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The main objective is to understand the mathematical foundationof electronic structure analysis, to develop and analysis efficient algorithms. W. E and J.F. Lu. The continuum limit and QM-continuum approximation of quantum mechanical models of solids. Comm. Math. Sci., vol. 5, no. 3, pp. 679-696, 2007. General issues in multiscale modeling S. Chen, W. E, Y. Liu and C.-W. Shu. A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations. J. Comput. Phys., vol. 225, no. 2, pp. 1314-1330, 2007. W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367-450, 2007.W. E and J.F. Lu. Seamless multiscale modeling via dynamics on fiber bundles. Comm. Math. Sci., vol. 5, no. 3, pp. 649-663, 2007. X. Yue and W. E. 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Effectiveness of implicit methods for stiff stochastic differential equations. Comm. Comput. Phys., vol. 3, no. 2, pp. 295-307, 2008. W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007. W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005. W. E, D. Liu and E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 1544-1585, 2005. W. E and X.-T. Li. Analysis of the heterogeneous multiscale method for gas dynamics. Methods Appl. Anal., vol. 11, no. 4, pp. 557-572, 2004. Stochastic chemical kinetic systems W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007. W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005. Multiscale modeling of solids W. Guo, T. P. Schulze and W. E. Simulation of impurity diffusion in a strained nanowire using off-lattice KMC. Comm. Comput. Phys., vol. 2, no. 1, pp. 164-176, 2007.X. Li and W. E. Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath. Phys. Rev. B, vol 76, no. 10, 104107, 2007.J.Z. Yang and W. E. Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models. Phys. Rev. B, vol. 74, no 18, 184110, 2006. Y. Xiang, H. Wei, P.B. Ming and W. E. A generalized Peierls�Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu. Acta Materialia, in press, available online 14 January 2008.W. E, J.-F. Lu, J.Z. Yang. Uniform accuracy of the quasicontinuum method. Phys. Rev. B, vol. 74, 214115, 2006. X.-T. Li and W. E. Variational boundary conditions for molecular dynamics simulation of solids at low temperature. Comm. Comput. Phys., vol. 1, No. 1, 135-175, 2006. N. Choly, G. Lu, W. E and E. Kaxiras. Multiscale simulations in simple metals: A density-functional based methodology. Phys. Rev. B, vol. 71, 094101, 2005. X.-T. Li and W. E. Multiscale modeling of the dynamics of solids at finite temperature. J. Mech. Phys. Solids, vol. 53, 1650-1685, 2005. W. E and X.-T. Li. Multiscale modeling of crystalline solids. Handbook of Materials Modeling, Part A, edited by S. Yip., pp. 1491-1506, Springer Netherlands, 2005. Y. Xiang and W. E. Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B, vol. 69, no. 3, 035409, 2004. Y. Xiang, D.J. Srolovitz, L.-T. 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Phys., vol. 104, no. 1-2, pp. 221-253, 2001. M.I. Mendelev, D.J. Srolovitz and W. E. Grain-boundary migration in the presence of diffusing impurities: simulations and analytical models. Philos. Mag. A, vol. 81, no. 9, pp. 2243-2269, 2001. T. Schulze and W. E. A continuum model for the growth of epitaxial films. J. Crystal Growth, vol. 222, no. 1-2, pp. 414-425, 2001. W. E and N.K. Yip. Continuum limits of step flow models. EQUADIFF 99 Proc. Intl. Conf. Differential Equations, vol. 1, 2 (Berlin, 1999), pp. 448-453, World Sci. Publishing, River Edge, NJ, 2000. R. Caflisch, W. E, M. Gyure, B. Merriman and C. Ratsch. Kinetic model for a step edge in epitaxial growth. Phys. Rev. E, vol. 59, no. 6, pp. 6879-6887, 1999. Multiscale modeling of complex fluids D. Hu, P. Zhang and W. E. Continuum theory of a moving membrane. Phys. Rev. E, vol. 75, no. 4, 041605, 2007. W. E and P. Zhang. A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Methods Appl Anal., vol 13, no. 2, pp. 181-198, 2006. D. Zhou, P. Zhang and W. E. Modified models of polymer phase separation. Phys. Rev. E, vol. 73, 061801, 2006. W. Ren and W. E. Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys., vol. 204, no. 1, pp. 1-26, 2005.S. Succi, W. E and E. Kaxiras. Lattice boltzmann methods for multiscale fluid problems. Handbook of Materials Modeling, Part B, pp. 2475-2486, Springer Netherlands, 2005.X. Nie, S. Chen, W. E and M. Robbins. Hybrid continuum-atomistic simulation of singular corner flow. Phys. Fluids, vol. 16, no. 10, pp. 3579-3591, 2004.T.-J. Li, E. Vanden-Eijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. Non-Newtonian Fluid Mechanics, vol. 121, no. 2-3, pp. 117-125, 2004. X. Nie, S. Chen, W. E and M.O. Robbins. A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. J. Fluid Mech., vol. 500, pp. 55-64, 2004. W. E, T.-J. Li and P.-W. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys., vol. 248, no. 2, pp. 409-427, 2004. T.-J. Li, E. Vanden-Eijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. Non-Newtonian Fluid Mechanics, vol. 121, 117-125, 2004. W. E, T.-J. Li, P.-W. Zhang. Convergence of a stochastic method for the modeling of polymeric fluids. Acta Math. Appl. Sin., vol. 18, no. 4, pp. 529-536, 2002. C.B. Muratov and W. E. Theory of phase separation kinetics in polymer-liquid crystal systems. J. Chem. Phys., vol. 116, no. 11, pp. 4723-4734, 2002. P. Palffy-Muhoray, T. Kosa and W. E. Brownian motors in the photoalignment of liquid crystals. Appl. Phys. A, vol. 75, no. 2, pp. 293-300, 2002. Q. Wang, W. E, C. Liu, P.-W. Zhang. Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. Phys. Rev. E, vol. 65, no. 5, 051504, 2002. W. E and P. Palffy-Muhoray. Dynamics of filaments during the isotropic-smectic A phase transition. J. Nonlin. Sci., vol. 9, no. 4, pp. 417-437, 1999. W. E and P. Palffy-Muhoray. Orientational ratchets and angular momentum balance in the Janossy effect. Mol. Cryst. Liq. Cryst., vol. 320, no. 1, pp. 193-206, 1998. W. E. Nonlinear continuum theory of smectic-A liquid crystals. Arch. Rat. Mech. Anal., vol. 137, no. 2, pp. 159-175, 1997. W. E and P. Palffy-Muhoray. Phase separation in incompressible systems. Phys. Rev. E, vol. 55, no. 4, pp. R3844-R3846 , 1997. F. Otto and W. E. Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys., vol. 107, no. 23, pp. 10177-10184, 1997.Multiscale methods for multiscale PDEs X. Yue and W. E. The local micro-scale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size. J. Comput. Phys., vol. 222, no. 2, pp. 556-572, 2007.W. E and B. Engquist. The heterogeneous multi-scale method for homogenization problems. Multiscale Methods in Sci. and Eng., pp. 89-110. Lect. Notes in Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005. W. E, P.B. Ming and P.-W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121-156, 2005.X. Yue and W. E. Numerical methods for multiscale transport equations and application to two-phase porous media flow. J. Comput. Phys., vol. 210, no. 2, pp. 656-675, 2005. A. Abdulle and W. E. Finite difference heterogeneous multi-scale method for homogenization problems. J. Comput. Phys., vol. 191, no. 1 pp. 18-39, 2003. The moving contact line problem and micro-fluidicsW. Ren and W. E. Boundary conditions for the moving contact line problem. Phys. Fluids, vol. 19, 022101, 2007. Homogenization theory B. Engquist and W. E. Large time behavior and homogenization of solutions of two-dimensional conservation laws. Comm. Pure Appl. Math., vol. 46, no. 1, pp. 1-26, 1993. W. E and C.-W. Shu. Effective equations and the inverse cascade theory for Kolmogorov flows. Phys. Fluids A, vol. 5, no. 4, pp. 998-1010, 1993. W. E. Propagation of oscillations in the solutions of 1-D compressible fluid equations. Comm. Partial Differential Equations, vol. 17, no. 3-4, pp. 545-552, 1992. W. E. Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math., vol. 45, no. 3, pp. 301-326, 1992. W. E. Homogenization of scalar conservation laws with oscillatory forcing terms. SIAM J. Appl. Math., vol. 52, no. 4, pp. 959-972, 1992. W. E and D. Serre. Correctors for the homogenization of conservation laws with oscillatory forcing terms. Asymptotic Anal., vol. 5, no. 4, pp. 311-316, 1992. W. E. A class of homogenization problems in the calculus of variations. Comm. Pure Appl. Math., vol. 44, no. 7, pp. 733-759, 1991. W. E and R.V. Kohn. The initial value problem for measure-valued solutions of a canonical 2x2 system with linearly degenerate fields. Comm. Pure Appl. Math., vol. 44, no. 8-9, pp. 981-1000, 1991. W. E and H. Yang. Numerical study of oscillatory solutions of the gas-dynamic equations. Stud. Appl. Math., vol. 85, no. 1, pp. 29-52, 1991. W. E and T.Y. Hou. Homogenization and convergence of the vortex method for 2-D Euler equations with oscillatory vorticity fields. Comm. Pure Appl. Math., vol. 43, no. 7, pp. 821-855, 1990. Analysis and modeling of stochastic problemsAnalysis of stochastic partial differential equations W. E and D. Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys., vol. 108, no. 5-6, pp. 1125-1156, 2002. W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 27-46. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001. W. E and J.C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 1386-1402, 2001. W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83-106, 2001. W. E. Stochastic hydrodynamics. Current Developments in Mathematics, 2000, pp. 109-147, Intl. Press, Somerville, MA, 2000. W. E, K. Khanin, A. Mazel and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math., vol. 151, no. 3, pp. 877-960, 2000. W. E and Ya. Sinai. Recent results on mathematical and statistical hydrodynamics. Russ. Math. Survey, vol. 55, no. 4, 635-666, 2000. W. E, Yu. Rykov and Ya. Sinai. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys., vol. 177, no. 2, pp. 349-380, 1996.Rare events: String method, minimum action method and transition path theory W. E, W. Ren, E. Vanden-Eijnden. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys., vol. 126, no. 16, 164103, 2007.T. Qian, W. Ren, J. Shi, W. E and P. Sheng. Numerical study of metastability due to tunneling: The quantum string method. Phys. A, vol. 379, no. 2, pp. 491-502, 2007. W. E and E. Vanden-Eijnden. Towards a theory of transition paths. J. Stat. Phys., vol. 123, No. 3, 503-523, 2006. W. E, W. Ren and E. Vanden-Eijnden. Transition pathways in complex systems: Reaction coordinates, iso-committor surfaces and transition tubes. Chem. Phys. Lett., vol. 413, no. 1-3, 242-247, 2005. W. Ren, E. Vanden-Eijnden, P. Maragakis and W. E. Transition pathways in complex systems: Application of the finite temperature string method to the alanine dipeptide. J. Chem. Phys., vol. 123, 134109, 2005. W. E, W. Ren and E. Vanden-Eijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109, 6688-6693, 2005. W. E, W. Ren, E. Vanden-Eijnden. Minimum action method for the study of rare events. Comm. Pure Appl. Math., vol. 57, no. 5, pp. 637-656, 2004.W. E and E. Vanden-Eijnden. Metastability, conformation dynamics, and transition pathways in complex systems. Multiscale Modelling and Simulation, pp. 35-68, Lect. Notes Comput. Sci. Eng., vol. 39, Springer, Berlin, 2004. W. E, W. Ren and E. Vanden-Eijnden. Energy landscape and thermally activated switching of submicron-sized ferromagnetic elements. J. Appl. Phys., vol. 93, no. 4, pp. 2275-2282, 2003. W. E, W. Ren and E. Vanden-Eijnden. String method for the study of rare events. Phys. Rev. B, vol. 66, no. 5, 052301, 2002. W. E, W. Ren and E. Vanden-Eijnden. Energy landscapes and rare events. ICM Report, vol. 1, pp. 621-630, Higher Ed. Press, Beijing, 2002. Stochastic chemical kinetic systems W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007. W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005. ``Burgers turbulence'' and passive scalarturbulence W. E and E. Vanden-Eijnden. A note on generalized flows. Phys. D, vol. 183, no. 3-4, pp. 159-174, 2003.W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 27-46. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001. W. E and E. Vanden-Eijnden. Turbulent Prandtl number effect on passive scalar advection. Phys. D, vol. 152-153, pp. 636-645, 2001. W. E and E. Vanden-Eijnden. Statistical theory for the stochastic Burgers equation in the inviscid limit. Comm. Pure Appl. Math., vol. 53, no. 7, pp. 852-901, 2000. W. E and E. Vanden-Eijnden. Another note on forced Burgers turbulence. Phys. Fluids, vol. 12, no. 1, pp. 149-154, 2000. W. E and E. Vanden-Eijnden. Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl. Acad. Sci., vol. 97, no. 15, pp. 8200-8205, 2000. W. E and E. Vanden-Eijnden. On the statistical solution of the Riemann equation and its implications for Burgers turbulence. Phys. Fluids, vol. 11, no. 8, pp. 2149-2153, 1999. W. E and E. Vanden-Eijnden. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett., vol. 83, no. 13, pp. 2572-2575, 1999. W. E, K. Khanin, A. Mazel and Ya. Sinai. Probability distribution functions for the random forced Burgers equation. Phys. Rev. Lett., vol. 78, no. 10, pp. 1904-1907, 1997. M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys., vol. 172, no. 1, pp. 13-38, 1995. M. Avellaneda, R. Ryan and W. E. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids, vol. 7, no. 12, pp. 3067-3071, 1995. General issues in stochastic modelingC.B. Muratov, E. Vanden-Eijnden, W. E. Noise can play an organizing role for the recurrent dynamics in excitable media. Proc. Natl. Acad. Sci., vol. 104, no. 3, pp. 702-707, 2007. C.B. Muratov, E. Vanden-Eijnden and W. E. Self-induced stochastic resonance in excitable systems. Phys. D, vol. 210, no. 3-4, pp. 227-240, 2005. P. Palffy-Muhoray, T. Kosa, W. E. Brownian ratchets and the photoalignment of liquid crystals. Braz. J. Phys., vol.32 no.2b, pp. 552-563, Sao Paulo, 2002. P. Palffy-Muhoray, T. Kosa and W. E. Dynamics of a Light Driven Molecular Motor. Mol. Cryst. Liq. Cryst., vol. 375, no. 1, pp. 577-592, 2002. T. Kosa, W. E and P. Palffy-Muhoray. Brownian motors in the photoalignment of liquid crystals. Intl J. Eng. Sci., vol. 38, no. 9-10, pp. 1077-1084, 2000.W. E and P. Palffy-Muhoray. Domain size in the presence of random fields. Phys. Rev. E, vol. 57, no. 1, pp. 135-137, 1998. Other topicsIncompressible flow: Projection methods, vorticity-based methods and gauge methods W. E and J.-G. Liu. Gauge method for viscous incompressible flows. Comm. Math. Sci., vol. 1, no. 2, pp. 317-332, 2003. W. E and J.-G. Liu. Projection method III: Spatial discretization on the staggered grid. Math. Comp., vol. 71, no. 237, pp. 27-47, 2002.W. E. Numerical methods for viscous incompressible flows: some recent advances. Advances in scientific computing, p. 29, Science Press, 2001. J.-G. Liu and W. E. Simple finite element method in vorticity formulation for incompressible flows. Math. Comp., vol. 70, no. 234, pp. 579-593, 2001. W. E and J.-G. Liu. Gauge finite element method for incompressible flows. Intl. J. Numer. Methods in Fluids, vol. 34, no. 8, pp. 701-710, 2000. W. E and J.-G. Liu. Finite difference schemes for incompressible flows in the velocity-impulse density formulation. J. Comput. Phys., vol. 130, no. 1, 67-76, 1997. W. E and J.-G. Liu. Finite difference methods for 3D viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids. J. Comput. Phys., vol. 138, no. 1, 57-82, 1997. W. E and J.-G. Liu. Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys., vol. 124, no. 2, pp. 368-382, 1996. W. E and J.-G. Liu. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys., vol. 126, no. 1, pp. 122-138, 1996. W. E and J.-G. Liu. Projection method II: Godunov-Ryabenki analysis. SIAM J. Numer. Anal., vol. 33, no. 4, pp. 1597-1621, 1996. W. E and J.-G. Liu. Finite difference schemes for incompressible flows in vorticity formulations. Vortex flows and related numerical methods, II (Montreal, PQ, 1995), pp. 181-195, ESAIM Proc., vol. 1, Soc. Math. Appl. Indust., Paris, 1996. W. E and J.-G. Liu. Projection method I: Convergence and numerical boundary layers. SIAM J. Numer. Anal., vol. 32, no. 4, pp. 1017-1057, August, 1995.Z.-T. Chen and W. E. Convergence of Legendre methods for Navier-Stokes equations. J. Comput. Math., vol. 12, no. 4, pp. 298-311, 1994.W. E and C.-W. Shu. A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. J. Comput. Phys., vol. 110, no. 1, pp. 39-46, 1994.W. E. Convergence of Fourier methods for the Navier-Stokes equations. SIAM J. Numer. Anal., vol. 30, no. 3, pp. 650-674, 1993.W. E. Convergence of spectral methods for Burgers' equation. SIAM J. Numer. Anal., vol. 29, no. 6, pp. 1520-1541, 1992. A posterior error estimates Work done in Master degree thesis, under the guidance ofProfessor Hongci Huang at the Chinese Academy of Sciences. The mainfocus is on finite element for problems with corner singularities.Issues discussed include: A posterior error estimates, direct andinverseerror estimates on locally refined domains, convergence of multi-grid methods on such domains, etc. 2b1af7f3a8