Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena (eBook)

Preface 5
References 8
Contents 9
Computation of Invariant Manifolds 12
A New Model Reduction Method for Nonlinear Dynamical Systems Using Singular PDE Theory 13
1 Introduction 13
2 Mathematical Preliminaries 15
3 Main Results 16
4 Conclusions 23
References 23
A Versatile Algorithm for Computing Invariant Manifolds 26
1 Introduction 26
2 Invariant Manifolds 28
3 Discrete Sections 33
4 The Discrete Graph Transform 36
5 Numerical Implementation 40
6 An Application 43
References 44
Covering an Invariant Manifold with Fat Trajectories 47
1 Introduction 47
2 Basic Definitions 49
3 Fat Trajectories 51
4 Flying Disks 52
5 Interpolation 56
6 Example 57
References 61
Ghost ILDM-Manifolds and Their Identification 63
1 Introduction 63
2 Theoretical Background 64
3 Ghost ILDM-Manifolds Examples 75
4 Criteria for Ghost -Manifolds Identification 82
5 Conclusions 85
References 85
Dynamic Decomposition of ODE Systems: Application to Modelling of Diesel Fuel Sprays 88
1 Introduction 88
2 Dynamic Fast-Slow Decomposition: Underlying Philosophy 90
3 Decomposition of the System of Equations 92
4 Choice of Decomposition 95
5 Application 97
6 Conclusions 102
References 102
Model Reduction of Multiple Time Scale Processes in Non- standard Singularly Perturbed Form 105
1 Introduction 105
2 Standard Singularly Perturbed Form 107
3 Nonstandard Singularly Perturbed Form 109
4 Application 114
5 Conclusion 118
References 118
Coarse-Graining and Ideas of Statistical Physics 120
Basic Types of Coarse-Graining 121
1 Introduction 121
2 The Ehrenfests’ Coarse-Graining 127
3 Coarse-Graining by Filtering 153
4 Errors of Models, e-trajectories and Stable Properties of Structurally Unstable Systems 166
5 Conclusion 173
References 175
Renormalization Group Methods for Coarse- Graining of Evolution Equations 181
1 Introduction and Basic Formalism 181
2 RSRG for the Selection of Relevant Degrees of Freedom 186
3 DMRG and the Time-Evolution of Strongly Correlated Many- Body Systems 194
4 Conclusions 207
References 207
A Stochastic Process Behind Boltzmann’s Kinetic Equation and Issues of Coarse Graining 211
1 Motivation and Problem 211
2 Markov Processes 214
3 Nonlinear Fokker-Planck Equations 215
4 Boltzmann’s Kinetic Equation 217
5 Boltzmann Process 218
6 Gaussian Boltzmann Process 220
7 Application: Diffusion Coefficient 223
8 Perspectives 225
References 226
Finite Difference Patch Dynamics for Advection Homogenization Problems 229
1 Introduction 229
2 Model Problems 233
3 Patch Dynamics 234
4 Convergence Results 237
5 Numerical Results for Advection Problems 241
6 Conclusions 248
References 248
Coarse-Graining the Cyclic Lotka-Volterra Model: SSA and Local Maximum Likelihood Estimation 251
1 Introduction 251
2 The Lattice Lotka-Volterra Model 252
3 Equation Free Computation 253
4 Estimation Procedure 254
5 Illustrations of Equation-Free Computation 258
6 Discussion 263
References 268
Relations Between Information Theory, Robustness and Statistical Mechanics of Stochastic Uncertain Systems via Large Deviation Theory 272
1 Introduction 272
2 Thermodynamics and Statistical Mechanics 276
3 Robustness of Stochastic Uncertain Systems: General Setting 279
4 Robustness of Stochastic Uncertain Systems: an Energy Constraint Formulation 281
5 Robustness of Stochastic Uncertain Systems: a Relative Entropy Constraint Formulation 286
6 The Large Deviations Principle Applied to Diffusion Processes 291
7 Conclusion 293
References 293
Kinetics and Model Reduction 296
Exactly Reduced Chemical Master Equations 297
1 Introduction 297
2 Stochastic Population Modeling and the Chemical Master Equation 300
3 Methods and Results 307
4 Conclusions 314
References 315
Model Reduction in Kinetic Theory 318
1 Introduction 318
2 Basic Kinetic Theory 319
3 Chapman-Enskog Method 321
4 Grad Moment Method 323
5 Combining the Chapman-Enskog and Grad Methods 325
6 Order of Magnitude Method 327
7 Relations Between the Various Sets of Equations 330
8 Applications 331
9 Conclusions and Outlook 337
References 339
Novel Trajectory Based Concepts for Model and Complexity Reduction in ( Bio) Chemical Kinetics 343
1 Introduction 343
2 Model Reduction: Constrained Relaxation of Chemical Forces and Minimal Entropy Production Trajectories 345
3 Complexity Reduction of Biochemical Reaction Networks 352
References 362
Dynamics of the Plasma Sheath 365
1 Introduction 365
2 The Euler Equations with Planar, Radical, and Spherical symmetry 366
3 Collisional and Collisionless Plasmas 366
4 Dynamics of the Plasma Sheath 367
5 Generalization to Non-Symmetric Case 369
References 371
Mesoscale and Multiscale Modeling 372
Construction of Stochastic PDEs and Predictive Control of Surface Roughness in Thin Film Deposition 373
1 Introduction 373
2 Preliminaries 375
3 Model Construction 381
4 Predictive Control 388
5 Conclusions 397
References 398
Lattice Boltzmann Method and Kinetic Theory 401
1 Introduction 401
2 Minimal Kinetic Model 403
3 Grad’s Moment System and the Kinetic Model: Linear Case 404
4 Lattice Boltzmann Method 408
5 Flow in a Lid-Driven Micro-Cavity 409
6 Reduced Description of the Flow 412
7 Application: Outflow Condition in Lattice Boltzmann Simulations 413
8 Discussion 417
References 419
Numerical and Analytical Spatial Coupling of a Lattice Boltzmann Model and a Partial Differential Equation 421
1 Introduction 421
2 Models for One-Dimensional Diffusive Systems 423
3 Constrained Runs Scheme 427
4 Spatial Coupling 429
5 Numerical Results 432
6 Conclusions and Future Work 437
References 438
Modelling and Control Considerations for Particle Populations in Particulate Processes Within a Multi- Scale Framework 440
1 Introduction 440
2 Population Balance Models 443
3 Solution Techniques for Population Balance Models 445
4 Distribution Control Considerations 452
5 Summary and conclusions 457
References 459
Diagnostic Goal-Driven Reduction of Multiscale Process Models 462
1 Introduction 462
2 The Model Reduction Problem 464
3 Prediction-Based Diagnosis and Loss Prevention 468
4 Multiscale Process Models and Model Reduction 471
5 Case Study: Model Reduction of a Granule Bed for Diagnosis 473
6 Conclusions 481
References 483
Understanding Macroscopic Heat/ Mass Transfer Using Meso- and Macro- Scale Simulations 485
1 Introduction 485
2 Numerical Methodology 487
3 Conductive Transport – the Case of Composite Materials 490
4 Convective Transport – the Case of Laminar Flow 494
5 Convective Transport – the Case of Turbulent Transport 500
6 Summary and Conclusions 503
References 504
An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation 510
1 Introduction 510
2 Problem Formulation 512
3 In Situ Adaptive Tabulation 513
4 Applications 517
5 Conclusion 525
References 527
A Reduced Input/Output Dynamic Optimisation Method for Macroscopic and Microscopic Systems 529
1 Introduction 529
2 Reduced Dynamic Optimisation for Input/Output Simulators 532
3 Multiple Shooting Approach for Dynamic Optimization 534
4 The Newton-Picard-Based Dynamic Optimisation Scheme 536
5 Numerical Examples 540
6 Conclusions 548
References 550