Spectral Methods for Uncertainty Quantification (eBook)

Preface 7
Acknowledgements 9
Contents 11
Introduction: Uncertainty Quantification and Propagation 17
Introduction 17
Simulation Framework 19
Uncertainties 20
Uncertainty Propagation and Quantification 21
Objectives 21
Probabilistic Framework 22
Data Uncertainty 22
Approach to UQ 23
Monte Carlo Methods 24
Spectral Methods 25
Overview 26
Basic Formulations 30
Spectral Expansions 31
Karhunen-Loève Expansion 32
Problem Formulation 32
Properties of KL Expansions 34
Practical Determination 35
Rational Spectra 35
Non-rational Spectra 38
Numerical Resolution 38
Gaussian Processes 41
Polynomial Chaos Expansion 42
Polynomial Chaos System 44
One Dimensional PC Basis 45
Multidimensional PC Basis 45
Truncated PC Expansion 47
Generalized Polynomial Chaos 49
Independent Random Variables 49
Chaos Expansions 51
Dependent Random Variables 51
Spectral Expansions of Stochastic Quantities 53
Random Variable 53
Random Vectors 54
Stochastic Processes 55
Application to Uncertainty Quantification Problems 57
Non-intrusive Methods 59
Non-intrusive Spectral Projection 61
Orthogonal Basis 61
Orthogonal Projection 61
Simulation Approaches for NISP 62
Monte Carlo Method 62
Improved Sampling Strategies 63
Deterministic Integration Approach for NISP 65
Quadrature Formulas 65
Gauss Quadratures 65
Nested Quadratures 67
Tensor Product Formulas 69
Sparse Grid Cubatures for NISP 70
Sparse Grid Construction 71
Adaptive Sparse Grids 73
Dimension-Adaptive Sparse Grid 74
General Adaptive Sparse Grid Method 74
Least Squares Fit 77
Least Squares Minimization Problem 78
Selection of the Minimization Points 79
Weighted Least Squares Problem 81
Collocation Methods 82
Approximation Problem 82
Polynomial Interpolation 83
Sparse Collocation Method 85
Closing Remarks 85
Galerkin Methods 87
Stochastic Problem Formulation 88
Model Equations and Notations 88
Deterministic Problem 88
Stochastic Problem 88
Functional Spaces 89
Case of Discrete Deterministic Problems 90
Weak Form 91
Stochastic Discretization 91
Stochastic Basis 92
Data Parametrization and Solution Expansion 93
Spectral Problem 94
Stochastic Residual 94
Galerkin Method 95
Comments 95
Linear Problems 96
General Formulation 96
Structure of Linear Spectral Problems 97
Case of Deterministic Operator 97
General Case 98
Solution Methods for Linear Spectral Problems 101
Nonlinearities 103
Polynomial Nonlinearities 104
Galerkin Product 104
Higher-Order Polynomial Nonlinearity 105
Galerkin Inversion and Division 106
Square Root 109
Absolute Values 110
Min and Max Operators 111
Integration Approach 113
Other Types of Nonlinearities 117
Taylor Expansion 117
Non-intrusive Projection 117
Closing Remarks 118
Detailed Elementary Applications 120
Heat Equation 121
Deterministic Problem 121
Variational Formulation 122
Finite Element Approximation 122
Stochastic Problem 123
Stochastic Variational Formulation 124
Deterministic Discretization 124
Stochastic Discretization 125
Spectral Problem 126
Example 1: Uniform Conductivity 129
Trivial Cases 130
Validation 131
Example 2: Nonuniform Conductivity 135
Setup 135
Mean and Standard Deviation 136
Analysis of the Solution Modes 137
Probability Density Functions 139
Example 3: Uncertain Boundary Conditions 139
Treatment of Uncertain Boundary Conditions 139
Test Case 142
Simulations 143
Variance Analysis 150
Functional Decomposition 151
Application 152
Stochastic Viscous Burgers Equation 154
Deterministic Problem 154
Spatial Discretization 155
Discrete Deterministic Problem 156
Stochastic Problem 157
Stochastic Discretization 157
Stochastic Galerkin Projection 158
Numerical Example 159
Convergence of the Stochastic Approximation 160
Non-intrusive Spectral Projection 161
Quadrature Formula 161
Comparison with the Galerkin Projection 162
Monte-Carlo Method 163
Monte-Carlo Sampling 164
First- and Second-Order Estimates 165
Determination of Percentiles 167
Application to Navier-Stokes Equations 170
SPM for Incompressible Flow 171
Governing Equations 172
Intrusive Formulation and Solution Scheme 173
Numerical Examples 176
Example 1 176
Example 2 180
Example 3 187
Boussinesq Extension 194
Deterministic Problem 196
Stochastic Formulation 197
Stochastic Expansion and Solution Scheme 198
Boundary Conditions 199
Solution Method 199
Validation 200
Deterministic Prediction 200
Convergence Analysis 200
Analysis of Stochastic Modes 211
Velocity Modes 211
Temperature Modes 214
Comparison with NISP 214
Gauss-Hermite Quadrature 216
Latin Hypercube Sampling 220
Uncertainty Analysis 223
Low-Mach Number Solver 225
Zero-Mach-Number Model 225
Solution Method 227
Stochastic System 227
Boundary Conditions 228
Solution Method 229
Galerkin and Pseudo-spectral Evaluation of Nonlinear Terms 230
Pressure Solvability Constraints 231
Validation 232
Boussinesq Limit 232
Non-Boussinesq Regime 234
Uncertainty Analysis 236
Heat Transfer Characteristics 236
Mean Fields 238
Standard Deviations 240
Remarks 241
Stochastic Galerkin Projection for Particle Methods 242
Particle Method 244
Boussinesq Equations in Rotation Form 244
Particle Formulation 245
Approximation of Diffusion and Buoyancy Terms 247
Acceleration of Velocity Computation 249
Remeshing 250
Stochastic Formulation 251
Stochastic Basis and PC Expansion 251
Straightforward Particle Formulation 253
Particle Discretization of the Stochastic Flow 254
Validation 258
Diffusion of a Circular Vortex 258
Convection of a Passive Scalar 261
Application to Natural Convection Flow 266
Remarks 273
Mulitphysics Example 276
Physical Models 277
Momentum 277
Species Concentrations 278
Electrostatic Field Strength 280
Stochastic Formulation 280
Implementation 281
Data Structure 281
Spatial Discretization 282
Electroneutrality 282
Electrostatic Field Strength 282
Time Integration 283
Estimates of Nonlinear Transformations 285
Validation 285
Protein Labeling in a 2D Microchannel 290
Concluding Remarks 295
Advanced Topics 297
Solvers for Stochastic Galerkin Problems 298
Krylov Methods for Linear Models 299
Krylov Methods for Large Linear Systems 300
GMRes Method 301
Conjugate Gradient Method 302
Bi-Conjugate Gradient Method 302
Preconditioning 302
Jacobi Preconditioner 303
ILU Preconditioners 304
Preconditioners for Galerkin Systems 305
Block-Jacobi Preconditioners 305
Operator Expectation Preconditioning 306
Specialized Block Diagonal Preconditioners 307
Multigrid Solvers for Diffusion Problems 308
Spectral Representation 309
Continuous Formulation and Time Discretization 311
Stochastic Galerkin Projection 311
Boundary and Initial Conditions 311
Implicit Time Discretization 312
Finite Difference Discretization 312
Spatial Discretization 312
Treatment of Boundary Conditions 313
Iterative Method 314
Outer Iterations 314
Inner Iterations 315
Convergence of the Iterative Scheme 316
Multigrid Acceleration 316
Definition of Grid Levels 317
Projection and Prolongation Procedures 317
Multigrid Cycles 318
Implementation of the Multigrid Scheme 318
Results 320
Multigrid Acceleration 320
Influence of Stochastic Representation Parameters 322
Effects of Diffusivity Field Statistics 323
Selection of Multigrid Parameters 326
Stochastic Steady Flow Solver 327
Governing Equations and Integration Schemes 328
Stochastic Spectral Problem 329
Resolution of Steady Stochastic Equations 331
Newton Iterations 332
Stochastic Increment Problem 333
Matrix Free Solver 334
Test Problem 335
Problem Definition 335
Unsteady Simulations 336
Newton Iterations 337
Influence of the Stochastic Discretization 340
Computational Time 343
Unstable Steady Flow 345
Uncertainty Settings 345
Flow Equations and Stochastic Decoupling 346
Results 347
Closing Remarks 350
Wavelet and Multiresolution Analysis Schemes 353
The Wiener-Haar expansion 355
Preliminaries 355
Haar Scaling Functions 355
Haar Wavelets 356
Wavelet Approximation of a Random Variable 357
Multidimensional Case 358
Comparison with Spectral Expansions 359
Applications of WHa Expansion 360
Dynamical System 360
Solution Method 361
Results 363
Rayleigh-Bénard Instability 370
WLe Expansion 373
WHa Expansion 374
Continuous Problem 380
Multiresolution Analysis and Multiwavelet Basis 383
Change of Variable 384
Multiresolution Analysis 385
Vector Spaces 385
Multiwavelet Basis 385
Construction of the psij's 386
MW Expansion 388
Expansion of the Random Process 389
The Multidimensional Case 390
Mean and variance 391
Application to Lorenz System 392
h-p Convergence of the MW Expansion 392
Solution Method 392
Convergence Results 393
Comparison with Monte Carlo Sampling 397
Classical Sampling Strategy 397
Latin Hypercube Sampling 398
Closing Remarks 398
Adaptive Methods 400
Adaptive MW Expansion 401
Algorithm for Iterative Adaptation 402
Application to Rayleigh-Bénard Flow 403
Adaptive Partitioning of Random Parameter Space 405
Partition of the Random Parameter Space 406
Local Expansion Basis 406
Error Indicator and Refinement Strategy 408
Example 409
Two-Dimensional Problem 409
Higher Dimensional Problems 415
A posteriori Error Estimation 415
Variational Formulation 418
Deterministic Variational Problem 418
Stochastic Variational Problem 418
Probability Space 419
Stochastic Discretization 419
Spatial Discretization 421
Approximation Space Uh 421
Dual-based a posteriori Error Estimate 422
A posteriori Error 422
Posterior Error Estimation 423
Methodology 425
Refinement Procedure 426
Global and Local Error Estimates 426
Refinement Strategies 426
Application to Burgers Equation 428
Uncertainty Settings 428
Variational Problems 429
Isotropic hxi Refinement 430
Isotropic hxi,x Refinement 433
Anisotropic h/q Refinement 438
Generalized Spectral Decomposition 442
Variational Formulation 444
Stochastic Discretization 444
General Spectral Decomposition 445
Definition of an Optimal Pair (U,lambda) 445
A Progressive Definition of the Decomposition 447
Algorithms for Building the Decomposition 448
Extension to Affine Spaces 450
Application to Burgers Equation 451
Variational Formulation 451
Implementation of Algorithms 1 and 2 452
Spatial Discretization 454
Stochastic Discretization 455
Solvers 456
Results 458
Application to a Nonlinear Stationary Diffusion Equation 469
Application of GSD Algorithms 470
Results 473
Closing Remarks 483
Epilogue 486
Extensions and Generalizations 486
Open Problems 487
New Capabilities 490
Appendix A Essential Elements of Probability Theory and Random Processes 491
Probability Theory 491
Measurable Space 491
Probability Measure 492
Probability Space 492
Measurable Functions 493
Induced Probability 493
Random Variables 493
Measurable Transformations 494
Integration and Expectation Operators 494
Integrability 494
Expectation 495
L2 Space 496
Random Variables 497
Distribution Function of a Random Variable 497
Density Function of a Random Variable 497
Moments of a Random Variable 498
Convergence of Random Variables 498
Random Vectors 499
Joint Distribution and Density Functions 499
Independence of Random Variables 501
Moments of a Random Vector 502
Gaussian Vector 503
Stochastic Processes 503
Motivation and Basic Definitions 503
Properties of Stochastic Processes 504
Finite Dimensional Distributions and Densities 505
Second Moment Properties 505
Appendix B Orthogonal Polynomials 507
Classical Families of Continuous Orthogonal Polynomials 508
Legendre Polynomials 508
Hermite Polynomials 509
Laguerre Polynomials 511
Gauss Quadrature 512
Gauss-Legendre Quadrature 513
Gauss-Hermite Quadratures 513
Gauss-Laguerre Quadrature 516
Askey Scheme 517
Jacobi Polynomials 518
Discrete Polynomials 519
Appendix C Implementation of Product and Moment Formulas 522
One-Dimensional Polynomials 522
Moments of One-Dimensional Polynomials 523
Multidimensional PC Basis 523
Multi-Index Construction 523
Moments of Multidimensional Polynomials 524
Implementation Details 525
References 526
Index 537