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Xianjin Chen

Postdoctoral Associate

Institute for Mathematics and Its Applications
University of Minnesota
114 Lind Hall, 207 Church Street SE
Minneapolis, MN 55455, USA

Office: 358 Lind Hall
Phone: (612) 626-0563
Fax: 1-612-626-7370
Email: xchen@ima.umn.edu
URL: http://www.ima.umn.edu/~xchen


---Recent papers


My research is in the area of computational and applied mathematics with a main focus on analysis, development and implementation of numerical methods for multiple unstable solutions (i.e., saddle points) to nonlinear partial differential equations (PDEs) and differential systems arising in physics (e.g., condensed matter physics, nonlinear optics), biology (e.g., pattern formation), etc. We aim to develop and establish some computational theory and methods for multiple solution problems in nonlinear PDEs. In particular, we investigate those unstable solutions in three aspects: characterizations, instability analysis and numerical computations. A little more detail of my research can be found here. Recently, we found some solutions to the Liouville-Gelfand problem using a local min-orthogonal method developed.  


Education

2008,   Ph.D.  Mathematics,  Texas A&M University (TAMU), College Station, TX 
1999,   M.S.   Applied Math., Huazhong University of Science and Technology(HUST), Wuhan, P.R. China
1997,   B.S.   Mathematics,    Huazhong University of Science and Technology, Wuhan, P.R. China

Research Interests

 Numerical methods for multiple (unstable) solution problems
 Nonlinear Schrodinger systems
 Nonlinear elliptic systems
 Scientific computing and computer simulation

Project Experience

"Computational theory and methods for solving multiple solution problems",
      sponsored by US National Science Foundation, #DMS-0311905. Advisor: Jianxin Zhou.
"Modeling and 3-D computer simulations of leaking oxygen in sealed chambers"
      (with professor Huan Qi), 1999-2001
"An optimization-based decision support system for ship scheduling and computer simu-
      lations of operation of permanent sluices in Three-Gorge Dam"
       (with professor Huan Qi and other team members), 1998-2000
  "Simulations and optimal design of Floating Raft Isolation Systems"
       (with professor Huan Qi), 1998-1999

Publications/Preprints

"A numerical method for finding multiple co-existing solutions to nonlinear cooperative
     systems" (with J. Zhou and X. Yao), Applied Numerical Mathematics, pdf
"On homotopy continuation method for computing multiple solutions to the Henon
     equation" (with J. Zhou), Num. Meth. PDEs,   pdf

 "Estimates of Morse index via a local min-orthogonal method" (with J. Zhou), preprint

 "A Min-Max-Orthogonal method for multiple solutions of noncooperative semilinear
     elliptic systems" (with J. Zhou), to appear on Math. Comp.             

          "A numerical simulation on gas leakage and dispersion" (with H. Qi and C. Wu),
               Computer Simulation, Vol. 20, 2(2003), 24-27.
          "3-D simulation of oxygen's leak and diffusion in a sealed chamber" (with H. Qi and
                C. Wu), Computer Simulation, Vol. 19, 6(2002), 15-18
          "Optimal design of a two-stage mounting isolation system by maximum entropy approach"
               (with H. Qi and C. Wu), Journal of Sound & Vibration, U.K., Vol. 243, 4(2001), 591-599
          "System simulation on Three Gorges permanent shiplock’s running" (with F. Lu
                 and H. Qi), Proceedings of Chinese Control and Decision Conference 2000, P.R. China, 709-712
          "Fuzzy random analysis of reliability of communication network" (with S.H. Zhang and
                M. Chen), J. of Huazhong Univ. of Sci. & Tech., Vol. 28, 10(2000), 83-84
          "A state-space approach for simulating a floating raft isolation system"
                (with H. Qi and C. Wu), Noise and Vibration Control, 1(1999), 15-18

Professional Activities

           ”A numerical method and its demo on finding multiple unstable solutions
               to nonlinear variational systems”,
               Seventh Mississippi State-UAB Conference on Differential
               Equations & Computational Simulations, Birmingham, AL November 1-3, 2007

           ”A local min-max-orthogonal method and its convergence”,
               Variational and Topological Methods: Theory, Applications, Numerical Simulations, and
               Open Problems, Northern Arizona University, Flagstaff, AZ, May 23-26, 2007

           ”Saddle point characterization and computation for strongly indefinite functionals”,
                AMS Session on PDEs, Joint Mathematics Meetings, New Orleans, LA, January 5-8, 2007

          ”On characterization of multiple solutions to two classes of nonlinear elliptic systems”,
               Special Session on Nonlinear Elliptic and Parabolic Equations,
               AMS Sectional Meeting, Storrs, CT, October 28-29, 2006

  ”A min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems”,
      Special Session on Nonlinear Differential Equations: Methods and Applications,
      AMS Sectional Meeting, Salt Lake City, UT, October 7-8, 2006

  ”A stable method to solve nonlinear elliptic PDE systems for multiple vector solitons”,
      AMS-SIAM Special Session on Nonlinear Dynamical Systems, Joint Mathematics Meetings,
      San Antonio, TX, January 12-15, 2006

  ”Finding multiple co-existing solutions to semilinear elliptic systems via a local min-orthogonal method”,
     Applied Mathematics Seminar, Department of Mathematics, Texas A&M University,
     College Station, TX, November, 2005

        29th Texas PDE conference, University of Texas at Arlington, 03/2006
        27th Texas PDE conference, TAMU, 04/2004
        Texas PDE conference, University of Texas at San Antonio, 02/2002

Computer Skills

       Operating systems: Unix/Linux, Windows
       Languages: C/C++, Visual C++, MATLAB/FEMLAB, Latex, HTML
       Packages developed:
            -- A nonlinear solver package for several classes of semilinear elliptic problems
               including semilinear elliptic equations or systems, nonlinear biharmonic equations
            -- A 2-D symmetric triangular mesh generator (written in Matlab) for symmetrical
               domains such as rectangles, circles/ellipses, annuli, etc

Numerical Demos

       "A real-time demo on solving nonlinear PDEs for multiple solutions",
            Variational and Topological Methods: Theory, Applications, Numerical Simulations,
            and Open Problems, Northern Arizona University, Flagstaff, AZ, May 2007
       "A real-time demo on finding multiple solutions to the Henon equation" ,
            Seminar in Applied Math, Texas A&M University, TX, March 2007
       "A 3-D numerical simulation on oxygen leak and dispersion in a sealed chamber",
            Wuhan Ship Design Institute, Wuhan, China, May 2001

Technical reports

       (Co-author) Modeling and 3-D simulation of oxygen's leak and diffusion in a sealed chamber, 2001
       (Co-author) A ship scheduling optimization decision support system and computer simulation
                                on running of Three Gorges permanent ship lock, 2000


 

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Last modified: 10-20-2008