjie He is already an associate professor at Univ. of Missouri, St. Louis. Dr. Xiangmming Xu is a chief scientist in a company working on signal and image communications. The research work which forms a part of Dr. Awanou's dissertation won him a best student paper award. V. Baramidze obtained a GRADUATE STUDENT EXCELLENCE IN RESEARCH AWARD from the University of Georgia. Mr. J. Zhou and Ms. O. Cho are graduated in the summer, 2006 and Mr. H. P. Liu and Mr. J. B. Wu graduated in the summer, 2007. Ms. Bree Ettinger graduated in the summer, 2009. Mr. Louis Y. Liu graduated in the summer, 2010.
• I encouraged and helped Bree Ettinger to work as an intern at Boeing company in summer, 2004 and Jeremy Praissman to work as an intern at NASA Langley Research Center in summer, 2007.
• I supervised 7 undergraduate students Katie Agle, Dustin Burns, Coop Cunliffe, Grant Fiddyment, Max Mautner, and Tarik Trent working on research problems (an NSF REU program) in summer, 2008. Their research work can be found here.
• I supervised a VIGRE seminar on Numerical Analysis in Fall, 2008 and Spring, 2009.

Service Profiles:

I have been members of various departmental, college level and university level committees.

Finally, my personal profiles: I came from Hangzhou, Zhejiang, China. I am very proud of Hangzhou, because it is one of the most beautiful cities in China. Seeing is believing.