Contents -- Preface -- Zhaojun Bai, Wenbin Chen, Richard Scalettar, Ichitaro Yamazaki: Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model -- Abstract -- 1 Hubbard model and QMC simulations -- 1.1 Hubbard model -- 1.1.1 Hubbard model with no hopping -- 1.1.2 Hubbard model without interaction -- 1.2 Determinant QMC -- 1.2.1 Computable approximation of distribution operator P -- 1.2.2 Algorithm -- 1.2.3 Physical measurements -- 1.3 Hybrid QMC -- 1.3.1 Computable approximation of distribution operator P -- 1.3.2 Algorithm -- 1.3.3 Physical measurements -- 2 Hubbard matrix analysis -- 2.1 Hubbard matrix -- 2.2 Basic properties -- 2.3 Matrix exponential B = et rK -- 2.4 Eigenvalue distribution of M -- 2.4.1 The case U = 0 -- 2.4.2 The case U = 0 -- 2.5 Condition number of M -- 2.5.1 The case U = 0 -- 2.5.2 The case U = 0 -- 2.6 Condition number of M(k) -- 3 Self-adaptive direct linear solvers -- 3.1 Block cyclic reduction -- 3.2 Block structural orthogonal factorization -- 3.3 A hybrid method -- 3.4 Self-adaptive reduction factor k -- 3.5 Self-adaptive block cyclic reduction method -- 3.6 Numerical experiments -- 4 Preconditioned iterative linear solvers -- 4.1 Iterative solvers and preconditioning -- 4.2 Previous work -- 4.3 Cholesky factorization -- 4.4 Incomplete Cholesky factorizations -- 4.4.1 IC -- 4.4.2 Modified IC -- 4.5 Robust incomplete Cholesky preconditioners -- 4.5.1 RICl -- 4.5.2 RIC2 -- 4.5.3 RIC3 -- 4.6 Performance evaluation -- 4.6.1 Moderately interacting systems -- 4.6.2 Strongly interacting systems -- Appendix A. Updating algorithm in DQMC -- A.1 Rank-one updates -- A.2 Metropolis ratio and Green's function computations -- Appendix B Particle-hole transformation -- B.1 Algebraic identities -- B.2 Particle-hole transformation in DQMC -- B.3 Particle-hole transformation in the HQMC.

B.4 Some identities of matrix exponentials -- Acknowledgments -- References -- Albert C. Fannjiang: Introduction to Propagation, Time Reversal and Imaging in Random Media -- 1 Scalar diffraction theory -- 1.1 Introduction -- 1.2 Kirchhoff's theory of diffraction -- 1.3 Huygens-Fresnel principle -- 1.4 Fresnel and Fraunhofer diffraction -- 1.5 Focal spot size and resolution -- 2 Approximations: weak fluctuation -- 2.1 Born approximation -- 2.2 Rytovapproximation -- 2.3 The extended Huygens-Fresnel principle -- 2.4 Paraxial approximation -- 3 The Wigner distribution -- 4 Markovian approximation -- 4.1 White-noise scaling -- 4.2 Markovian limit -- 5 Two-frequency transport theory -- 5.1 Paraxial waves -- 5.1.1 Two-frequency radiative transfer equations -- 5.1.2 The longitudinal and transverse cases -- 5.2 Spherical waves -- 5.2.1 Geometrical radiative transfer -- 5.2.2 Spatial (frequency) spread and coherence bandwidth -- 5.2.3 Small-scale asymptotics -- 6 Application: time reversal -- 6.1 Spherical wave -- 6.2 Paraxial wave -- 6.3 Anomalous focal spot -- 6.4 Duality and turbulence-induced aperture -- 6.5 Coherence length -- 6.6 Broadband time reversal communications -- 7 Application: imaging in random media -- 7.1 Imaging of phase objects -- 7.2 Long-exposure imaging -- 7.3 Short-exposure imaging -- 7.4 Coherent imaging of multiple point targets in Rician media -- 7.4.1 Differential scattered field in clutter -- 7.4.2 Imaging functions -- 7.4.3 Numerical simulation with a Rician medium -- 7.5 Coherent imaging in a Rayleigh medium -- Acknowledgement -- References -- Thomas Y. Hou: Multiscale Computations for Flow and Transport in Porous Media -- Abstract -- 1 Introduction -- 2 Review of homogenization theory -- 2.1 Homogenization theory for elliptic problems -- 2.2 Homogenization for hyperbolic problems -- 2.3 Convection of microstructure.

3 Numerical homogenization based on sampling techniques -- 3.1 Convergence of the particle method -- 3.2 Vortex methods for incompressible flows -- 4 Numerical upscaling based on multiscale finite element methods -- 4.1 Multiscale finite element methods for elliptic PDEs -- 4.2 Error estimates (h €) -- 4.4 The over-sampling technique -- 4.5 Performance and implementation issues -- 4.6 Applications -- 4.7 Brief overview of mixed finite element and finite volume element methods -- 4.8 MsFEM using limited global information -- 4.9 Analysis -- 5 Multiscale finite element methods for nonlinear partial differential equations -- 5.1 Multiscale finite volume element method (MsFVEM) -- 5.2 Examples of V h 10 -- 5.3 Convergence of MsFEM for nonlinear partial differential equations -- 5.4 Multiscale finite element methods for nonlinear parabolic equations -- 5.5 Numerical results -- 5.6 Generalizations of MsFEM and some remarks -- 6 Multiscale simulations of two-phase immiscible flow in adaptive coordinate system -- 6.1 Numerical averaging across streamlines -- 6.2 N urnerical results -- 7 Conclusions -- References -- Chun Liu: An Introduction of Elastic Complex Fluids: An Energetic Variational Approach -- Abstract -- 1 Introduction -- 2 Calculus of variations -- 2.1 Euler-Lagrange equations -- 2.2 Direct methods -- 2.3 Convexity -- 2.4 Dynamics -- 2.5 Hamilton's principle -- 2.5.1 Flow map and deformation tensor -- 2.5.2 Variation of the domain v.s. variation of the function -- 2.5.3 Least action principle -- 2.6 Constraint problems -- 2.6.1 Harmonic maps -- 2.6.2 Liquid crystals -- 2.6.3 Methods of penalty -- 3 Navier-Stokes equation -- 3.1 Newtonian fluids -- 3.1.1 Existence of global weak solution -- 3.1.2 Existence of classical solution -- 3.1.3 Regularity -- 3.1.4 Partial regularity -- 4 Viscoelastic materials.

4.1 Flow map and deformation tensor -- 4.2 Force balance and Oldroyd-B systems -- 4.3 Energetic variational formulation -- 5 Liquid crystal flows -- 5.1 Ericksen-Leslie theory -- 5.2 Existence and regularity -- 6 Free interface motion in mixtures -- 6.1 An energetic variational approach with phase field method -- 6.2 Marangoni-Benard convection -- 6.3 Mixtures involving liquid crystals -- 7 Magneto hydrodynamics (MHD) -- 7.1 Introduction -- 7.2 The evolution of the magnetic field -- 7.3 The energy law -- 7.4 The linear momentum equation -- 7.5 The dynamics of magnetic field lines -- References -- Qi Wang: Introduction to Kinetic Theory for Complex Fluids -- Abstract -- 1 Introduction -- 2 A primer for equilibrium thermodynamics -- 3 Basics of statistical mechanics -- 4 Equilibrium distribution of the end-toend vector in simple polymer models -- 5 Kinetic theory for polymers -- 5.1 Langevin equation -- 5.2 System of constraints -- 5.3 Bead-spring (Rouse) chain model -- References.