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 < e)
4.3 Error estimates (h > ย )
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