Ming-Jun Lai's Collaborators are listed below in alphabetical order. Awanou, G. Awanou, G. and Lai, M. -J., Trivariate Spline Approximations of 3D Navier-Stokes Equations, Mathematics of Computation, vol. 74 (2005) pp. 585--601 Awanou, G. and Lai, M. -J., On Convergence Rate of the Augmented Lagrangian Algorithm for Nonsymmetric Saddle Point Problems, Journal of Applied Numerical Mathematics, vol. 54 (2005) pp. 122--134 Awanou, G. and Lai, M. -J., $C^1$ Quintic Spline Interpolation over Tetrahedral Partitions, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 1--16 Awanou, G., Lai, M. -J. and Wenston, P., The Multivariate Spline Method for Scattered Data Fitting and Numerical Solution of Partial Differential Equations, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 24--74 Baramidze, V. Baramidze, V. and Lai, M. -J., Nonnegative Data Interpolation by Spherical Splines, J. Applied and Comput. Math., vol. 342 (2018) pp. 463--477 Baramidze, V., Lai, M. -J. and Shum, C. K., Spherical Splines for Data Interpolation and Fitting, SIAM Journal of Scientific Computing, vol. 28 (2006) pp. 241--259 Lai, M. -J., Shum, C. K., Baramidze, V. and Wenston, P., Triangulated Spherical Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp. 695--708 Baramidze, V. and Lai, M. -J., Volume Data Interpolation by Tensor Products of Spherical and Radial Splines, Advances in Constructive Approximation, Nashboro Press, (2004) edited by M. Neamtu and E. Saff pp. 75--88 Baramidze, V. and Lai, M. -J., Error Bounds for Minimal Energy Interpolatory Spherical Splines, Approximation Theory XI, Nashboro Press, (2005) pp. 25--50 Baramidze, V. and Lai, M. -J., Spherical Spline Solution to a PDE on the Sphere, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 75--92 Baramidze, V. and Lai, M. -J., Convergence of Discrete and Penalized Least Squares of Spherical Splines, Journal of Approximation Theory, (2011) pp. 1091--1106 Bracco, A.. Meile, C., Lai, M. -J., Bracco, A.., Luo, H. and Joye, S., Interpretation of oxygen profiles in the aftermath of the BP/Deepwater Horizon hydrocarbon discharge , submitted, (2014) Cassidy, P. Lai, M. -J., Lian, J. A. and Cassidy, P., Removal of Gaps among Compound C^1 Bi-Cubic Parametric B-spline Surfaces, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 287--313 Chen, G. R. Chen, G. R. and Lai, M. -J., Wavelets and Splines, Athens, 2005, Nashboro Press, (2006) Chen, G. R., Chui, C. K. and Lai, M. -J., Construction of Real-Time Spline Quasi-Interpolation Scheme, Journal of Approximation Theory and its Application, vol. 4 (1988) pp. 61--75 Cho, O. Cho, O. and Lai, M. -J., A Class of Compactly Supported Orthonormal B-Spline wavelets, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 123--151 Chui, C. K. Chui, C. K. and Lai, M. -J., Computation of Box Splines and B-splines on Triangulations of Nonuniform Rectangular Partitions, Journal of Approximation Theory and its Application, vol. 3-4 (1987) pp. 37--62 Chui, C. K. and Lai, M. -J., Multivariate Analog of Marsden's Identity and a Quasi-Interpolation Scheme, Constructive Approximation, vol. 3 (1987) pp. 111--122 This is one of my best works. We present a polynomial representation by using translates of a compactly supported box spline function. This result can be not improved any further and thus is the best. Chen, G. R., Chui, C. K. and Lai, M. -J., Construction of Real-Time Spline Quasi-Interpolation Scheme, Journal of Approximation Theory and its Application, vol. 4 (1988) pp. 61--75 Chui, C. K. and Lai, M. -J., Multivariate Vertex Splines and Finite Elements, Journal of Approximation Theory, vol. 60 (1990) pp. 245--343 Chui, C. K. and Lai, M. -J., On Bivariate Super Vertex Splines, Constructive Approximation, vol. 6 (1990) pp. 399--419 Chui, C. K. and Lai, M. -J., Algorithms for Generating B-nets and Graphically Displaying Box Spline Surfaces, Computer Aided Geometric Design, vol. 8 (1992) pp. 479--493 Chui, C. K. and Lai, M. -J., Filling Polygonal Holes using $C^1$ Cubic Triangular Spline Patches, Computer Aided Geometric Design, vol. 17 (2000) pp. 297--307 Chui, C. K., Lai, M. -J. and Lian, J. A., Algorithms for $G\sp 1$ Connection of Multiple Parametric Bicubic NURBS Surfaces, Numerical Algorithms, vol. 23 (2000) pp. 285--313 Chui, C. K. and Lai, M. -J., On Bivariate Vertex Splines, Multivariate Approximation Theory III, Birkhauser, (1985) edited by W.Schempp and K.Zeller pp. 84--115 Chui, C. K. and Lai, M. -J., VanderMonde Determinants and Lagrange Interpolation in R^s, Nonlinear and Convex analysis, Marcel Dekker, (1987) edited by B.L.Lin and S.Simons pp. 23--35 Chui, C. K. and Lai, M. -J., On Multivariate Vertex Splines and Applications, Topics in Multivariate Approximation, Academic Press, (1987) edited by Chui, C.K., L.L. Schumaker, and F. Utreras pp. 19--36 Davison, I.. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Davulcu, H. 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) Deng, Chongyang Deng, Chongyang and Lai, M. -J., On e-Locality of Harmonic Generalized Barycentric Coordinatesi and Their Application to Solution of the Poisson Equation, accepted , Pure and Applied Functional Analysis , (2022) Deng, Chongyang, Hu, Tsung-Wei and Lai, M. -J., The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex, submitted , (2022) Deng, Chongyang, Hong, Q. F., Lai, M. -J., Mersmann, Clayton and Xu, Yidong, Multivariate Splines for Curve and Surface Interpolation and Fitting, submitted , (2021) 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) Deng, W. 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. Ettinger, B. Ettinger, B., Guillas, S. and Lai, M. -J., Bivariate Splines for Functional Regression Models with Application to Ozone Forecasting, Environmetrics, vol. 23 (2012) pp. 317--328 Fan, W. 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) This is one of my best works. We have established the most efficient algorithm to find matrix completion with similar or better accuracy than any other available algorithms. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Farmer, K. W. Farmer, K. W. and Lai, M. -J., Scattered Data Interpolation by C^2 Quintic Splines Using Energy Minimization, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 47--54 Feng, G. J. Lai, M. -J. and Feng, G. J., On the Uniform Convergence of the Birkhoff Interpolation with Two Points, Math. Numer. Sinica, vol. 6 (1984) pp. 222--224 Feng, R. Z. Feng, R. Z., Huang, A., Lai, M. -J. and Shen, Z. M., Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method, Journal of Computational Math, vol. 41 (2023) pp. 18--38. Floater, M. 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. Download Code Foucart, S. Foucart, S. and Lai, M. -J., Sparsest Solutions of Underdetermined Linear Systems via $\ell_q$-minimization for $0\le q \le 1$, Applied and Computational Harmonic Analysis, vol. 26 (2009) pp. 395--407 Foucart, S. and Lai, M. -J., Sparse Recovery with Pre-Gaussian Random Matrices, Studia Mathematica, vol. 200 (2010) pp. 91--102 Gao, Fuchang Gao, Fuchang and Lai, M. -J., A new H2 regularity condition of the solution to Dirichlet problem of the Poisson equation and its applications, Acta Mathematica Sinica, vol. 36 (2020) pp. 21--39 Gao, L. Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Complex Domains, Statistica Sinica, vol. 30 (2020) pp. 347--360 Geronimo, J. Geronimo, J. and Lai, M. -J., Factorization of Multivariate Positive Laurent Polynomials, Journal of Approximation Theory, vol. 139 (2006) pp. 327--345 Guillas, S. Guillas, S. and Lai, M. -J., Bivariate Splines for Spatial Functional Regression Models, Journal of Nonparametric Statistics, vol. 22 (2010) pp. 477--497 Ettinger, B., Guillas, S. and Lai, M. -J., Bivariate Splines for Functional Regression Models with Application to OLiu, 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. zone Forecasting, Environmetrics, vol. 23 (2012) pp. 317--328 Guo, W. H. 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. Gutierrez, J. 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. An expanded and corrected version can be found below. 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. Han, D. Hu, X., Han, D. and Lai, M. -J., Bivariate Splines of Various Degrees for Numerical Solution of PDE, SIAM Journal of Scientific Computing, vol. 29 (2007) pp. 1338--1354 Zhou, T., Han, D. and Lai, M. -J., Energy Minimization Method for Scattered Data Hermite Interpolation, Journal of Applied Numerical Mathematics, vol. 58 (2008) pp. 646--659 He, W. He, W. and Lai, M. -J., On Digital Filters Associated with Bivariate Box Spline Wavelets, Journal of Electronic Imaging, vol. 6 (1997) pp. 453--466 He, W. and Lai, M. -J., Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities, Applied and Computational Harmonic Analysis, vol. 6 (1999) pp. 53--74 This is one of my best works. We found a natural extension of biorthogonal wvelets from the univariate setting to the bivariate setting. Several researchers including C. K. Chui, J. Stoeckler, A. Cohen, S. Riemenschneider, Z. W. Shen have tried to do such an extension. Our extension is the best. He, W. and Lai, M. -J., Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets, IEEE Transactions on Image Processing, vol. 9 (2000) pp. 949--953 He, W. and Lai, M. -J., Construction of Trivariate Compactly Supported Biorthogonal Box Spline Wavelets, Journal of Approximation Theory, vol. 120 (2003) pp. 1--19 He, W. and Lai, M. -J., Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets, Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 3169 (1997) pp. 303--314 He, W. and Lai, M. -J., Bivariate Box Spline Wavelets in Sobolev Spaces, Wavelet Applications in Signal and Image Processing VI, proceedings of SPIE, vol. 3458 (1998) pp. 56--66 He, W. and Lai, M. -J., A New Sufficient Condition for the Orthonormality of Refineable Functions, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 121--128 Hong, Q. Hong, Q., Lai, M. -J. and Wang, J., Convergence Analysis of a Finite Difference Scheme for the Gradient Flow associated with the ROF Model, submitted, (2012) Hong, Q. and Lai, M. -J., A Bivariate Spline Approach for Image Enhancement, manuscript, (2011) Hong, Q. F. Deng, Chongyang, Hong, Q. F., Lai, M. -J., Mersmann, Clayton and Xu, Yidong, Multivariate Splines for Curve and Surface Interpolation and Fitting, submitted , (2021) Hong, Q. Y. Hong, Q. Y., Lai, M. -J. and Messi, L. M., The Rayleigh-Ritz Method for Total Variation Minimization Using Bivariate Splines Functions on Triangulation, submitted, (2014) Hong, Q. Y. Hong, Q. Y., Lai, M. -J. and Messi, L. M., The Rayleigh-Ritz Method for Total Variation Minimization Using Bivariate Splines Functions on Triangulation, submitted, (2014) Hong, Qianying Hong, Qianying, Lai, M. -J., Messi, Leopold Matamba and Wang, Jingyue, Galerkin method with splines for total variation minimization, J. Algorithms Computational Technology , vol. 13 (2019) pp. 1--16 Hu, Tsung-Wei Deng, Chongyang, Hu, Tsung-Wei and Lai, M. -J., The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex, submitted , (2022) Hu, X. Hu, X., Han, D. and Lai, M. -J., Bivariate Splines of Various Degrees for Numerical Solution of PDE, SIAM Journal of Scientific Computing, vol. 29 (2007) pp. 1338--1354 Huang, A. Feng, R. Z., Huang, A., Lai, M. -J. and Shen, Z. M., Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method, Journal of Computational Math, vol. 41 (2023) pp. 18--38. Huang, Meng 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 Joye, S. Meile, C., Lai, M. -J., Bracco, A.., Luo, H. and Joye, S., Interpretation of oxygen profiles in the aftermath of the BP/Deepwater Horizon hydrocarbon discharge , submitted, (2014) Kang, Hongmei Kang, Hongmei, Lai, M. -J. and Li, Xin, An economecal representation of PDE solution by using compressive sensing approach, J. Computer Aided Design , vol. 115 (2019) pp. 78--86 Kapita, S. Kapita, S. and Lai, M. -J., A bivariate spline Soluton to the Exterior Helmholtz Equation and Its Applications, submitted , (2019) Kersey, S. Kersey, S. and Lai, M. -J., Convergence of Local Variational Spline Interpolation, Journal of Mathematical Analysis and Applications, vol. 314 (2008) pp. 398--415 Lai, M. -J. Lai, M. -J., Xie, Jiaxin and Xu, Zhiqiang, Graph Sparsification by Universal Greedy Algorithms, Journal of Computational Math, vol. ?? (2023) pp. Published onlne Shen, Z. M., Lai, M. -J. and Li, S.., Graph-based Semi-supervised Local Clutering with Few Labeled Nodes, accepted, IJCAI , (2023) Lai, M. -J. and Lee, J., Trivariate Spline based Collocation Methods for Numerical Solution to 3D Monge-Ampere Equations, Published Onlne , Journal of Scientific Computing , (2023) Deng, Chongyang and Lai, M. -J., On e-Locality of Harmonic Generalized Barycentric Coordinatesi and Their Application to Solution of the Poisson Equation, accepted , Pure and Applied Functional Analysis , (2022) Deng, Chongyang, Hu, Tsung-Wei and Lai, M. -J., The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex, submitted , (2022) Lai, M. -J. and Lee, J., A Multivariate Spline based Collocation Method for Numerical Solution of PDEs, SIAM J. Numerical Analysis, vol. 60 (2022) pp. 2405--2434 Lai, M. -J. and Shen, Z. M., The Kolmogorov Superposition theorem can break the curse of dimensionality when approximating high dimensional functions, submitted, (2022) Lai, M. -J. and Wang, Y., Sparse Solutions of Underdetermined Linear Systems and Their Applications, SIAM Publication, (2021) 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. Feng, R. Z., Huang, A., Lai, M. -J. and Shen, Z. M., Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method, Journal of Computational Math, vol. 41 (2023) pp. 18--38. 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 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. Deng, Chongyang, Hong, Q. F., Lai, M. -J., Mersmann, Clayton and Xu, Yidong, Multivariate Splines for Curve and Surface Interpolation and Fitting, submitted , (2021) Lai, M. -J. and Shen, Z., An effective approach to semi-supervised cluster extraction , to appear , Journal of Scientific Computing , (2022) 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) Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Complex Domains, Statistica Sinica, vol. 30 (2020) pp. 347--360 Hong, Qianying, Lai, M. -J., Messi, Leopold Matamba and Wang, Jingyue, Galerkin method with splines for total variation minimization, J. Algorithms Computational Technology , vol. 13 (2019) pp. 1--16 Kang, Hongmei, Lai, M. -J. and Li, Xin, An economecal representation of PDE solution by using compressive sensing approach, J. Computer Aided Design , vol. 115 (2019) pp. 78--86 Gao, Fuchang and Lai, M. -J., A new H2 regularity condition of the solution to Dirichlet problem of the Poisson equation and its applications, Acta Mathematica Sinica, vol. 36 (2020) pp. 21--39 Kapita, S. and Lai, M. -J., A bivariate spline Soluton to the Exterior Helmholtz Equation and Its Applications, submitted , (2019) Lai, M. -J. and Mersmann, C., A bivariate spline Soluton of Helmholtz Equation with large wave number, submitted , (2019) Lai, M. -J. and Lanterman, J., Construction of C1 Polygonal Splines over Quadrilateral Partition, Computer Aided Geometric Design, vol. 92 (2022) pp. Paper No.102063 Lai, M. -J. and Lee, J., A Multivariate Spline based Collocation Method for Numerical Solution of PDEs, submitted , (2021) Wen, Jinming, Zhou, Zhengchun, Liu, Zilong, Lai, M. -J. and Tang, Xiaohu, Sharp sufficient conditions for stable recovery of block sparse signals b y block orthogonal matching pursuit, Applied and Computational Harmonic Analysis, (2019) pp. 948--974 Lai, M. -J. and Varghese, Abraham, On Convergence of the Alternating Projection Method for Matrix Completion and Sparse Recovery Problems , submitted , (2017) 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. Lai, M. -J., Max-Norm Estimate of Spline Solution to Second Order Elliptic PDE in Nondivergence Form, submitted, (2017) 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. Download Code 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. An expanded and corrected version can be found below. 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) This is one of my best works. We have established the most efficient algorithm to find matrix completion with similar or better accuracy than any other available algorithms. Hong, Q. Y., Lai, M. -J. and Messi, L. M., The Rayleigh-Ritz Method for Total Variation Minimization Using Bivariate Splines Functions on Triangulation, submitted, (2014) This is one of my best works. We use bivariate splines to do image enhancements such as denoising, enlargement, impainting, and wrinkle removing. Numerical results are very satisfactory. Check it out. Meile, C., Lai, M. -J., Bracco, A.., Luo, H. and Joye, S., Interpretation of oxygen profiles in the aftermath of the BP/Deepwater Horizon hydrocarbon discharge , submitted, (2014) Lai, M. -J. and Messi, L. M., Multiscale Hierarchical Decompositon: Modes and Rates of Convergence, submitted, (2014) Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Wang, Z., Lai, M. -J., Lu, Z. and Ye, J., Orthogonal Rank One Matrix Pursuit for Matrix Completion, appeared, International Conference on Machine Learning (ICML) , (2014) 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. This is one of my best works. We are able to find the edges of images very accurately. Comparing with many methods, our method is the best to find the skeleton of an image. Seeing is believing. Check the paper out. 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. This is one of my best works. We established the convergence rate of argumented $\ell_1$ minimization algorithms for compressed sensing and matrix completion. Lai, M. -J., Li, S., Liu, L. Y. and Wang, H., Two Results on the Schatten p-quasi-norm minimization for low rank matrix recovery , submitted, (2012) Hong, Q., Lai, M. -J. and Wang, J., Convergence Analysis of a Finite Difference Scheme for the Gradient Flow associated with the ROF Model, submitted, (2012) Lai, M. -J. and Meile, C., Wator Column Oxygen Anomalies in the Aftermath of the BP Oil Spill, a poster in a Gulf of Mexico Research Initiative conference, (2013) Lai, M. -J. and Meile, C., Oxygen Anomalies in the Aftermath of the BP oil spill, a poster in a conference, (2012) 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 Lai, M. -J. and Schumaker, L. L., Spline Functions over Triangulations, Cambridge University Press, (2007) This is one of my best works. This book summarizes a lot of properties of multivariate splines over triangulations or tetrahedral partitions. No other book nor papers contain so much proerties of spline functions over triangulations. Chen, G. R. and Lai, M. -J., Wavelets and Splines, Athens, 2005, Nashboro Press, (2006) Lai, M. -J., On Estimations for the Exact Bounds of the Coefficients of Approximation by Cubic Spline Interpolation, Math. Numer. Sinica, vol. 6 (1984) pp. 105--108 Lai, M. -J. and Wang, X. H., A Note to the Remainder of a Multivariate Interpolation Polynomial, Journal of Approximation Theory and its Application, vol. 1 (1984) pp. 57--63 Lai, M. -J. and Feng, G. J., On the Uniform Convergence of the Birkhoff Interpolation with Two Points, Math. Numer. Sinica, vol. 6 (1984) pp. 222--224 Lai, M. -J., Exact Error Bounds for Cubic Birkhoff Spline Interpolation, Numerical Math. J. Chinese Univ., vol. 7 (1985) pp. 369--372 Lai, M. -J. and Wang, X. H., On Multivariate Newtonian Interpolation, Scientia Sinica, vol. 29 (1986) pp. 23--32 Chui, C. K. and Lai, M. -J., Computation of Box Splines and B-splines on Triangulations of Nonuniform Rectangular Partitions, Journal of Approximation Theory and its Application, vol. 3-4 (1987) pp. 37--62 Chui, C. K. and Lai, M. -J., Multivariate Analog of Marsden's Identity and a Quasi-Interpolation Scheme, Constructive Approximation, vol. 3 (1987) pp. 111--122 This is one of my best works. We present a polynomial representation by using translates of a compactly supported box spline function. This result can be not improved any further and thus is the best. Chen, G. R., Chui, C. K. and Lai, M. -J., Construction of Real-Time Spline Quasi-Interpolation Scheme, Journal of Approximation Theory and its Application, vol. 4 (1988) pp. 61--75 Lai, M. -J., A Remark on Integer Translates of a Box Spline, Journal of Approximation Theory and its Application, vol. 5 (1989) pp. 97--104 Lai, M. -J., On Construction of Bivariate and Triavariate Vertex Splines, Texas AM University, (1989) pp. Dept. of Mathematics Chui, C. K. and Lai, M. -J., Multivariate Vertex Splines and Finite Elements, Journal of Approximation Theory, vol. 60 (1990) pp. 245--343 Chui, C. K. and Lai, M. -J., On Bivariate Super Vertex Splines, Constructive Approximation, vol. 6 (1990) pp. 399--419 Lai, M. -J., On Dual Functionals of Polynomials in B-form, Journal of Approximation Theory, vol. 67 (1991) pp. 19--37 Lai, M. -J., Fortran Subroutines for B-nets of Box Splines on Three and Four Directional Meshes, Numerical Algorithms, vol. 2 (1992) pp. 33--38 Chui, C. K. and Lai, M. -J., Algorithms for Generating B-nets and Graphically Displaying Box Spline Surfaces, Computer Aided Geometric Design, vol. 8 (1992) pp. 479--493 Lai, M. -J., A Characteristic Theorem of Multivariate Splines in Blossom Form, Computer Aided Geometric Design, vol. 8 (1992) pp. 513--521 Lai, M. -J., Asymptotic Formulae of Multivariate Bernstein Approximation, Journal of Approximation Theory, vol. 70 (1992) pp. 229--242 Lai, M. -J., Some Sufficient Conditions for Convexity of Multivariate Bernstein-Bezier Polynomials and Box Spline Surfaces, Studia Scientiarum Math. Hungarica, vol. 28 (1993) pp. 363--374 Lai, M. -J., A Serendipity Family of Locally Supported Splines in $S^2_8(\triangle)$, Journal of Approximation Theory and its Application, vol. 10 (1993) pp. 43--53 Lai, M. -J., On Computation of Battle-Lemarie's Wavelets, Mathematics of Computation, vol. 63 (1994) pp. 689--699 Lai, M. -J., On Stromberg's Spline Wavelets, Applied and Computational Harmonic Analysis, vol. 1 (1994) pp. 188--193 Lai, M. -J., Approximation Order from Bivariate $C^1$ Cubics on a Four-Directional Mesh is Full, Computer Aided Geometric Design, vol. 11 (1994) pp. 215--223 Lai, M. -J., On the Digital Filter Associated with Daubechies' Wavelets, IEEE Transactions on Signal Processing, vol. 43 (1995) pp. 2203--2205 Lai, M. -J., Scattered Data Interpolation and Approximation by $C^1$ Piecewise Cubic Polynomials, Computer Aided Geometric Design, vol. 13 (1996) pp. 81--88 Lai, M. -J., On the Fundamental Solutions for Multivariate Singular Interpolation, Journal of Approximation Theory and its Application, vol. 12 (1996) pp. 73--92 Lai, M. -J. and Wenston, P., On Multilevel Bases for Elliptic Boundary Value Problems, Journal of Computational and Applied Mathematics, vol. 71 (1996) pp. 95--113 Lai, M. -J., On $C^2$ Quintic Spline Functions over Triangulations of Powell-Sabin's Type, Journal of Computational and Applied Mathematics, vol. 73 (1996) pp. 135--155 Lai, M. -J., Geometric Interpretation of Smoothness Conditions of Triangular Polynomial Patches, Computer Aided Geometric Design, vol. 14 (1997) pp. 191--199 Lai, M. -J. and Schumaker, L. L., Scattered Data Interpolation using Piecewise Polynomials of Degree Six, SIAM Journal on Numerical Analysis, vol. 34 (1997) pp. 905--921 He, W. and Lai, M. -J., On Digital Filters Associated with Bivariate Box Spline Wavelets, Journal of Electronic Imaging, vol. 6 (1997) pp. 453--466 Lai, M. -J. and Schumaker, L. L., Approximation Power of Bivariate Splines, Advances in Computational Mathematics, vol. 9 (1998) pp. 251--279 He, W. and Lai, M. -J., Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities, Applied and Computational Harmonic Analysis, vol. 6 (1999) pp. 53--74 This is one of my best works. We found a natural extension of biorthogonal wvelets from the univariate setting to the bivariate setting. Several researchers including C. K. Chui, J. Stoeckler, A. Cohen, S. Riemenschneider, Z. W. Shen have tried to do such an extension. Our extension is the best. Lai, M. -J. and Schumaker, L. L., On the Approximation Power of Splines on Triangulated Quadrangulations, SIAM Journal on Numerical Analysis, vol. 36 (1999) pp. 143--159 Lai, M. -J. and Wenston, P., On Schwarz's Domain Decomposition Methods for Elliptic Boundary Value Problems, Numerische Mathematik, vol. 84 (2000) pp. 475--495 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Numerical Solution of Navier-Stokes Equations over Polygons in Stream Function Formulation, Numerical Methods for P.D.E., vol. 16 (2000) pp. 147--183 He, W. and Lai, M. -J., Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets, IEEE Transactions on Image Processing, vol. 9 (2000) pp. 949--953 Chui, C. K. and Lai, M. -J., Filling Polygonal Holes using $C^1$ Cubic Triangular Spline Patches, Computer Aided Geometric Design, vol. 17 (2000) pp. 297--307 Lai, M. -J., Convex Preserving Scattered Data Interpolation using Bivariate $C^1$ Cubic Splines, Journal of Computational and Applied Mathematics, vol. 119 (2000) pp. 249--258 Chui, C. K., Lai, M. -J. and Lian, J. A., Algorithms for $G\sp 1$ Connection of Multiple Parametric Bicubic NURBS Surfaces, Numerical Algorithms, vol. 23 (2000) pp. 285--313 Lai, M. -J. and Schumaker, L. L., Macro-Elements and Stable Local Bases for Splines on Clough-Tocher Triangulations, Numerische Mathematik, vol. 88 (2001) pp. 105--119 Wang, X. H., Li, C. and Lai, M. -J., An Unified Convergence Theory for Newton's Type Methods for Zeros of Nonlinear Operators in Banach spaces, BIT, vol. 42 (2002) pp. 206--213 Lai, M. -J. and Schumaker, L. L., Quadrilateral Macro-Elements, SIAM Journal on Mathematical Analysis, vol. 33 (2002) pp. 1107--1116 Von Golitschek, M., Lai, M. -J. and Schumaker, L. L., Bounds for Minimal Energy Bivariate Polynomial Splines, Numerische Mathematik, vol. 93 (2002) pp. 315--331 Lai, M. -J. and Schumaker, L. L., Macro-Elements and Stable Local Bases for Splines on Powell-Sabin Triangulations, Mathematics of Computation, vol. 72 (2003) pp. 335--354 Lai, M. -J., Liu, C. and Wenston, P., Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation, Numerical Methods for P.D.E., vol. 19 (2003) pp. 776--827 He, W. and Lai, M. -J., Construction of Trivariate Compactly Supported Biorthogonal Box Spline Wavelets, Journal of Approximation Theory, vol. 120 (2003) pp. 1--19 Lai, M. -J. and Wenston, P., $L^1$ Spline Methods for Scattered Data Interpolation and Approximation, Advances in Computational Mathematics, vol. 21 (2004) pp. 293--315 Lai, M. -J. and Le Mehaute, A., A New Kind of Trivariate C^1 Spline Space, Advances in Computational Mathematics, vol. 21 (2004) pp. 273--292 Lai, M. -J., Liu, C. and Wenston, P., On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability, Applicable Analysis, vol. 83 (2004) pp. 541--562 Lai, M. -J., Liu, C. and Wenston, P., Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations, Applicable Analysis, vol. 83 (2004) pp. 563--577 Lai, M. -J. and Wenston, P., Bivariate Splines for Fluid Flows, Computers and Fluids, vol. 33 (2004) pp. 1047--1073 This is one of my best works. We use bivariate splines to numerically solve 2D Navier-Stokes equations based on stream function formulation. Many fluid flows over different domains are simulated. Seeing is believing. Check the paper out. Wang, X. H., Lai, M. -J. and Yang, S., On Divided Differences of the Remainder of Polynomial Interpolation, Journal of Approximation Theory, vol. 127 (2004) pp. 193--197 Awanou, G. and Lai, M. -J., Trivariate Spline Approximations of 3D Navier-Stokes Equations, Mathematics of Computation, vol. 74 (2005) pp. 585--601 Awanou, G. and Lai, M. -J., On Convergence Rate of the Augmented Lagrangian Algorithm for Nonsymmetric Saddle Point Problems, Journal of Applied Numerical Mathematics, vol. 54 (2005) pp. 122--134 Wang, X. H., Wang, H. and Lai, M. -J., Some Results on Numerical Divided Difference Formulas, Scientia Sinica, vol. Ser. A., 35 (2005) pp. 712--720 Lai, M. -J., Construction of Multivariate Compactly Supported Orthonormal Wavelets, Advances in Computational Mathematics, vol. 25 (2006) pp. 41--56 Baramidze, V., Lai, M. -J. and Shum, C. K., Spherical Splines for Data Interpolation and Fitting, SIAM Journal of Scientific Computing, vol. 28 (2006) pp. 241--259 Geronimo, J. and Lai, M. -J., Factorization of Multivariate Positive Laurent Polynomials, Journal of Approximation Theory, vol. 139 (2006) pp. 327--345 Lai, M. -J. and Stoeckler, J., Construction of Multivariate Compactly Supported Tight Wavelet Frames, Applied and Computational Harmonic Analysis, vol. 21 (2006) pp. 324--348 Lai, M. -J., Construction of Multivariate Compactly Supported Prewavelets in $L_2$ spaces and pre-Riesz Basis in Sobolev Spaces, Journal of Approximation Theory, vol. 142 (2006) pp. 83--115 Lai, M. -J., Le Mehaute, A. and Sorokina, T., An Octahedral C^2 Macro-Element, Computer Aided Geometric Design, vol. 23 (2006) pp. 640--654 Lai, M. -J. and Schumaker, L. L., Trivariate $C^r$ Polynomial Macro-elements, Constructive Approximation, vol. 26 (2007) pp. 11--28 Lai, M. -J. and Petukhov, A., The Method of Virtual Components for Constructing Wavelet Frames, Applied and Computational Harmonic Analysis, vol. 22 (2007) pp. 304--318 Lai, M. -J., Convergence of three $L_1$ Spline Methods for Data Interpolation and Fitting, Journal of Approximation Theory, vol. 145 (2007) pp. 196--211 Hu, X., Han, D. and Lai, M. -J., Bivariate Splines of Various Degrees for Numerical Solution of PDE, SIAM Journal of Scientific Computing, vol. 29 (2007) pp. 1338--1354 Zhou, T., Han, D. and Lai, M. -J., Energy Minimization Method for Scattered Data Hermite Interpolation, Journal of Applied Numerical Mathematics, vol. 58 (2008) pp. 646--659 Kersey, S. and Lai, M. -J., Convergence of Local Variational Spline Interpolation, Journal of Mathematical Analysis and Applications, vol. 314 (2008) pp. 398--415 Lai, M. -J. and Nam, K., On the Number of Tight Wavelet Framelets Associated with Multivariate Box Splines, Journal of Approximation Theory and its Application, (2008) Lai, M. -J. and Schumaker, L. L., Domain Decomposition Method for Scattered Data Fitting, SIAM Journal on Numerical Analysis, vol. 47 (2009) pp. 911-928 Foucart, S. and Lai, M. -J., Sparsest Solutions of Underdetermined Linear Systems via $\ell_q$-minimization for $0\le q \le 1$, Applied and Computational Harmonic Analysis, vol. 26 (2009) pp. 395--407 Lai, M. -J., Shum, C. K., Baramidze, V. and Wenston, P., Triangulated Spherical Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp. 695--708 Lai, M. -J. and Petukhov, A., Method of Virtual Components in the Multivariate Setting, Journal of Fourier Analysis and Its Applications, vol. 16 (2010) pp. 471--494 Foucart, S. and Lai, M. -J., Sparse Recovery with Pre-Gaussian Random Matrices, Studia Mathematica, vol. 200 (2010) pp. 91--102 Lai, M. -J., On Sparse Solution of Underdetermined Linear Systems, Journal of Concrete and Applicable Mathematics, vol. 8 (2010) pp. 296--327 Guillas, S. and Lai, M. -J., Bivariate Splines for Spatial Functional Regression Models, Journal of Nonparametric Statistics, vol. 22 (2010) pp. 477--497 Lai, M. -J., Pan, R. and Zhao, K., Initial Boundary Value Problem for 2D Viscous Boussinesq Equation, Arch Rational Mech Anal, vol. 199 (2011) pp. 739--760 Lai, M. -J. and Wang, J., An Unconstrained l_q Minimization for Sparse Solution of Underdetermined Linear Systems, SIAM Journal of Optimization, vol. 21 (2011) pp. 82--101 Lai, M. -J. and Liu, L. Y., The Null Space Property for Sparse Recovery From Multiple Measurement Vectors, Applied and Computational Harmonic Analysis, vol. 30 (2011) pp. 402--406 Chui, C. K. and Lai, M. -J., On Bivariate Vertex Splines, Multivariate Approximation Theory III, Birkhauser, (1985) edited by W.Schempp and K.Zeller pp. 84--115 Chui, C. K. and Lai, M. -J., VanderMonde Determinants and Lagrange Interpolation in R^s, Nonlinear and Convex analysis, Marcel Dekker, (1987) edited by B.L.Lin and S.Simons pp. 23--35 Chui, C. K. and Lai, M. -J., On Multivariate Vertex Splines and Applications, Topics in Multivariate Approximation, Academic Press, (1987) edited by Chui, C.K., L.L. Schumaker, and F. Utreras pp. 19--36 Lai, M. -J., A Matrix Approach to Computations of Various Wavelets, Proceedings of IMACS World Congress, vol. 1 (1994) pp. 284--286 Lai, M. -J., Bivariate Spline Spaces on FVS-triangulations, Approximation Theory VIII, Academic Press, (1995) edited by Chui, C. K. and Schumaker, L. L. pp. 309--316 Lai, M. -J., Bivariate Box Splines for Image Processing, Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 2825 (1996) pp. 474--487 He, W. and Lai, M. -J., Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets, Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, vol. 3169 (1997) pp. 303--314 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Numerical Solution of Steady State Navier-Stokes Equations over Polygons in Stream Function Formulation, Advances in Computational Mathematics, New York, (1998) edited by Z. Chen, Y. Li, C. Micchelli, and Y. Xu, Marcel Dekker pp. 245--277 He, W. and Lai, M. -J., Bivariate Box Spline Wavelets in Sobolev Spaces, Wavelet Applications in Signal and Image Processing VI, proceedings of SPIE, vol. 3458 (1998) pp. 56--66 He, W. and Lai, M. -J., A New Sufficient Condition for the Orthonormality of Refineable Functions, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 121--128 Farmer, K. W. and Lai, M. -J., Scattered Data Interpolation by C^2 Quintic Splines Using Energy Minimization, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 47--54 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Navier-Stokes Equations: Domain Decomposition Technique, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 153--160 Lai, M. -J. and Roach, D. W., Nonseparable Symmetric Wavelets with Short Support, Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII, July, vol. 3813 (1999) pp. 132-146 Lai, M. -J. and Wenston, P., Trivariate C^1 Cubic Splines for Numerical Solution of Biharmonic Equations, Trends in Approximation Theory, Vanderbilt University Press, (2001) edited by K. Kopotun, T. Lyche, and M. Neamtu pp. 224--234 Lai, M. -J. and Roach, D. W., The Nonexistence of Bivariate Symmetric Wavelets with Short Support and Two Vanishing Moments, Trends in Approximation Theory, Vanderbilt University Press, (2001) edited by K. Kopotun, T. Lyche, and M. Neamtu pp. 213--223 Lai, M. -J. and Roach, D. W., Parameterizations of Univariate Orthogonal Wavelets with Short Support, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 369--384 Lai, M. -J., Wenston, P. and Ying, L. A., Bivariate Splines for Exterior Biharmonic Equations, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 385--404 Awanou, G. and Lai, M. -J., $C^1$ Quintic Spline Interpolation over Tetrahedral Partitions, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 1--16 Lai, M. -J., Methods for Constructing Nonseparable Compactly Supported Orthonormal Wavelets, Wavelet Analysis: Twenty Year's Development, (2002) edited by D. X. Zhou, World Scientific pp. 231--251 Baramidze, V. and Lai, M. -J., Volume Data Interpolation by Tensor Products of Spherical and Radial Splines, Advances in Constructive Approximation, Nashboro Press, (2004) edited by M. Neamtu and E. Saff pp. 75--88 Baramidze, V. and Lai, M. -J., Error Bounds for Minimal Energy Interpolatory Spherical Splines, Approximation Theory XI, Nashboro Press, (2005) pp. 25--50 Awanou, G., Lai, M. -J. and Wenston, P., The Multivariate Spline Method for Scattered Data Fitting and Numerical Solution of Partial Differential Equations, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 24--74 This is one of my best works. We explain how to use multivariate splines of arbitrary degree, arbitrary smoothness over any triangulation or tetrahedral partitions for scattered data interpolation and numerical solution of partial differential equations. This describes a veratile tool for numerical approximation of any known and unkown functions in the multivariate setting. Baramidze, V. and Lai, M. -J., Spherical Spline Solution to a PDE on the Sphere, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 75--92 Cho, O. and Lai, M. -J., A Class of Compactly Supported Orthonormal B-Spline wavelets, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 123--151 Lai, M. -J., Lian, J. A. and Cassidy, P., Removal of Gaps among Compound C^1 Bi-Cubic Parametric B-spline Surfaces, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 287--313 Lai, M. -J. and Nam, K., Tight Wavelet Frames over Bounded Domains, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 314--327 Lai, M. -J., Multivariate splines for Data Fitting and Approximation, the conference proceedings of the 12th Approximation Theory, San Antonio, Nashboro Press, (2008) edited by M. Neamtu and Schumaker, L. L. pp. 210--228 Lai, M. -J., Lucier, B. and Wang, J., The Convergence of a Central Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising, Proceedings of SSVM, (2009) edited by X. C. Tai et al pp. 514--526 Baramidze, V. and Lai, M. -J., Convergence of Discrete and Penalized Least Squares of Spherical Splines, Journal of Approximation Theory, (2011) pp. 1091--1106 Hong, Q. and Lai, M. -J., A Bivariate Spline Approach for Image Enhancement, manuscript, (2011) Ettinger, B., Guillas, S. and Lai, M. -J., Bivariate Splines for Functional Regression Models with Application to Ozone Forecasting, Environmetrics, vol. 23 (2012) pp. 317--328 Lanterman, J. Lai, M. -J. and Lanterman, J., Construction of C1 Polygonal Splines over Quadrilateral Partition, Computer Aided Geometric Design, vol. 92 (2022) pp. Paper No.102063 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. Le Mehaute, A. Lai, M. -J. and Le Mehaute, A., A New Kind of Trivariate C^1 Spline Space, Advances in Computational Mathematics, vol. 21 (2004) pp. 273--292 Lai, M. -J., Le Mehaute, A. and Sorokina, T., An Octahedral C^2 Macro-Element, Computer Aided Geometric Design, vol. 23 (2006) pp. 640--654 Lee, J. Lai, M. -J. and Lee, J., Trivariate Spline based Collocation Methods for Numerical Solution to 3D Monge-Ampere Equations, Published Onlne , Journal of Scientific Computing , (2023) Lai, M. -J. and Lee, J., A Multivariate Spline based Collocation Method for Numerical Solution of PDEs, SIAM J. Numerical Analysis, vol. 60 (2022) pp. 2405--2434 Lai, M. -J. and Lee, J., A Multivariate Spline based Collocation Method for Numerical Solution of PDEs, submitted , (2021) Li, C. Wang, X. H., Li, C. and Lai, M. -J., An Unified Convergence Theory for Newton's Type Methods for Zeros of Nonlinear Operators in Banach spaces, BIT, vol. 42 (2002) pp. 206--213 Li, Q. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Li, S. 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. Lai, M. -J., Li, S., Liu, L. Y. and Wang, H., Two Results on the Schatten p-quasi-norm minimization for low rank matrix recovery , submitted, (2012) Li, S.. Shen, Z. M., Lai, M. -J. and Li, S.., Graph-based Semi-supervised Local Clutering with Few Labeled Nodes, accepted, IJCAI , (2023) Li, Xin Kang, Hongmei, Lai, M. -J. and Li, Xin, An economecal representation of PDE solution by using compressive sensing approach, J. Computer Aided Design , vol. 115 (2019) pp. 78--86 Lian, J. A. Chui, C. K., Lai, M. -J. and Lian, J. A., Algorithms for $G\sp 1$ Connection of Multiple Parametric Bicubic NURBS Surfaces, Numerical Algorithms, vol. 23 (2000) pp. 285--313 Lai, M. -J., Lian, J. A. and Cassidy, P., Removal of Gaps among Compound C^1 Bi-Cubic Parametric B-spline Surfaces, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 287--313 Lin, B. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Liu, C. Lai, M. -J., Liu, C. and Wenston, P., Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation, Numerical Methods for P.D.E., vol. 19 (2003) pp. 776--827 Lai, M. -J., Liu, C. and Wenston, P., On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability, Applicable Analysis, vol. 83 (2004) pp. 541--562 Lai, M. -J., Liu, C. and Wenston, P., Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations, Applicable Analysis, vol. 83 (2004) pp. 563--577 Liu, L. Y. 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., Li, S., Liu, L. Y. and Wang, H., Two Results on the Schatten p-quasi-norm minimization for low rank matrix recovery , submitted, (2012) Lai, M. -J. and Liu, L. Y., The Null Space Property for Sparse Recovery From Multiple Measurement Vectors, Applied and Computational Harmonic Analysis, vol. 30 (2011) pp. 402--406 Liu, X. 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. Liu, Y. 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. Liu, Zilong Wen, Jinming, Zhou, Zhengchun, Liu, Zilong, Lai, M. -J. and Tang, Xiaohu, Sharp sufficient conditions for stable recovery of block sparse signals b y block orthogonal matching pursuit, Applied and Computational Harmonic Analysis, (2019) pp. 948--974 Lu, Z. 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) This is one of my best works. We have established the most efficient algorithm to find matrix completion with similar or better accuracy than any other available algorithms. Wang, Z., Lai, M. -J., Lu, Z. and Ye, J., Orthogonal Rank One Matrix Pursuit for Matrix Completion, appeared, International Conference on Machine Learning (ICML) , (2014) Lucier, B. Lai, M. -J., Lucier, B. and Wang, J., The Convergence of a Central Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising, Proceedings of SSVM, (2009) edited by X. C. Tai et al pp. 514--526 Luo, H. Meile, C., Lai, M. -J., Bracco, A.., Luo, H. and Joye, S., Interpretation of oxygen profiles in the aftermath of the BP/Deepwater Horizon hydrocarbon discharge , submitted, (2014) Matt, M. 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. Mckenzie, Daniel. 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. Meile, C. 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. Meile, C., Lai, M. -J., Bracco, A.., Luo, H. and Joye, S., Interpretation of oxygen profiles in the aftermath of the BP/Deepwater Horizon hydrocarbon discharge , submitted, (2014) Lai, M. -J. and Meile, C., Wator Column Oxygen Anomalies in the Aftermath of the BP Oil Spill, a poster in a Gulf of Mexico Research Initiative conference, (2013) Lai, M. -J. and Meile, C., Oxygen Anomalies in the Aftermath of the BP oil spill, a poster in a conference, (2012) Mersmann, C. Lai, M. -J. and Mersmann, C., A bivariate spline Soluton of Helmholtz Equation with large wave number, submitted , (2019) 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. Mersmann, Clayton Deng, Chongyang, Hong, Q. F., Lai, M. -J., Mersmann, Clayton and Xu, Yidong, Multivariate Splines for Curve and Surface Interpolation and Fitting, submitted , (2021) Messi, L. M. Hong, Q. Y., Lai, M. -J. and Messi, L. M., The Rayleigh-Ritz Method for Total Variation Minimization Using Bivariate Splines Functions on Triangulation, submitted, (2014) This is one of my best works. We use bivariate splines to do image enhancements such as denoising, enlargement, impainting, and wrinkle removing. Numerical results are very satisfactory. Check it out. Lai, M. -J. and Messi, L. M., Multiscale Hierarchical Decompositon: Modes and Rates of Convergence, submitted, (2014) 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 Messi, Leopold Matamba Hong, Qianying, Lai, M. -J., Messi, Leopold Matamba and Wang, Jingyue, Galerkin method with splines for total variation minimization, J. Algorithms Computational Technology , vol. 13 (2019) pp. 1--16 Nam, K. Lai, M. -J. and Nam, K., On the Number of Tight Wavelet Framelets Associated with Multivariate Box Splines, Journal of Approximation Theory and its Application, (2008) Lai, M. -J. and Nam, K., Tight Wavelet Frames over Bounded Domains, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 314--327 Pan, R. Lai, M. -J., Pan, R. and Zhao, K., Initial Boundary Value Problem for 2D Viscous Boussinesq Equation, Arch Rational Mech Anal, vol. 199 (2011) pp. 739--760 Peng, Z. 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. Petukhov, A. Lai, M. -J. and Petukhov, A., The Method of Virtual Components for Constructing Wavelet Frames, Applied and Computational Harmonic Analysis, vol. 22 (2007) pp. 304--318 Lai, M. -J. and Petukhov, A., Method of Virtual Components in the Multivariate Setting, Journal of Fourier Analysis and Its Applications, vol. 16 (2010) pp. 471--494 Roach, D. W. Lai, M. -J. and Roach, D. W., Nonseparable Symmetric Wavelets with Short Support, Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII, July, vol. 3813 (1999) pp. 132-146 Lai, M. -J. and Roach, D. W., The Nonexistence of Bivariate Symmetric Wavelets with Short Support and Two Vanishing Moments, Trends in Approximation Theory, Vanderbilt University Press, (2001) edited by K. Kopotun, T. Lyche, and M. Neamtu pp. 213--223 Lai, M. -J. and Roach, D. W., Parameterizations of Univariate Orthogonal Wavelets with Short Support, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 369--384 Schumaker, L. L. Lai, M. -J. and Schumaker, L. L., Spline Functions over Triangulations, Cambridge University Press, (2007) This is one of my best works. This book summarizes a lot of properties of multivariate splines over triangulations or tetrahedral partitions. No other book nor papers contain so much proerties of spline functions over triangulations. Lai, M. -J. and Schumaker, L. L., Scattered Data Interpolation using Piecewise Polynomials of Degree Six, SIAM Journal on Numerical Analysis, vol. 34 (1997) pp. 905--921 Lai, M. -J. and Schumaker, L. L., Approximation Power of Bivariate Splines, Advances in Computational Mathematics, vol. 9 (1998) pp. 251--279 Lai, M. -J. and Schumaker, L. L., On the Approximation Power of Splines on Triangulated Quadrangulations, SIAM Journal on Numerical Analysis, vol. 36 (1999) pp. 143--159 Lai, M. -J. and Schumaker, L. L., Macro-Elements and Stable Local Bases for Splines on Clough-Tocher Triangulations, Numerische Mathematik, vol. 88 (2001) pp. 105--119 Lai, M. -J. and Schumaker, L. L., Quadrilateral Macro-Elements, SIAM Journal on Mathematical Analysis, vol. 33 (2002) pp. 1107--1116 Von Golitschek, M., Lai, M. -J. and Schumaker, L. L., Bounds for Minimal Energy Bivariate Polynomial Splines, Numerische Mathematik, vol. 93 (2002) pp. 315--331 Lai, M. -J. and Schumaker, L. L., Macro-Elements and Stable Local Bases for Splines on Powell-Sabin Triangulations, Mathematics of Computation, vol. 72 (2003) pp. 335--354 Lai, M. -J. and Schumaker, L. L., Trivariate $C^r$ Polynomial Macro-elements, Constructive Approximation, vol. 26 (2007) pp. 11--28 Lai, M. -J. and Schumaker, L. L., Domain Decomposition Method for Scattered Data Fitting, SIAM Journal on Numerical Analysis, vol. 47 (2009) pp. 911-928 Shen, Z. Lai, M. -J. and Shen, Z., An effective approach to semi-supervised cluster extraction , to appear , Journal of Scientific Computing , (2022) Shen, Z. M. Shen, Z. M., Lai, M. -J. and Li, S.., Graph-based Semi-supervised Local Clutering with Few Labeled Nodes, accepted, IJCAI , (2023) Lai, M. -J. and Shen, Z. M., The Kolmogorov Superposition theorem can break the curse of dimensionality when approximating high dimensional functions, submitted, (2022) Feng, R. Z., Huang, A., Lai, M. -J. and Shen, Z. M., Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method, Journal of Computational Math, vol. 41 (2023) pp. 18--38. Shum, C. K. Baramidze, V., Lai, M. -J. and Shum, C. K., Spherical Splines for Data Interpolation and Fitting, SIAM Journal of Scientific Computing, vol. 28 (2006) pp. 241--259 Lai, M. -J., Shum, C. K., Baramidze, V. and Wenston, P., Triangulated Spherical Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp. 695--708 Slavov, G. Lai, M. -J. and Slavov, G., On Recursive Refinement of Convex Polygons, Computer Aided Geometric Design, vol. 45 (2016) pp. 83--90. 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. An expanded and corrected version can be found below. 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. Sorokina, T. Lai, M. -J., Le Mehaute, A. and Sorokina, T., An Octahedral C^2 Macro-Element, Computer Aided Geometric Design, vol. 23 (2006) pp. 640--654 Stoeckler, J. Lai, M. -J. and Stoeckler, J., Construction of Multivariate Compactly Supported Tight Wavelet Frames, Applied and Computational Harmonic Analysis, vol. 21 (2006) pp. 324--348 Sun, Q. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Tang, Xiaohu Wen, Jinming, Zhou, Zhengchun, Liu, Zilong, Lai, M. -J. and Tang, Xiaohu, Sharp sufficient conditions for stable recovery of block sparse signals b y block orthogonal matching pursuit, Applied and Computational Harmonic Analysis, (2019) pp. 948--974 Varghese, Abraham 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 Lai, M. -J. and Varghese, Abraham, On Convergence of the Alternating Projection Method for Matrix Completion and Sparse Recovery Problems , submitted , (2017) Von Golitschek, M. Von Golitschek, M., Lai, M. -J. and Schumaker, L. L., Bounds for Minimal Energy Bivariate Polynomial Splines, Numerische Mathematik, vol. 93 (2002) pp. 315--331 Wang, C. M. 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 Wang, G. Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Complex Domains, Statistica Sinica, vol. 30 (2020) pp. 347--360 Wang, H. 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. Lai, M. -J., Li, S., Liu, L. Y. and Wang, H., Two Results on the Schatten p-quasi-norm minimization for low rank matrix recovery , submitted, (2012) Wang, X. H., Wang, H. and Lai, M. -J., Some Results on Numerical Divided Difference Formulas, Scientia Sinica, vol. Ser. A., 35 (2005) pp. 712--720 Wang, J. Hong, Q., Lai, M. -J. and Wang, J., Convergence Analysis of a Finite Difference Scheme for the Gradient Flow associated with the ROF Model, submitted, (2012) Lai, M. -J. and Wang, J., An Unconstrained l_q Minimization for Sparse Solution of Underdetermined Linear Systems, SIAM Journal of Optimization, vol. 21 (2011) pp. 82--101 Lai, M. -J., Lucier, B. and Wang, J., The Convergence of a Central Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising, Proceedings of SSVM, (2009) edited by X. C. Tai et al pp. 514--526 Wang, Jingyue Hong, Qianying, Lai, M. -J., Messi, Leopold Matamba and Wang, Jingyue, Galerkin method with splines for total variation minimization, J. Algorithms Computational Technology , vol. 13 (2019) pp. 1--16 Wang, L. Wang, L., Wang, G., Lai, M. -J. and Gao, L., Efficient Estimation of Partially Linear Models for Spatial Data over Complex Domains, Statistica Sinica, vol. 30 (2020) pp. 347--360 Lai, M. -J. and Wang, L., Bivariate penalized splines for regression, Statistica Sinica, vol. 23 (2013) pp. 1399--1417 Wang, X. H. Lai, M. -J. and Wang, X. H., A Note to the Remainder of a Multivariate Interpolation Polynomial, Journal of Approximation Theory and its Application, vol. 1 (1984) pp. 57--63 Lai, M. -J. and Wang, X. H., On Multivariate Newtonian Interpolation, Scientia Sinica, vol. 29 (1986) pp. 23--32 Wang, X. H., Li, C. and Lai, M. -J., An Unified Convergence Theory for Newton's Type Methods for Zeros of Nonlinear Operators in Banach spaces, BIT, vol. 42 (2002) pp. 206--213 Wang, X. H., Lai, M. -J. and Yang, S., On Divided Differences of the Remainder of Polynomial Interpolation, Journal of Approximation Theory, vol. 127 (2004) pp. 193--197 Wang, X. H., Wang, H. and Lai, M. -J., Some Results on Numerical Divided Difference Formulas, Scientia Sinica, vol. Ser. A., 35 (2005) pp. 712--72 Wang, Y. Lai, M. -J. and Wan Wang, Z. 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) This is one of my best works. We have established the most efficient algorithm to find matrix completion with similar or better accuracy than any other available algorithms. Wang, Z., Lai, M. -J., Lu, Z. and Ye, J., Orthogonal Rank One Matrix Pursuit for Matrix Completion, appeared, International Conference on Machine Learning (ICML) , (2014) Wen, Jinming Wen, Jinming, Zhou, Zhengchun, Liu, Zilong, Lai, M. -J. and Tang, Xiaohu, Sharp sufficient conditions for stable recovery of block sparse signals b y block orthogonal matching pursuit, Applied and Computational Harmonic Analysis, (2019) pp. 948--974 Wenston, P. Lai, M. -J. and Wenston, P., On Multilevel Bases for Elliptic Boundary Value Problems, Journal of Computational and Applied Mathematics, vol. 71 (1996) pp. 95--113 Lai, M. -J. and Wenston, P., On Schwarz's Domain Decomposition Methods for Elliptic Boundary Value Problems, Numerische Mathematik, vol. 84 (2000) pp. 475--495 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Numerical Solution of Navier-Stokes Equations over Polygons in Stream Function Formulation, Numerical Methods for P.D.E., vol. 16 (2000) pp. 147--183 Lai, M. -J., Liu, C. and Wenston, P., Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation, Numerical Methods for P.D.E., vol. 19 (2003) pp. 776--827 Lai, M. -J. and Wenston, P., $L^1$ Spline Methods for Scattered Data Interpolation and Approximation, Advances in Computational Mathematics, vol. 21 (2004) pp. 293--315 Lai, M. -J., Liu, C. and Wenston, P., On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability, Applicable Analysis, vol. 83 (2004) pp. 541--562 Lai, M. -J., Liu, C. and Wenston, P., Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations, Applicable Analysis, vol. 83 (2004) pp. 563--577 Lai, M. -J. and Wenston, P., Bivariate Splines for Fluid Flows, Computers and Fluids, vol. 33 (2004) pp. 1047--1073 This is one of my best works. We use bivariate splines to numerically solve 2D Navier-Stokes equations based on stream function formulation. Many fluid flows over different domains are simulated. Seeing is believing. Check the paper out. Lai, M. -J., Shum, C. K., Baramidze, V. and Wenston, P., Triangulated Spherical Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp. 695--708 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Numerical Solution of Steady State Navier-Stokes Equations over Polygons in Stream Function Formulation, Advances in Computational Mathematics, New York, (1998) edited by Z. Chen, Y. Li, C. Micchelli, and Y. Xu, Marcel Dekker pp. 245--277 Lai, M. -J. and Wenston, P., Bivariate Spline Method for Navier-Stokes Equations: Domain Decomposition Technique, Approximation Theory IX: Computational Aspects, Vanderbilt University Press, (1998) edited by Charles K. Chui and Larry L. Schumaker pp. 153--160 Lai, M. -J. and Wenston, P., Trivariate C^1 Cubic Splines for Numerical Solution of Biharmonic Equations, Trends in Approximation Theory, Vanderbilt University Press, (2001) edited by K. Kopotun, T. Lyche, and M. Neamtu pp. 224--234 Lai, M. -J., Wenston, P. and Ying, L. A., Bivariate Splines for Exterior Biharmonic Equations, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 385--404 Awanou, G., Lai, M. -J. and Wenston, P., The Multivariate Spline Method for Scattered Data Fitting and Numerical Solution of Partial Differential Equations, Wavelets and Splines, Nashboro Press, (2006) edited by G. Chen and Lai, M. -J. pp. 24--74 Xie, Jiaxin Lai, M. -J., Xie, Jiaxin and Xu, Zhiqiang, Graph Sparsification by Universal Greedy Algorithms, Journal of Computational Math, vol. ?? (2023) pp. Published onlne Xu, Y. Y. 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 Xu, Yidong Deng, Chongyang, Hong, Q. F., Lai, M. -J., Mersmann, Clayton and Xu, Yidong, Multivariate Splines for Curve and Surface Interpolation and Fitting, submitted , (2021) Xu, Zhiqiang Lai, M. -J., Xie, Jiaxin and Xu, Zhiqiang, Graph Sparsification by Universal Greedy Algorithms, Journal of Computational Math, vol. ?? (2023) pp. Published onlne 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 Yang, S. Wang, X. H., Lai, M. -J. and Yang, S., On Divided Differences of the Remainder of Polynomial Interpolation, Journal of Approximation Theory, vol. 127 (2004) pp. 193--197 Ye, J. 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) This is one of my best works. We have established the most efficient algorithm to find matrix completion with similar or better accuracy than any other available algorithms. Lin, B., Li, Q., Sun, Q., Lai, M. -J., Davison, I.., Fan, W. and Ye, J., Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation, submitted, (2014) Wang, Z., Lai, M. -J., Lu, Z. and Ye, J., Orthogonal Rank One Matrix Pursuit for Matrix Completion, appeared, International Conference on Machine Learning (ICML) , (2014) Yin, W. T. 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 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. This is one of my best works. We established the convergence rate of argumented $\ell_1$ minimization algorithms for compressed sensing and matrix completion. 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 Ying, L. A. Lai, M. -J., Wenston, P. and Ying, L. A., Bivariate Splines for Exterior Biharmonic Equations, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, (2002) edited by Chui, C. K., Schumaker, L. L., Stoeckler, J. pp. 385--404 Zhao, K. Lai, M. -J., Pan, R. and Zhao, K., Initial Boundary Value Problem for 2D Viscous Boussinesq Equation, Arch Rational Mech Anal, vol. 199 (2011) pp. 739--760 Zhou, T. Lai, M. -J. and Zhou, T.,
