Theodoropoulos / Hans / Kazantzis Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

1. Auflage 2006
ISBN: 978-3-540-35888-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 560 Seiten

Model reduction and coarse-graining are important in many areas of science and engineering. How does a system with many degrees of freedom become one with fewer? How can a reversible micro-description be adapted to the dissipative macroscopic model? These crucial questions, as well as many other related problems, are discussed in this book. All contributions are by experts whose specialities span a wide range of fields within science and engineering.

Autoren/Hrsg.

Theodoropoulos, Constantinos

Hans,

Kazantzis, Nikolas

Konstantinos,

Inhaltsverzeichnis

1;Preface;5
1.1;References;8
2;Contents;9
3;Computation of Invariant Manifolds;12
3.1;A New Model Reduction Method for Nonlinear Dynamical Systems Using Singular PDE Theory;13
3.1.1;1 Introduction;13
3.1.2;2 Mathematical Preliminaries;15
3.1.3;3 Main Results;16
3.1.4;4 Conclusions;23
3.1.5;References;23
3.2;A Versatile Algorithm for Computing Invariant Manifolds;26
3.2.1;1 Introduction;26
3.2.2;2 Invariant Manifolds;28
3.2.3;3 Discrete Sections;33
3.2.4;4 The Discrete Graph Transform;36
3.2.5;5 Numerical Implementation;40
3.2.6;6 An Application;43
3.2.7;References;44
3.3;Covering an Invariant Manifold with Fat Trajectories;47
3.3.1;1 Introduction;47
3.3.2;2 Basic Definitions;49
3.3.3;3 Fat Trajectories;51
3.3.4;4 Flying Disks;52
3.3.5;5 Interpolation;56
3.3.6;6 Example;57
3.3.7;References;61
3.4;Ghost ILDM-Manifolds and Their Identification;63
3.4.1;1 Introduction;63
3.4.2;2 Theoretical Background;64
3.4.3;3 Ghost ILDM-Manifolds Examples;75
3.4.4;4 Criteria for Ghost -Manifolds Identification;82
3.4.5;5 Conclusions;85
3.4.6;References;85
3.5;Dynamic Decomposition of ODE Systems: Application to Modelling of Diesel Fuel Sprays;88
3.5.1;1 Introduction;88
3.5.2;2 Dynamic Fast-Slow Decomposition: Underlying Philosophy;90
3.5.3;3 Decomposition of the System of Equations;92
3.5.4;4 Choice of Decomposition;95
3.5.5;5 Application;97
3.5.6;6 Conclusions;102
3.5.7;References;102
3.6;Model Reduction of Multiple Time Scale Processes in Non- standard Singularly Perturbed Form;105
3.6.1;1 Introduction;105
3.6.2;2 Standard Singularly Perturbed Form;107
3.6.3;3 Nonstandard Singularly Perturbed Form;109
3.6.4;4 Application;114
3.6.5;5 Conclusion;118
3.6.6;References;118
4;Coarse-Graining and Ideas of Statistical Physics;120
4.1;Basic Types of Coarse-Graining;121
4.1.1;1 Introduction;121
4.1.2;2 The Ehrenfests’ Coarse-Graining;127
4.1.3;3 Coarse-Graining by Filtering;153
4.1.4;4 Errors of Models, e-trajectories and Stable Properties of Structurally Unstable Systems;166
4.1.5;5 Conclusion;173
4.1.6;References;175
4.2;Renormalization Group Methods for Coarse- Graining of Evolution Equations;181
4.2.1;1 Introduction and Basic Formalism;181
4.2.2;2 RSRG for the Selection of Relevant Degrees of Freedom;186
4.2.3;3 DMRG and the Time-Evolution of Strongly Correlated Many- Body Systems;194
4.2.4;4 Conclusions;207
4.2.5;References;207
4.3;A Stochastic Process Behind Boltzmann’s Kinetic Equation and Issues of Coarse Graining;211
4.3.1;1 Motivation and Problem;211
4.3.2;2 Markov Processes;214
4.3.3;3 Nonlinear Fokker-Planck Equations;215
4.3.4;4 Boltzmann’s Kinetic Equation;217
4.3.5;5 Boltzmann Process;218
4.3.6;6 Gaussian Boltzmann Process;220
4.3.7;7 Application: Diffusion Coefficient;223
4.3.8;8 Perspectives;225
4.3.9;References;226
4.4;Finite Difference Patch Dynamics for Advection Homogenization Problems;229
4.4.1;1 Introduction;229
4.4.2;2 Model Problems;233
4.4.3;3 Patch Dynamics;234
4.4.4;4 Convergence Results;237
4.4.5;5 Numerical Results for Advection Problems;241
4.4.6;6 Conclusions;248
4.4.7;References;248
4.5;Coarse-Graining the Cyclic Lotka-Volterra Model: SSA and Local Maximum Likelihood Estimation;251
4.5.1;1 Introduction;251
4.5.2;2 The Lattice Lotka-Volterra Model;252
4.5.3;3 Equation Free Computation;253
4.5.4;4 Estimation Procedure;254
4.5.5;5 Illustrations of Equation-Free Computation;258
4.5.6;6 Discussion;263
4.5.7;References;268
4.6;Relations Between Information Theory, Robustness and Statistical Mechanics of Stochastic Uncertain Systems via Large Deviation Theory;272
4.6.1;1 Introduction;272
4.6.2;2 Thermodynamics and Statistical Mechanics;276
4.6.3;3 Robustness of Stochastic Uncertain Systems: General Setting;279
4.6.4;4 Robustness of Stochastic Uncertain Systems: an Energy Constraint Formulation;281
4.6.5;5 Robustness of Stochastic Uncertain Systems: a Relative Entropy Constraint Formulation;286
4.6.6;6 The Large Deviations Principle Applied to Diffusion Processes;291
4.6.7;7 Conclusion;293
4.6.8;References;293
5;Kinetics and Model Reduction;296
5.1;Exactly Reduced Chemical Master Equations;297
5.1.1;1 Introduction;297
5.1.2;2 Stochastic Population Modeling and the Chemical Master Equation;300
5.1.3;3 Methods and Results;307
5.1.4;4 Conclusions;314
5.1.5;References;315
5.2;Model Reduction in Kinetic Theory;318
5.2.1;1 Introduction;318
5.2.2;2 Basic Kinetic Theory;319
5.2.3;3 Chapman-Enskog Method;321
5.2.4;4 Grad Moment Method;323
5.2.5;5 Combining the Chapman-Enskog and Grad Methods;325
5.2.6;6 Order of Magnitude Method;327
5.2.7;7 Relations Between the Various Sets of Equations;330
5.2.8;8 Applications;331
5.2.9;9 Conclusions and Outlook;337
5.2.10;References;339
5.3;Novel Trajectory Based Concepts for Model and Complexity Reduction in ( Bio) Chemical Kinetics;343
5.3.1;1 Introduction;343
5.3.2;2 Model Reduction: Constrained Relaxation of Chemical Forces and Minimal Entropy Production Trajectories;345
5.3.3;3 Complexity Reduction of Biochemical Reaction Networks;352
5.3.4;References;362
5.4;Dynamics of the Plasma Sheath;365
5.4.1;1 Introduction;365
5.4.2;2 The Euler Equations with Planar, Radical, and Spherical symmetry;366
5.4.3;3 Collisional and Collisionless Plasmas;366
5.4.4;4 Dynamics of the Plasma Sheath;367
5.4.5;5 Generalization to Non-Symmetric Case;369
5.4.6;References;371
6;Mesoscale and Multiscale Modeling;372
6.1;Construction of Stochastic PDEs and Predictive Control of Surface Roughness in Thin Film Deposition;373
6.1.1;1 Introduction;373
6.1.2;2 Preliminaries;375
6.1.3;3 Model Construction;381
6.1.4;4 Predictive Control;388
6.1.5;5 Conclusions;397
6.1.6;References;398
6.2;Lattice Boltzmann Method and Kinetic Theory;401
6.2.1;1 Introduction;401
6.2.2;2 Minimal Kinetic Model;403
6.2.3;3 Grad’s Moment System and the Kinetic Model: Linear Case;404
6.2.4;4 Lattice Boltzmann Method;408
6.2.5;5 Flow in a Lid-Driven Micro-Cavity;409
6.2.6;6 Reduced Description of the Flow;412
6.2.7;7 Application: Outflow Condition in Lattice Boltzmann Simulations;413
6.2.8;8 Discussion;417
6.2.9;References;419
6.3;Numerical and Analytical Spatial Coupling of a Lattice Boltzmann Model and a Partial Differential Equation;421
6.3.1;1 Introduction;421
6.3.2;2 Models for One-Dimensional Diffusive Systems;423
6.3.3;3 Constrained Runs Scheme;427
6.3.4;4 Spatial Coupling;429
6.3.5;5 Numerical Results;432
6.3.6;6 Conclusions and Future Work;437
6.3.7;References;438
6.4;Modelling and Control Considerations for Particle Populations in Particulate Processes Within a Multi- Scale Framework;440
6.4.1;1 Introduction;440
6.4.2;2 Population Balance Models;443
6.4.3;3 Solution Techniques for Population Balance Models;445
6.4.4;4 Distribution Control Considerations;452
6.4.5;5 Summary and conclusions;457
6.4.6;References;459
6.5;Diagnostic Goal-Driven Reduction of Multiscale Process Models;462
6.5.1;1 Introduction;462
6.5.2;2 The Model Reduction Problem;464
6.5.3;3 Prediction-Based Diagnosis and Loss Prevention;468
6.5.4;4 Multiscale Process Models and Model Reduction;471
6.5.5;5 Case Study: Model Reduction of a Granule Bed for Diagnosis;473
6.5.6;6 Conclusions;481
6.5.7;References;483
6.6;Understanding Macroscopic Heat/ Mass Transfer Using Meso- and Macro- Scale Simulations;485
6.6.1;1 Introduction;485
6.6.2;2 Numerical Methodology;487
6.6.3;3 Conductive Transport – the Case of Composite Materials;490
6.6.4;4 Convective Transport – the Case of Laminar Flow;494
6.6.5;5 Convective Transport – the Case of Turbulent Transport;500
6.6.6;6 Summary and Conclusions;503
6.6.7;References;504
6.7;An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation;510
6.7.1;1 Introduction;510
6.7.2;2 Problem Formulation;512
6.7.3;3 In Situ Adaptive Tabulation;513
6.7.4;4 Applications;517
6.7.5;5 Conclusion;525
6.7.6;References;527
6.8;A Reduced Input/Output Dynamic Optimisation Method for Macroscopic and Microscopic Systems;529
6.8.1;1 Introduction;529
6.8.2;2 Reduced Dynamic Optimisation for Input/Output Simulators;532
6.8.3;3 Multiple Shooting Approach for Dynamic Optimization;534
6.8.4;4 The Newton-Picard-Based Dynamic Optimisation Scheme;536
6.8.5;5 Numerical Examples;540
6.8.6;6 Conclusions;548
6.8.7;References;550