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