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, Helmholtz equation and Maxwell equations, image enhancements, surface design, and data forecasting, etc. 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 such as edge detection. 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. Mainly, I use GBC functions to construct continuous polygonal splines. See my joint paper with Michael Floater. One of major results 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. Recently, Jay Lanterman and I used Washspress coordinates to construct C^1 smooth vertex splines.
My current research interests are 1) GBC(generalized barycentric coordinates) for polygonal splines; 2) Matrix and Data Tensor Completion; 3) Numerical Solution of Helmholtz equation and Maxwell equations; 4) Graph Clustering and Semi-supervised Learning, as well as 5) Scattered Data Fitting.
AUGUST 1, 2015 P.I. on a research grant (#DMS-1521537) from the National Science Foundation for three years (2015--2018) for the amount $150,336.
SEPT. 1, 2013 Received a collaboration grant from Simons Foundation for 5 years for an amount of $35,000.
AUG. 23, 2011 P. I. on a research grant from the U.S. Army Research Office for one year for the amount $49,998.
SEPTEMBER 1, 2007 P.I. on a research grant (#DMS-0713807) from the National Science Foundation for three years (2007--2010) for the amount $215,855.
SEPTEMBER 15, 2004 Received a conference grant from the National Science Foundation for the amount $18,000.
APRIL 12, 2004 Received a conference grant from the U.S. Army Research Office for the amount $13,293.
SEPTEMBER 1, 2003 P.I. on a collaborative mathematical research grant from the National Science Foundation (#EAR-0327577) for four years, 2003-2007, for the amount $250,166.
JULY 1, 2003 Received a computer equipment grant from the U.S. Army Research Office (#44659-MA-RIP) for the amount $54,179.
MARCH 18, 2002 Received a conference grant from the U.S. Army Research Office for the amount $4,448.
JANUARY 10, 2001 Co.-P.I. on a VIGRE grant from the National Science Foundation (#DMS-0089927) for five years, 2001-2005, for the amount $2,450,000.
AUGUST 1, 1998 P.I on a research grant from the National Science Foundation (#9870178) for three years 1998-2001 for the amount $70,334.
SEPTEMBER 10, 1996 Received an equipment supplement from the National Science Foundation for the amount $4,896 and a matching amount from the State of Georgia.
JULY 15, 1994 Received an equipment supplement from the National Science Foundation for the amount $11,200 and a matching amount from the State of Georgia.
SEPTEMBER 8, 1993 P.I on a research grant foam the National Science Foundation (#DMS-9303121) for three years, 1993-1996, for the amount $60,000.
 Gao, Fuchang and Lai, M. -J., New regularity conditions for the solution to Dirichlet problem of the Poisson equation and their applications , accepted, (2019).
 Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Com plex Domains, accepted, Statistica Sinica, (2018) .
 Baramidze, V. and Lai, M. -J., Nonnegative Data Interpolation by Spherical Splines, J. Applied and Comput. Math., vol. 342 (2018) pp. 463--477.
 Lai, M. -J. and Wang, C. M., A bivariate spline method for 2nd order elliptic equations in non-diverge nce form, Journal of Scientific Computing , (2018) pp. 803--829.
 Lai, M. -J. and Mersmann, C., Adaptive Triangulation Methods for Bivariate Spline Solutions of PDEs, Approximation Theory XV: San Antonio, 2016, Springer Verlag, (2017) edited by G. Fasshauer and L. L. Schumaker pp. 155--175.
 Lai, M. -J. and Lanterman, J., A polygonal spline method for general 2nd order elliptic equations and its applications, Approximation Theory XV: San Antonio, 2016, Springer Verlag, (2017) edited by G. Fasshauer and L. L. Schumaker pp. 119--154.
 Deng, W., Lai, M. -J., Peng, Z. and Yin, W. T., Parallel Multi-Block ADMM with o(1/k) Convergence, Journal of Scientific Computing>, vol. 71 (2017) pp. 712--736.
 Lai, M. -J. and Slavov, G., On Recursive Refinement of Convex Polygons, Computer Aided Geometric Design, vol. 45 (2016) pp. 83--90.
 Floater, M. and Lai, M. -J., Polygonal spline spaces and the numerical solution of the Poisson equation, SIAM Journal on Numerical Analysis, (2016) pp. 797--824.
 Liu, X. , Guillas, S. and Lai, M. -J., Efficient spatial modeling using the SPDE approach with bivariate splines, Journal of Computational and Graphical Statistics , vol. 25 (2016) pp. 1176--1194.
 Gutierrez, J. , Lai, M. -J. and Slavov, G., Bivariate Spline Solution of Time Dependent Nonlinear PDE for a Population Density over Irregular Domains, Mathematical Biosciences, vol. 270 (2015) pp. 263--277.
 Lai, M. -J. and Meile, C., Scattered data interpolation with nonnegative preservation using bivariate splines, Computer Aided Geometric Design, vol. 34 (2015) pp. 37--49.
 Wang, Z., Lai, M. -J., Lu, Z., Fan, W., Davulcu, H. and Ye, J., Orthogonal Rank-One Matrix Pursuit for Low Rank Matrix Completion, A488--A514, SIAM Journal of Scientific Computing, vol. 37 (2015)
 Lai, M. -J. and Liu, L. Y., The Probabilitic Estimates on the Largest and Smallest $q$-singular Values of Random Matrices, Mathematics of Computation, vol. 84 (2015) pp. 1775--1794.
 Lai, M. -J. and Matt, M., A Cr Trivariate Macro-Element Based on Alfeld Split, Journal of Approximation Theory, vol. 175 (2013) pp. 114--131.
 Guo, W. H. and Lai, M. -J., Box Spline Wavelet Frames for Image Edge Analysis, SIAM Journal Imaging Sciences, vol. 6 (2013) pp. 1553--1578.
 Lai, M. -J. and Yin, W. T., Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm, SIAM Journal Imaging Sciences, vol. 6 (2013) pp. 1059--1091.
 Lai, M. -J. and Zhou, T., Scattered data interpolation by bivariate splines with higher approximation order, Journal of Computational and Applied Mathematics, vol. 242 (2013) pp. 125--140
 Lai, M. -J. and Wang, L., Bivariate penalized splines for regression, Statistica Sinica, vol. 23 (2013) pp. 1399--1417
 Lai, M. -J. and Messi, L. M., Piecewise Linear Approximation of the continuous Rudin-Osher-Fatemi model for image denoising, SIAM Journal on Numerical Analysis, vol. 50 (2013) pp. 2446--2466
 Lai, M. -J., Xu, Y. Y. and Yin, W. T., Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed lp Minimization , SIAM Journal on Numerical Analysis, vol. 51 (2013) pp. 927--957
I have been working with many graduate students and supervise them to get a master degree and/or a Ph.D.. Since 1998, 16 Ph.D. have been graduated under my direction. They are
 ABRAHM VARGHESE Ph.D. working on matrix completion. He graduated in May, 2018. He is now an assistant professor at Shenandoah University, Virginia.
 JAMES LANTERMAN Ph.D. working on constructing smooth vertex splines over quadrilatral partitions. He graduated in May, 2018 and works at IHG at Atlanta, Georgia.
 GEORGE SLAVOV Ph.D. working on numerical solution of diffusion-reaction equation and predator-prey system of PDE by using multivariate splines. He graduated in Aug. 2016. His dissertation title is "Bivariate Spline Solution to a Class of Reaction-Diffussion equations." He is a software engineer working at a Hedge fund company in Bulgaria.
 LEOPOLD MATAMBA MESSI Ph.D. working on numerical solution of nonlinear PDE by using multivariate splines. He graduated in Aug. 2012. His dissertation title is "Theoretical and Numerical Approximation of the Rudin-Osher-Fetami model for image denoising in the continuous setting." He is a postdoc at Mathematical Biology Institute, Ohio State University for three years and now works at a bank as a financial analyst.
 QIANGYING HONG Ph.D., graduated in Aug. 2011. Her dissertation title is Bivariate Splines in variational models for image enhancements. She is now a lecturer at University of Kansas, Lawrenceville, Kansas.
 LOUIS YANG LIU Ph.D., graduated in August, 2010. His dissertation is Non-convex Optimization for Linear System with Pre-Gaussian Matrices and Recovery from Multiple Measurements. He was an assistant professor at the College of William and Mary in fall, 2011 and Spring, 2012. He is now a visiting assisitant professor at Michigan State University.
 BREE ETTINGER Ph.D., graduated in August, 2009. Her dissertation title is Bivariate Splines for Ozone Concentration Prediction. She is a postdoctorate at a university in Milan, Italy. She is now a teaching assistant professor at Department of Mathematics and Computer Science, Emory University.
 HAIPENG LIU Ph.D., graduated in August, 2007. His dissertation title is Prewavelets for Numerical Solution of PDE's. He is a lecturer at Georgia State University.
 JIANBAO WU Ph.D., graduated in August, 2007. His dissertation title is Spherical Splines for Hermite Interpolation and Surface Design. He is a bio-statistian working for a company.
 JIE ZHOU Ph.D., graduated in August, 2006. His dissertation title is Construction of Orthonormal Wavelets of Dilation Factor 3 with Application in Image Compression and A New Construction of Multivariate Compactly Supported Tight Frame. He is a tenure track assistant professor at Coastal Carolina University in the fall, 2007.
 KYUNGLIN NAM Ph.D., graduated in August, 2005. Her dissertation title is "Tight Wavelet Frame Construction and Its Application for Image Processing". She was a visiting assistant professor at University of Toledo, Toledo, Ohio. Now she is a Lecturer at Baylor University.
 GERARD AWANOU Ph.D., graduated in August, 2003. His dissertation title is "Energy Methods for Numerical Solution of 3D Navier-Stokes Equations by Trivariate Splines". After post-doc working at IMA (Institute for Mathematics and its Applications) at the University of Minnesota for two years, he is now an associate professor at North Illinois University. He has been a full professor at University of Illinois at Chicago since Fall, 2015.
 XIANGMING XU Ph.D., graduated in May, 2001. His dissertation title is Construction of Two-Dimensional Non-separable of Orthonormal Wavelets of Short Support. He is a research scientist at Telchemy Inc., Sewanee, GA.
 WENJIE HE Ph.D., graduated in August, 1998. His dissertation title is Compactly Supported Multivariate Multiwavelets: Theory and Constructions. He is now an associate professor at the Department of Mathematics and Computer Sciences, the University of Missouri at St. Louis, Missouri.
Certainly, 5 Ph.D. students are under my supervision. They are
DANIEL MCKENZIE Ph.D. student working on graph/network clustering.
YIDONG XU Ph.D. student working on optimization.
CLAYTON MERSMANN Ph.D. student working on numerical solution of nonlinear PDE.
Zhaiming Shen, Ph.D. student working on graph/network clustering and semi-supervised learning.
Kenneth Allen, Ph.D. student working matrix/tensor completion.