Multiscale and Stochastic Modeling in Computational Science
Developing systematic mathematical framework and computational methodologies for stochastic and multiscale modeling in science and engineering.
The mathematical aspects include :
- Analysis of Stochastic Partial Differential Equations
- Mathematical Foundation of Molecular Dynamics
- Mathematical Theory of Solids: From Atomistic to Macroscopic Scales
The numerical aspects include :
- Design of multiscale algorithms
- Analysis of multiscale algorithms
- Boundary conditions in molecular dynamics
- Stochastic simulation algorithms
- String method for studying rare events
Other particular areas of application include:
- Fluids: complex polymeric and/or liquid crystalline flows, micro-fluidics, contact line dynamics.
- Solids: material defects such as phase boundaries, dislocations and cracks; interaction of microscopic mechanisms and macroscopic deformation, nucleation and thermal effects.
- Chemistry: reaction rates and pathways.
- Biology: multiscale modeling of biopolymers such as proteins and DNA, membrane deformation and dynamics.
1993-1995 Alfred P. Sloan Foundation Fellowship
1997-2002 Presidential Faculty Fellowship
1999 Feng Kang Prize in Scientific Computing
X. Li and W. E, "Multiscale modeling of dynamics of solids at finite temperature", J. Mech. Phys. Solids, accepted for publication.
W. E, B. Engquist, X. Li, W. Ren, E. Vanden-Eijnden, "The Heterogeneous Multiscale Method: A Review".
W. Ren and W. E, "Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics", J. Comput. Phys., in press.
W. E, D. Liu, and E. Vanden-Eijnden, "Analysis of Multiscale Methods for Stochastic Di erential Equations", Comm. on Pure and Applied Math., 1-48 (2003).
W. E and X.T. Li, "Multiscale Modeling of Crystalline Solids", submitted to the Handbook of Computational Material Science.
W. E, X. Li, E. Vanden-Eijnden, "Some Recent Progress in Multiscale Modeling".
T. Li, E. Vanden-Eijnden, P. Zhang and W. E, "Stochastic models of polymeric fluids at small Deborah number", submitted to J. Non-Newtonian Fluids.
A. Abdulle and W. E, "Finite difference heterogeneous multi-scale method for homogenization problems", J. Comput. Phys., 191, 18-39 (2003).
L. T. Cheng and W. E, "The heterogeneous multiscale method for interface dynamics", to appear in Contemporary Mathematics: A Special Volume in Honour of Stan Osher, S. Y. Cheng, C. W. Shu and T. Tang eds.
C. Muratov and W. E, "Theory of phase separation kinetics in polymer-liquid crystal systems", J. Chem. Phys., 116(11), 4723-4734 (2002).
W. E, "Analysis of the heterogeneous multiscale method for ordinary differential equations", Comm. Math. Sci., 1(3), 423-436 (2003).
W. E and B. Engquist, "The heterogeneous multi-scale methods", Comm. Math. Sci., 1, 87-133 (2003).
W. E and B. Engquist, "The heterogeneous multi-scale method for homogenization problems", submitted to SIAM J. Multiscale Modeling and Simulations.
W. E and B. Engquist, "Multiscale Modeling and Computation", Notices of the AMS, 50(9), 1062-1070 (2003).
W. E, B. Engquist and Z. Huang, "Heterogeneous multi-scale method -- a general methodology for multi-scale modeling", Phys. Rev. B, 67(9), 092101 (2003).
W. E and Z. Huang, "Matching conditions in atomistic-continuum modeling of materials", Phys. Rev. Lett., 87 (13), 135501 (2001).
W. E and Z. Huang, "A dynamic atomistic-continuum method for the simulation of crystalline materials", J. Comput. Phys., 182, 234-261 (2002).
W. E, P. B. Ming and P. W. Zhang, "Analysis of the heterogeneous multiscale method for elliptic homogenization problems", preprint.
P. B. Ming and P. W. Zhang, "Analysis of the heterogeneous multiscale method for dynamic homogenization problems", preprint.
T. Schulze, P. Smereka and W. E, "Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth", J. Comput. Phys., 189, 197-211 (2003).
Editorial Boards: Journal of American Mathematical Society, Acta Mathematica Sinica, Journal of Computational Mathematics, International Mathematics Research Notes, Communications of Contemporary Mathematics, Methods of Analysis and Applications.