Research interests


Numerical analysis, scientific computing, numerical solutions of partial differential equations, anisotropic diffusion problems, mesh adaptation, image processing, parallel computing, mathematical modeling and simulation

Current Projects


Moving Mesh Finite Element Method for Phase-field Modeling of Brittle Fractures
crack propagation    multiple crack propagation
See F. Zhang, W. Huang, X. Li, and S. Zhang, Moving Mesh Finite Element Simulation for Phase-Field Modeling of Brittle Fracture and Convergence of Newton's Iteration, Journal of Computational Physics 356: 127-149,  2018. (DOI: 10.1016/j.jcp.2017.11.033) or (arXiv:1706.05449).

Anisotropic Mesh Adaptation for Partial Differential Equations
Partial Differential Equations are used to describe many phynomena in science and engineering. Anisotropic mesh adaptation is a powerful tool to improve the efficiency and accuracy of the computations in finding numerical solutions of those mathematical models. Meanwhile, some physical problems do not have negative values. So it is important to preserve the nonnegativity of the numerical solution. We develop mesh adaptation techniques to solve the mathematical models efficiently while preserving the nonnegativity of the solutions.

Anisotropic Mesh Adaptation in Image Processing
Triangular meshes have gained much interest in image representation and have been widely used in image processing. In this project, we investigate the application of anisotropic mesh adaptation (AMA) methods to image processing, including representation, scaling, smoothing, segmentation, and etc.

GRANTs


  1. Mathematical Studies on Problems in Disease Modeling and Image Processing in Contex of South Africa, Co-PI with Patidar (PI) and Vaidya (Co-PI), University of Missouri South African Education Program (UMSAEP), Summer 2018, $4,900.
  2. Mathematical Foundation of Finite Element Methods II, Co-PI with He (PI), Chen (Co-PI), and Montgomery-Smith (Co-PI), University of Missouri Inter-campus Course Sharing Grant, 09/01/2018 – 05/31/2019, $9,731.
  3. Bringing the Field of Applied Mathematics from Shadowed State to the Frontline Discipline, PI (with Bani-Yaghoub, Rhee, and Vaidya), Funding for Excellence Program, University of Missouri - Kansas City, 01/01/2016-12/31/2016, $30,000.
  4. Numerical Computations in Image Processing, PI, University of Missouri Research Board, 06/01/2015-05/31/2016, $18,837.
  5. Modeling and Analysis of the Next Life Event - Relating Theory and Data, PI (with Bani-Yaghoub), Development and Research Grant, H&R Block, 11/01/2014-05/30/2015, $60,000.
  6. Mathematical Foundation of Finite Element Methods, Co-PI with He (PI) and Chen (Co-PI), University of Missouri Inter-campus Course Sharing Grant, 09/01/2014 – 05/31/2015, $7,769.

Publications


  1. X. Li and J. Martinez, "Quantitative comparison of white matter segmentation for brain MR images", (accepted), 2019.
  2. M.S. Richman, X. Li and A.N. Caruso, "Deficiency of the extrapolation-length method for modeling the interface of a ferroelectric-graphene heterostructure", (submitted), 2019.
  3. F. Zhang, W. Huang, X. Li and S. Zhang, "A study on phase-field models for brittle fracture", (submitted), 2019.
  4. N.K. Vaidya, X. Li and F.B. Wang, "Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics", Discrete & Continuous Dynamical Systems - B, 24(1): 321-349, 2019. (doi:10.3934/dcdsb.2018099.)
  5. F. Zhang, W. Huang, X. Li and S. Zhang, "Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of Newton’s iteration", Journal of Computational Physics, 356: 127-149, 2018.
  6. F. Zhang, W. Huang, X. Li and S. Zhang, "A study on moving mesh finite element solution of phase-field models for hydraulic fracturing", International Journal of Chemical Engineering and Applications, 9(2): 51-57, 2018. (ICPPE 2018 conference paper.) (doi:10.18178/ijcea.2018.9.2.698.)
  7. X. Li, "Anisotropic mesh adaptation for finite element solution of anisotropic porous medium equation", Computers & Mathematics with Applications, 75: 2086–2099, 2018. (doi:10.1016/j.camwa.2017.08.005.)
  8. X. Li and W. Huang, "Anisotropic mesh adaptation for 3D anisotropic diffusion problems with application to fractured reservoir simulation", Numerical Mathematics: Theory, Methods and Applications, 10(4): 913-940, 2017. (doi:10.4208/nmtma.2017.m1625.)
  9. X. Li and W. Huang, "A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations", Applied Mathematics and Computation, 309: 49-67, 2017. (doi:10.1016/j.amc.2017.03.038.)
  10. X. Li, "Fourier series for functions defined on arbitrary limited intervals with polynomial expansion", American Review of Mathematics and Statistics, 4(2): 10-17, 2016. (doi:10.15640/arms.v4n2a2.)
  11. X. Li, "Anisotropic mesh adaptation for image representation", J. Image Video Proc. 2016: 26, 2016. (doi:10.1186/s13640-016-0132-7.)
  12. X. Li and W. Huang, "Maximum principle for the finite element solution of time-dependent anisotropic diffusion problems", Numer. Meth. PDEs, 29(6): 1963-1985, 2013. (doi:10.1002/num.21784.)
  13. X. Li and W. Huang, "An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems", J. Comput. Phys., 229: 8072-8094, 2010.
  14. W. Huang and X. Li, "Anisotropic mesh adaptation method for the finite element solution of variational problems", Fin. Elem. Anal. Des., 46: 61-73, 2010.
  15. W. Huang, L. Kamenski, and X. Li, "Anisotropic mesh adaptation for variational problems using error estimation based on hierarchical bases", Canadian Applied Mathematics Quarterly (Special issue for the 30th anniversary of CAIMS), 17: 501-522, 2009.
  16. S. McCool, X. Li and G. P. Wilhite, "Flow of a Polyacrylamide/Chromium Acetate System in a Long Conduit", SPE Journal, 14(1): 54-66, 2009.
  17. X. Li, D. Svyatskiy, and M. Shashkov, "Mesh adaptation and discrete maximum principle for 2D anisotropic diffusion problems", LANL technical report, LA-UR 10-01227, 2007.
  18. S. McCool, X. Li and G. P. Wilhite, "Effect of shear on flow properties during placement and on syneresis after placement of a polyacrylamide-chromium acetate gelant", Society of Petroleum Engineers, SPE 106059-MS, 2007. (doi:10.2118/106059-MS.)
  19. Q. Wen, S. Zhang, L. Wang, Y. Liu and X. Li, "The effect of proppant embedment upon the long-term conductivity of fractures", Journal of Petroleum Science and Engineering, 55: 221-227, 2007.

Presentations


 • SIAM Central States Section 3rd Annual Meeting, Colorado State University, October, 2017.
 • SIAM Central States Section 2nd Annual Meeting, University of Arkansas at Little Rock, Oct. 1st, 2016.
 • SIAM Central States Section 2nd Annual Meeting, University of Arkansas at Little Rock, Oct. 2nd, 2016
• Computational and Applied Math Seminar, University of Kansas, Dec. 2015.
 • Colloquium talk, Department of Physics and Astronomy, University of Missouri-Kansas City, Apr. 2015
 • SIAM Central States Section 1st Annual Meeting, Missouri University of Science and Technology, Apr. 2015
 • AMS Central Spring Sectional Meeting, Michigan State University, Mar. 2015
 • 2014 AARMS-CRM Workshop on Adaptive Methods for PDEs, Memorial University, St. John’s, NL, Canada, Aug. 2014
 • Colloquium Talk, Department of Mathematics and Statistics, Missouri University of Science and Technology, Feb. 2014
 • First Central Region Conference on Numerical Analysis and Dynamical Systems, University of Kansas, May 2013
 • South Central Conference on Advanced Numerical Methods and Applications, University of Arkansas, Little Rock, Apr. 2013
 • AMS Special Session on Numerical Analysis and Finite Element Methods, Joint Mathematics Meetings, San Diego, CA, Jan. 2013
 • AMS Spring Central Section Meeting, special session on Numerical Analysis and Scientific Computing, Univ. of Kansas, Mar. 2012
 • Finite Element Circus Fall 2011, University of Connecticut at Avery Point, Oct. 2011
 • Mathematics Seminar, University of Central Arkansas, Oct. 2011
 • Applied Mathematics Seminar, University of Arkansas at Little Rock, Sept. 2011
 • Midwest Numerical Analysis Day 2010, Iowa State University, Ames, Iowa, Apr. 2010

Softwares


AMAimage (in progress)