Numerical Analysis of Multiscale Computations

Name: Numerical Analysis of Multiscale Computations | Proceedings of a Winter Workshop at the Banff International Research Station 2009
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Proceedings of a Winter Workshop at the Banff International Research Station 2009

Björn Engquist Olof Runborg Yen-Hsi R. Tsai(Editor)

Springer (Publisher)

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Published on 14. October 2011

X, 430 pages

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Intro
Numerical Analysis of Multiscale Computations
Preface
Acknowledgements
Contents
Explicit Methods for Stiff Stochastic Differential Equations
1 Introduction
2 Stiff Stochastic Systems and Stability
3 Chebyshev Methods
4 The S-ROCK Methods
4.1 Construction of the S-ROCK Methods
4.2 Accuracy of the S-ROCK Methods
5 Extended Mean-Square Stability and Damping
6 Numerical Illustrations
References
Oscillatory Systems with Three Separated Time Scales:Analysis and Computation
1 Introduction
2 Effective Behavior Across Different Time Scales
2.1 Slowly Changing Quantities
2.2 Multiscale Charts
3 A Homogenization Approach
3.1 A Two Scales Example
3.2 Three Scales: Example 1
3.3 Three Scales: Example 2
3.4 Observations
4 Numerical Algorithms
4.1 Accuracy and Efficiency
5 Examples
5.1 Harmonic Oscillators
5.2 An Example Motivated by the Fermi-Pasta-Ulam (FPU) Problem
6 Summary
References
Variance Reduction in Stochastic Homogenization: The Technique of Antithetic Variables
1 Introduction
1.1 Homogenization Theoretical Setting
1.2 Numerical Approach
1.3 The Technique of Antithetic Variables
1.4 A Brief Summary of Our Former Results
2 Variance Reduction for Problems Involving Correlations or Anisotropy
2.1 Correlated Cases
2.1.1 Influence of Correlation: Identical Local Behaviour
2.1.2 Centered vs Equidistributed Correlation Structure
2.1.3 Longer Correlation Lengths
2.2 Anisotropic Cases
2.2.1 Test Cases
2.2.2 Numerical Results
3 Variance Reduction for Eigenproblems
References
A Stroboscopic Numerical Method for HighlyOscillatory Problems
1 Introduction
2 A Modified Equation Approach to Averaging
3 A Numerical Method
4 Examples
5 Numerical Experiments
5.1 Constant Step-Sizes
5.2 Variable Step-Sizes
References
The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint
1 Introduction
2 Electronic Structure Models for Finite Systems
2.1 The N-Body Schrödinger Model
2.2 The N-Body Schrödinger Model for Non-interacting Electrons
2.3 Density Operators
2.4 The Hartree Model and Other Density Operator Models of Electronic Structures
3 The Hartree Model for Crystals
3.1 Basics of Fourier and Bloch-Floquet Theories
3.2 Perfect Crystals
3.3 Crystals with Local Defects
4 Dielectric Response of a Crystal
4.1 Series Expansion of the Time-Independent Response
4.2 Properties of Qm,0F and m,0F for Small Amplitude Defects
4.3 Dielectric Operator and Macroscopic DielectricPermittivity
4.4 Time-Dependent Response
References
Fast Multipole Method Using the Cauchy Integral Formula
1 Introduction
2 Cauchy's Integral Formula and Low-Rank Approximations
3 Connection with Fourier and Laplace Transforms
4 Construction of Fast Methods
5 Interpolation and Anterpolation
6 Error Analysis
7 Detailed Error Analysis
8 Preliminary Numerical Results
9 Conclusion
References
Tools for Multiscale Simulation of Liquids Using Open Molecular Dynamics
1 Introduction
2 Open MD: Molecular Dynamics for Open Systems
2.1 Open MD Setup
2.2 Imposition of the Macro-State
2.2.1 State-Coupling Based on Constrained Dynamics
2.2.2 The Flux-Based Scheme
2.3 Mass and Density Profile at the Buffer
2.3.1 Distribution of the External Force
2.3.2 The Buffer Mass: Particle Insertion and Deletion
3 Using Adaptive Resolution: The Mesoscopic Interface
4 HybridMD: Particle-Continuum Hybrid Scheme
4.1 Time Coupling
4.2 Hybrids Based on State Coupling
4.3 Hybrids Based on Flux Exchange
4.4 Mass Transfer and Continuity in Flux Based Schemes
5 Conclusions and Perspectives
References
Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time
1 Introduction
2 HMM for the Wave Equation and Finite Time
2.1 One Numerical Example
3 HMM for the Wave Equation over Long Time
3.1 The HMM Algorithm for Long Time
3.2 A Long Time Computation with HMM
3.3 Non-periodic Material
3.4 Theory
3.5 Stability Analysis of the Macro Scheme for the Long Time Effective Equation
4 Conclusions
References
Numerical Homogenization via Approximation of the SolutionOperator
1 Introduction
1.1 Background
1.2 Mathematical Problem Formulation
1.3 Coarse-Graining of the Differential Operator (Homogenization)
1.4 Coarse-Graining of the Solution Operator
2 Data-Sparse Matrices
3 Case Study: A Discrete Laplace Equation on a Square
3.1 Problem Formulation
3.2 Model Problems
3.3 Compressibility of the Solution Operator
3.4 Techniques for Computing the Solution OperatorThat Fully Resolve the Micro-Structure
3.5 Techniques Accelerated by Collecting Statisticsfrom a Representative Volume Element
3.6 Fusing a Homogenized Model to a Locally FullyResolved Region
4 Case Study: Two-Phase Media
5 Generalizations
6 Conclusions
References
Adaptive Multilevel Monte Carlo Simulation
1 Introduction
1.1 A Single Level Posteriori Error Expansion
2 Adaptive Algorithms and MultilevelVariance Reduction
3 A Stopped Diffusion Example
References
Coupled Coarse Graining and Markov Chain Monte Carlofor Lattice Systems
1 Introduction
2 MCMC Methods
2.1 Mixing Times and Speed of Convergence
3 The Coupled CGMC Method
3.1 The Algorithm
3.2 The Rate of Convergence
4 Extended Lattice Systems
4.1 Coarse-Graining of Microscopic Systems
4.2 Multiscale Decomposition and Splitting Methods for MCMC
4.3 Microscopic Reconstruction
5 Example: Short and Long-Range Interactions
6 Conclusions
References
Calibration of a Jump-Diffusion Process Using Optimal Control
1 Introduction
2 The Forward Equation
3 The Optimal Control Problem
4 Reconstructing the Volatility Surface and Jump Intensity
4.1 The Hamiltonian System
4.2 Discretization
4.3 The Newton Method
5 Numerical Examples
5.1 Artificial Data No Jumps
5.2 Artificial Data with Jumps
5.3 Real Data
References
Some Remarks on Free Energy and Coarse-Graining
1 Motivation
1.1 Coarse-Graining of Thermodynamic Quantities
1.2 Coarse-Graining of Dynamical Quantities
1.3 Outline of the Article
1.4 Notation
2 Computing Macroscopic Stress-Strain Relationsfor One-Dimensional Chains of Atoms
2.1 The Nearest Neighbour (NN) Case
2.1.1 Computing the Strain for a Given Stress
2.1.2 Computing the Stress for a Given Strain
2.1.3 Equivalence of Stress-Strain Relations in the Thermodynamic Limit
2.2 The Next-to-Nearest Neighbour (NNN) Case
2.2.1 Computing the Strain for a Given Stress
2.2.2 Computing the Stress for a Given Strain
2.2.3 Equivalence of Stress-Strain Relations in the Thermodynamic Limit
2.2.4 Numerical Computation of F' and Comparison with the Zero Temperature Model
3 A Coarse-Graining Procedure in the Dynamical Setting
3.1 Measuring Distances Between Probability Measures
3.2 Effective Dynamics
3.3 The Proof in a Simple Two-Dimensional Case
3.4 Numerical Results
3.4.1 A Three Atom Molecule
3.4.2 The Butane Molecule Case
References
Linear Stationary Iterative Methods for the Force-Based Quasicontinuum Approximation
1 Introduction
2 The QC Approximations and Their Stability
2.1 Function Spaces and Norms
2.2 The Atomistic Model
2.2.1 The Critical Strain F*
2.3 The Local QC Approximation (QCL)
2.4 The Force-Based QC Approximation (QCF)
2.5 The Original Energy-Based QCApproximation (QCE)
2.6 The Quasi-Nonlocal QC Approximation (QNL)
3 Stability and Spectrum of the QCF operator
3.1 Spectral Properties of LqcfF in U0,2=l2e
3.2 Spectral Properties of LqcfF in U1,2
4 Linear Stationary Iterative Methods
5 The Richardson Iteration (P=I)
5.1 Numerical Example for the Richardson Iteration
6 Preconditioning with QCL (P=LqclF=AF L)
6.1 Analysis of the QCL Preconditioner in U2,8
6.2 Analysis of the QCL Preconditioner in U1,8
6.3 Analysis of the QCL Preconditioner in U1,2
6.4 Numerical Example for QCL-Preconditioning
7 Preconditioning with QCE (P=LqceF): Ghost-ForceCorrection
References
Analysis of an Averaging Operator for Atomic-to-ContinuumCoupling Methods by the Arlequin Approach
1 Introduction
2 Particle and Continuum Model Problems
2.1 Particle Model
2.2 Continuum Model
2.3 Calibration of Continuum Model
3 Coupling Method with Averaging Operator
3.1 Energy of the Coupled System
3.2 Averaging Coupling Operator
3.3 Formulation of the Coupled Problem
4 Mathematical Analysis of the Coupling Method
5 Finite Element Formulation
6 Numerical Results
6.1 One-Dimensional Numerical Results
6.1.1 Overlap Region Composed of One RVE
6.1.2 Overlap Region Composed of Several RVE's
6.1.3 An Example with a Large Number of Particles
6.1.4 Simulation of a Defect
6.2 Two-Dimensional Numerical Results
7 Conclusion
References
A Coupled Finite Difference - Gaussian Beam Method for High Frequency Wave Propagation
1 Introduction
2 Gaussian Beams
3 Motivating Examples
4 Local Finite Difference Method
5 Hybrid Method
5.1 Example: Double Slit Experiment
5.2 Example: Sound Speed with Inclusion
6 Conclusion
References
Editorial Policy
Lecture Notes in Computational Science and Engineering
Monographs in Computational Science and Engineering
Texts in Computational Science and Engineering

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Numerical Analysis of Multiscale Computations

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