My major research areas are: **(1) Multivariate Splines and Their Applications and (2) Sparse Solutions of Linear System and Their Applications.**

**(1) 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 construction**, and **data forecasting**, etc. Details can be found in my publications.

(2) **Sparse solutions of Linear System** **and Their Applications. ** Sparse solution is a solution with small number of nonzero entries to a linear system. In particular, when a linear system is underdetermined, the system has many solutions. One particular solution is sparsest. One is interested in finding such one. This sparse solution has many applications. See my book with title: the sparse solution of underdetermined linear system as well as **matrix completion, graph clustering, phase retrieval, and etc.**.

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) **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 and used them for constructing smooth suitcase corners.

See a complete list of my publications at **alpha.math.uga.edu/~mjlai **

A graphical introduction to my research (some topics) can be found at **here.**

Recently, collaborated with Dr. Lin Mu and Dr. Weiwei Hu, I have organized an on-line conference on Saturday, Feb. 27, 2021 on **Geogia Scientific Computing Symposium**

## Research

**Research Areas:**

**Research Interests:**

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.**

**Grants:**

**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.

## Selected Publications

**Selected Publications:**

[1] Lai, M. -J., Liu, Y., Li, S. and Wang, H., On the Schatten p norm minimization for low rank matrix recovery, Applied Comput. Harmonic Analysis, vol. 51 (2021) pp. 157--170.

[2] Huang, Meng, Lai, M. -J., Varghese, Abraham and Xu, Zhiqiang, On DC based methods for Phase Retrieval, Approximation Theory XVI: San Antonio, 2019, Springer Verlag , (2021) edited by G. Fasshauer, M. Neamtu, and L. L. Schumaker, pp. 87--121.

[3] Lai, M. -J. and Mckenzie, Daniel., Compressive Sensing Approach to Cut Improvement and Local Clustering , SIAM J. Math. Data Science, vol. 2 (2020) pp. 368--395.

[4] Deng, Chongyang, Fan, X. Li. and Lai, M. -J., A minimization approach for constructing generalized barycentric coordinates and its computation, J. of Scientific Computing, vol. 84 (2020)

[5] Gao, Fuchang and Lai, M. -J., New regularity conditions for the solution to Dirichlet problem of the Poisson equation and their applications , Acta Mathematica Sinica, vol. 36 (2020) pp. 21--39.

[6] Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Com plex Domains, Statistica Sinica, vol. 30 (2020) pp. 347--360.

[7] Baramidze, V. and Lai, M. -J., Nonnegative Data Interpolation by Spherical Splines, J. Applied and Comput. Math., vol. 342 (2018) pp. 463--477.

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] Lai, M. -J. and Slavov, G., On Recursive Refinement of Convex Polygons, Computer Aided Geometric Design, vol. 45 (2016) pp. 83--90.

[13] 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.

[14] 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.

[15] 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.

[16] 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.

[17] 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)

[18] 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.

[19] 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.

[20] 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.

[21] 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.

[22] 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

[23] Lai, M. -J. and Wang, L., Bivariate penalized splines for regression, Statistica Sinica, vol. 23 (2013) pp. 1399--1417

[24] 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

[25] 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

## Education

**Education:**

I have been working with many graduate students and supervise them to get a master degree and/or a Ph.D.. Since 1998, 22 Ph.D.'s have been graduated under my direction. They are

**[22] Jinsil Lee**, Ph.D. graduated in Aug., 2023. She works on Numerical Solutions of Partial Differential Equations using Spline Collocation Methods.

**[21] Tsung-wei He**, Ph.D. graduated in May, 2023. He works on Image Deformation and Smooth Interpolating Curves and Surfaces Construction.

**[20] Kenneth Allen**, Ph.D. graduated in Dec. 2020. He works on matrix and tensor completion.

**[19] Yidong Xu **Ph.D. working on multivariate splines for smooth surface reconstruction and numerical solution to 3D partial differential equations. He graduated in Dec, 2019. He is now in Shanghai, China.

**[18] Daniel Mckenzie **Ph.D. working on graph clustering. He graduated in May, 2019. He is now a postdoc at University of California, Los Angeles.

**[17] Clayton Mersmann **Ph.D. working on multivariate spline solution to Helmholtz equations and Maxwell equations. He graduated in Aug, 2019. He is now an assistant professor at University of South Florida.

**[16] ABRAHM VARGHESE** Ph.D. working on matrix completion. He graduated in May, 2018. He is now an assistant professor at Shenandoah University, Virginia.

**[15] JAY LANTERMAN **Ph.D. working on constructing smooth vertex splines over quadrilatral partitions. He graduated in May, 2018 and works at IHG at Atlanta, Georgia.

**[14] 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.

**[13] 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.

**[12] 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.

**[11] 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.

**[10] 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.

**[9] 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.

**[8] 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.

**[7] OKKYUNG CHO** Ph.D., graduated in August, 2006.Her dissertation title is Construction of Compactly Supported Multiwavelets.She is an assistant professor at Montgomery College.

**[6] 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.

**[5] VICTORIA BARAMIDZE** Ph.D., graduated in August, 2005. Her dissertation title is Spherical Splines for Scattered Data Fitting. She is a full professor at West Illinois University as 2017.

**[4] 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.

**[3] 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.

**[2] 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.

**[1] 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, 4 Ph.D. students are under my supervision. They are

Zhaiming Shen, Ph.D. student working on graph/network clustering and semi-supervised learning.

Kenneth Allen, Ph.D. student working matrix/tensor completion.

Jinsil Lee, Ph.D. student working on multivariate spline collocation methods for numerical solution of partial differential equations

Tsung-Wei Hu Ph.D. student working 3D GBC functions and deformation.

## Other Information

**Courses Regularly Taught:**