Handbook of Uncertainty Quantification

Contents......Page 6
 Introduction to Uncertainty Quantification......Page 13
 Introduction to Sensitivity Analysis......Page 14
 Introduction to Software for Uncertainty Quantification......Page 15
Contributors......Page 16
Part I Introduction to Uncertainty Quantification......Page 23
  1 Introduction......Page 24
Part II Methodology......Page 28
  Contents......Page 29
  1 Introduction......Page 30
  2 Example: Rainfall Runoff Simulator......Page 31
  3 The Bayesian Analysis of Computer Simulators for Physical Systems......Page 32
  5 Emulation......Page 34
  6 Example: Emulating FUSE......Page 37
  7 Model Discrepancy......Page 40
  8 Example: Model Discrepancy for FUSE......Page 41
  9 History Matching......Page 42
  10 Example: History Matching FUSE......Page 44
   10.1 Introducing f2......Page 46
  11 Forecasting......Page 48
  12 Example: Forecasting for FUSE......Page 49
  Appendix: Internal Discrepancy Perturbations......Page 50
  References......Page 52
  Contents......Page 53
  1 Introduction......Page 54
   2.1 Bayesian Inference......Page 56
   2.2 Entropic Inference......Page 59
   2.3 Sufficient Statistics......Page 62
   2.4 Consistent Data Sets and Approximate Bayesian Computation......Page 63
  3 Algorithmic Structure......Page 64
   3.1 Remarks......Page 68
    4.1.1 Setup......Page 69
    4.1.2 Single Exponential Model Fit......Page 70
    4.1.3 Summary Statistics for Processed Data Products......Page 71
    4.1.4 PDP Pseudo-metric......Page 72
    4.1.5 Generating Consistent Data and Pooling the Posterior......Page 75
   4.2 Chemical System......Page 76
  5 Conclusion......Page 85
  References......Page 86
 4 Multi-response Approach to Improving Identifiability in Model Calibration......Page 88
  1 Introduction......Page 89
   2.1 Overview of Lack of Identifiability......Page 92
   2.2 Case Study: An Illustrative Simply Supported Beam Example......Page 95
   2.3 When Is Calibration Identifiability Possible?......Page 100
  3 Improving Identifiability in Model Calibration Using Multiple Responses......Page 104
   3.1 Multi-response Modular Bayesian Approach......Page 105
    3.1.1 Multi-response Gaussian Process Model for the Computer Model (Module 1)......Page 106
    3.1.2 Multi-response Gaussian Process Model for the Discrepancy Functions (Module 2)......Page 108
    3.1.4 Prediction of the Experimental Responses and Discrepancy Function (Module 4)......Page 109
    3.2.1 Identifiability with Single Responses......Page 110
    3.2.2 Identifiability with Multiple Responses......Page 112
   3.3 Remarks on the Multi-response Modular Bayesian Approach......Page 114
  4 Preposterior Analyses for Predicting Identifiability in Model Calibration......Page 116
   4.1 Preposterior Analysis......Page 117
    4.1.1 Step 1: Preliminaries......Page 119
    4.1.2 Step 2: Monte Carlo (MC) Loop (for i=1, … ,Nmc)......Page 120
    4.1.4 Step 4: Based on the Preposterior Covariances of All Subsets of Responses, Select a Subset to Measure Experimentally......Page 122
   4.2 A Modified Algorithm for Investigating the Behavior of the Preposterior Analysis......Page 123
   4.3 Fisher Information-Based Surrogate Preposterior Analysis......Page 124
   4.4 Single-Response Case Study: Simply Supported Beam Example......Page 125
    4.4.1 Computer Model, Physical Experiments, and Preliminaries......Page 127
    4.4.2 Comparing the Preposterior and Posterior Covariances......Page 128
   4.5 Using the Preposterior Analysis to Select Ne......Page 130
   4.6 Multi-response Case Study: Simply Supported Beam Example......Page 131
   4.7 Remarks on the Preposterior Analyses Method......Page 136
  5 Conclusions......Page 139
  Appendix A: Estimates of the Hyperparameters for the Computer Model MRGP......Page 140
  Appendix B: Posterior Distributions of the Computer Responses......Page 141
  Appendix C: Estimates of the Hyperparameters for the Discrepancy Functions MRGP......Page 142
  Appendix E: Posterior Distribution of the Experimental Responses......Page 143
  References......Page 144
 5 Validation of Physical Models in the Presence of Uncertainty......Page 147
  1 Introduction......Page 148
   1.1 Measuring Agreement Under Uncertainty......Page 149
   1.2 Different Uses of Models......Page 150
   2.1 Sources of Uncertainty in Validation Tests......Page 152
   2.2 Assessing Consistency of Models and Data......Page 154
    2.2.1 Single Observation of a Single Observable......Page 155
    2.2.2 Multiple Observations of a Single Observable......Page 157
    2.2.3 Defining Test Quantities......Page 158
    2.2.4 General Posterior Model Checks......Page 159
   2.3 Data for Validation......Page 161
   3.1 Mathematical Structure for Prediction......Page 163
    3.1.1 Simplest Case......Page 164
    3.1.2 Generalizations......Page 165
   3.2 Validation for Prediction......Page 166
    3.2.1 Accounting for Model Error......Page 167
    3.2.2 Predictive Assessment......Page 168
  4 Conclusions and Challenges......Page 172
  References......Page 173
 6 Toward Machine Wald......Page 175
  1 Introduction......Page 176
   2.1 C̆ebys̆ev, Markov, and Kreĭn......Page 178
   2.2 Optimal Uncertainty Quantification......Page 179
   2.4 Stochastic and Robust Optimization......Page 181
   2.5 C̆ebys̆ev Inequalities and Optimization Theory......Page 182
   3.1 From Game Theory to Decision Theory......Page 183
   3.2 The Optimization Approach to Statistics......Page 184
   3.3 Abraham Wald......Page 185
   3.5 Model Error and Optimal Models......Page 188
   3.6 Mean Squared Error, Variance, and Bias......Page 189
   3.9 Mixing Models......Page 190
   4.1 The Bayesian Approach......Page 191
   4.2 Relation Between Adversarial Model Error and Bayesian Error......Page 192
   4.3 Complete Class Theorem......Page 193
  5 Incorporating Complexity and Computation......Page 195
   5.2 Reduction Calculus......Page 196
   5.4 On the Borel-Kolmogorov Paradox......Page 197
   5.5 On Bayesian Robustness/Brittleness......Page 198
   5.6 Information-Based Complexity......Page 199
   Construction of πD......Page 200
   Proof of Theorem 2......Page 201
   Conditional Expectation as an Orthogonal Projection......Page 202
  References......Page 203
  Contents......Page 210
  1 Introduction......Page 211
  2 Hierarchical Modeling in the Presence of Uncertainty......Page 212
   2.1 Basic Hierarchical Structure......Page 213
   2.3 Process Models......Page 215
   2.4 Parameter Models......Page 216
   2.5 Bayesian Formulation......Page 217
  3 Dynamical Spatio-temporal Process Models......Page 218
   3.1 Linear DSTM Process Models......Page 219
   3.2 Nonlinear DSTM Process Models......Page 220
   3.3 Multivariate DSTM Process Models......Page 221
    3.4.1 Parameter Reduction......Page 222
    3.4.2 Process Rank Reduction......Page 223
  4 Example: Near-Surface Winds Over the Ocean......Page 224
   4.1 Surface Vector Wind Background......Page 225
    4.2.1 Data Models......Page 226
    4.2.2 Process Model......Page 227
   4.3 Implementation......Page 230
   4.4 Results......Page 231
  5 Conclusion......Page 232
  References......Page 234
  Contents......Page 236
  1 Introduction......Page 238
   2.1 What Is a Random Matrix?......Page 239
   2.2 What Is the Nonparametric Method for Uncertainty Quantification?......Page 240
   3.1 Random Matrix Theory (RMT)......Page 241
   3.2 Nonparametric Method for UQ and Its Connection with the RMT......Page 242
  4 Overview......Page 243
   5.4 Norms and Usual Operators......Page 244
   6.1 Volume Element and Probability Density Function (PDF)......Page 245
    6.1.2 Probability Density Function of a Symmetric Real Random Matrix......Page 246
    6.3.1 Available Information......Page 247
   7.1 Classical Definition ch5-1:Mehta2014......Page 248
   7.3 Decentering and Interpretation of Hyperparameter δ......Page 249
  8 Fundamental Ensembles for Positive-Definite Symmetric Real Random Matrices......Page 250
    8.1.1 Definition of  SG0+  Using the MaxEnt and Expression of the pdf......Page 251
    8.1.3 Invariance of Ensemble  SG0+  Under Real Orthogonal Transformations......Page 252
    8.1.5 Probability Density Function of the Random Eigenvalues......Page 253
   8.2 Ensemble SG+ of Positive-Definite Random Matrices with a Unit Mean Value and an Arbitrary Positive-Definite Lower Bound......Page 254
   8.3 Ensemble SG+b of Positive-Definite Random Matrices with Given Lower and Upper Bounds and with or without Given Mean Value......Page 255
    8.3.1 Definition of SG+b for a Non-given Mean Value Using the MaxEnt......Page 256
    8.3.2 Definition of SG+b for a Given Mean Value Using the MaxEnt......Page 257
    8.4.1 Definition of SG+λ Using the MaxEnt and Expression of the pdf......Page 258
  9 Ensembles of Random Matrices for the Nonparametric Method in Uncertainty Quantification......Page 259
    9.1.2 Expression of [A0] as a Transformation of [G0] and Generator of Realizations......Page 260
   9.2 Ensemble SE+ of Positive-Definite Random Matrices with a Given Mean Value and an Arbitrary Positive-Definite Lower Bound......Page 261
    9.2.2 Properties of Random Matrix [A]......Page 262
    9.3.2 Definition and Construction of Ensemble SE+0......Page 263
    9.4.2 Definition of Ensemble SErect......Page 264
    9.5.1 Defining the Deterministic Matrix Problem......Page 265
    9.5.2 Construction of a Nonparametric Stochastic Model......Page 266
  10 MaxEnt as a Numerical Tool for Constructing Ensembles of Random Matrices......Page 267
    10.1.1 Example of Parameterization......Page 268
  11 MaxEnt for Constructing the pdf of a Random Vector......Page 269
   11.1 Existence and Uniqueness of a Solution to the MaxEnt......Page 270
   11.2 Numerical Calculation of the Lagrange Multipliers......Page 273
   11.3 Generator for Random Vector Yλ and Estimation of the Mathematical Expectations in High Dimension......Page 274
    11.3.1 Random Generator and Estimation of Mathematical Expectations......Page 275
    11.3.2 Discretization Scheme and Estimating the Mathematical Expectations......Page 277
  12 Nonparametric Stochastic Model For Constitutive Equation in Linear Elasticity......Page 278
   12.1 Positive-Definite Matrices Having a Symmetry Class......Page 279
   12.3 Introducing Deterministic Matrices [A] and [S]......Page 280
   12.4 Nonparametric Stochastic Model for [C]......Page 281
    12.5.1 Defining the Available Information......Page 282
    12.5.3 Construction of [A] Using the Parameterization and Generator of Realizations......Page 283
   13.1 Methodology......Page 284
   13.3 Reduced-Order Model (ROM) of the Mean Computational Model......Page 285
    13.3.1 Convergence of the ROM with Respect to n Over Frequency Band of Analysis B......Page 286
    13.4.1 Available Information for Constructing a Prior Probability Model of [M], [D], and [K]......Page 287
    13.5.1 Mean Computational Model, ROM, and Convergence......Page 288
    13.5.3 Available Information for Constructing a Prior Probability Model of [M], [D(ω)], and [K(ω)]......Page 289
   13.6 Estimation of the Hyperparameters of the Nonparametric Stochastic Model of Uncertainties......Page 290
   14.1 Mean Nonlinear Computational Model in Structural Dynamics......Page 291
   14.2 Reduced-Order Model (ROM) of the Mean Nonlinear Computational Model......Page 292
    14.3.1 Methodology......Page 293
    14.3.2 Prior Probability Model of Y, Hyperparameters, and Generator of Realizations......Page 294
   14.4 Estimation of the Hyperparameters of the Parametric-Nonparametric Stochastic Model of Uncertainties......Page 295
   15.1 Propagation of Uncertainties Using Nonparametric or Parametric-Nonparametric Stochastic Models of Uncertainties......Page 296
   15.3 Additional Ingredients for the Nonparametric Stochastic Modeling of Uncertainties......Page 297
  16 Conclusions......Page 298
  References......Page 299
  Contents......Page 305
  1 Introduction......Page 306
   2.1 Sliced Latin Hypercube......Page 307
    2.1.1 Sampling Properties of SLHD......Page 308
   2.3 The Enhanced Stochastic Evolutionary Algorithm......Page 309
   2.4 An Alternate Construction Method for SLHDs......Page 310
  3 A Motivating Example......Page 311
  4 Construction of Maximin SLHD......Page 313
  5 Numerical Illustration......Page 314
   6.1 Evaluation of Multiple Computer Models......Page 318
   6.2 Cross Validation of Prediction Error......Page 319
  References......Page 324
 10 The Bayesian Approach to Inverse Problems......Page 326
  Contents......Page 327
  1 Introduction......Page 328
   1.1 Bayesian Inversion on Rn......Page 329
   1.2 Inverse Heat Equation......Page 331
   1.3 Elliptic Inverse Problem......Page 333
  2 Prior Modeling......Page 335
   2.1 General Setting......Page 336
   2.2 Uniform Priors......Page 337
   2.3 Besov Priors......Page 339
   2.4 Gaussian Priors......Page 345
   2.5 Random Field Perspective......Page 350
   2.6 Summary......Page 353
   2.7 Bibliographic Notes......Page 354
   3.1 Conditioned Random Variables......Page 355
   3.2 Bayes\' Theorem for Inverse Problems......Page 357
   3.3 Heat Equation......Page 358
    3.4.1 Forward Problem......Page 360
    3.4.2 Uniform Priors......Page 361
    3.4.3 Gaussian Priors......Page 363
  4 Common Structure......Page 364
   4.1 Well Posedness......Page 365
   4.2 Approximation......Page 369
   4.3 MAP Estimators and Tikhonov Regularization......Page 373
  5 Measure Preserving Dynamics......Page 379
   5.1 General Setting......Page 380
   5.2 Metropolis-Hastings Methods......Page 382
   5.3 Sequential Monte Carlo Methods......Page 386
   5.4 Continuous Time Markov Processes......Page 394
    5.5.1 Background Theory......Page 395
    5.5.2 Motivation for Equation (10.53)......Page 398
    5.6.1 Assumptions on Change of Measure......Page 399
    5.6.2 Finite-Dimensional Approximation......Page 401
    5.6.3 Main Theorem and Proof......Page 403
   5.7 Bibliographic Notes......Page 407
    A.1.1 p and Lp Spaces......Page 409
    A.1.3 Sobolev Spaces......Page 411
    A.1.4 Other Useful Function Spaces......Page 414
    A.1.5 Interpolation Inequalities and Sobolev Embeddings......Page 415
    A.2.1 Product Measure for i.i.d. Sequences......Page 417
    A.2.2 Probability and Integration on Separable Banach Spaces......Page 418
    A.2.4 Metrics on Probability Measures......Page 421
    A.2.5 Kolmogorov Continuity Test......Page 427
    A.3.1 Separable Banach Space Setting......Page 429
    A.3.2 Separable Hilbert Space Setting......Page 432
   A.4 Wiener Processes in Infinite-Dimensional Spaces......Page 434
   A.5 Bibliographical Notes......Page 437
  References......Page 439
 11 Multilevel Uncertainty Integration......Page 444
   1.1 Motivation......Page 445
   1.2 Multilevel System Models......Page 447
   1.3 Organization of the Chapter......Page 450
   2.1 Verification......Page 451
    2.1.1 Discretization Error......Page 453
    2.1.2 Surrogate Model Uncertainty......Page 454
   2.2 Calibration......Page 455
   2.3 Validation......Page 456
    2.3.1 Validation by Bayesian Hypothesis Testing......Page 457
    2.3.2 Model Validation by Model Reliability Metric......Page 459
   3.1 Description of the Problem......Page 462
   3.2 Verification, Validation, and Calibration......Page 463
   3.3 Integration and Overall Uncertainty Quantification......Page 464
  4 Multilevel Models with Type-I Interaction......Page 465
   4.1 Verification, Calibration, and Validation......Page 466
   4.2 Integration for Overall Uncertainty Quantification......Page 467
   4.3 Extension to Multiple Models......Page 468
  5 Numerical Example: Two Models with Type-I Interaction......Page 470
  6 Multilevel Models with Type-II Interaction......Page 472
   6.1 Verification, Calibration, and Validation......Page 473
   6.2 Relevance Analysis......Page 474
   6.3 Integration for Overall Uncertainty Quantification......Page 477
   7.1 Problem Description......Page 479
   7.2 Results and Analysis......Page 481
  8 Conclusion......Page 485
  References......Page 486
  Contents......Page 491
  1 Introduction......Page 492
  2 A Brief Review on Kriging......Page 494
   3.1 The Prior and Posterior Processes......Page 495
   3.2 Nugget Parameter......Page 497
  4 Simulation Study and Results......Page 498
  5 Application......Page 502
  6 Conclusions......Page 506
  References......Page 508
 13 Propagation of Stochasticity in Heterogeneous Media and Applications to Uncertainty Quantification......Page 510
  1 Introduction......Page 511
  2 Propagation of Stochasticity for Elliptic Equations......Page 512
   2.1 Asymptotic Random Fluctuations in One Dimension......Page 513
   2.2 Large Deviations in One Dimension......Page 515
   2.3 Some Remarks on the Higher-Dimensional Case......Page 518
   3.1 Perturbation Theory for Bounded Random Potentials......Page 519
   3.2 Homogenization Theory for Large Random Potentials......Page 523
   3.3 Convergence to Stochastic Limits for Long-Range Random Potentials......Page 525
  4 Applications to Uncertainty Quantification......Page 527
   4.2 Concentration Inequalities and Coupled PCE-MC Framework......Page 528
  5 Conclusions......Page 530
  References......Page 531
 14 Polynomial Chaos: Modeling, Estimation, and Approximation......Page 533
  1 Introduction......Page 534
  2 Mathematical Setup......Page 538
  3 Polynomial Chaos......Page 539
  4 Representation of Stochastic Processes......Page 544
  5 Polynomial Chaos with Random Coefficients: Model Error......Page 546
  6 Adapted Representations of PCE......Page 549
   7.1 Nonintrusive Evaluation of the Stochastic Galerkin Solution......Page 551
   7.2 Adapted Preconditioners for the Stochastic Galerkin Equations......Page 553
  8 Embedded Quadratures for Stochastic Coupled Physics......Page 554
  9 Constructing Stochastic Processes......Page 557
  References......Page 559
Part III Forward Problems......Page 564
  Contents......Page 565
  1 Introduction......Page 566
   2.1 Physical Models......Page 567
   2.2 Computer Emulators......Page 568
   2.3 The Uncertainty Propagation Problem......Page 569
    2.4.1 Bayesian Surrogates......Page 570
    2.5.1 Modeling Prior Knowledge About the Response......Page 571
    2.5.3 Treating Space and Time by Using Separable Mean and Covariance Functions......Page 576
    2.5.4 Treating Space and Time by Using Output Dimensionality Reduction......Page 577
   2.6 Training the Parameters of the Gaussian Process......Page 578
    2.6.3 Markov Chain Monte Carlo......Page 579
    2.7.1 Completely Independent Outputs......Page 580
    2.7.3 Linearly Correlated Outputs......Page 581
   2.8 Sampling Possible Surrogates......Page 582
    2.8.1 The Karhunen-Loève Approach for Constructing (·;θ)......Page 584
    2.8.2 The O\'Hagan Approach for Constructing (·;θ)......Page 586
    2.9.1 One-Dimensional Output with No Spatial or Time Inputs......Page 588
    2.9.2 One-Dimensional Output with Spatial and/or Time Inputs......Page 589
   3.1 Synthetic One-Dimensional Example......Page 590
   3.2 Dynamical System Example......Page 592
   3.3 Partial Differential Equation Example......Page 593
  4 Conclusions......Page 603
  References......Page 606
  Contents......Page 610
  1 Introduction: The Stochastic Finite Element Method......Page 611
   2.1 Multigrid Methods I......Page 615
   2.2 Multigrid Methods II: Mean-Based Preconditioning......Page 617
   2.3 Hierarchical Methods......Page 619
  3 Approaches for Other Formulations......Page 621
  References......Page 624
  Contents......Page 626
  1 Introduction......Page 627
  2 Theory and Algorithms......Page 628
  3 Example: Intrusive Propagation of Uncertainty Through ODEs......Page 634
  4 Challenges and Opportunities......Page 640
  6 Conclusions......Page 641
  References......Page 642
 18 Multiresolution Analysis for Uncertainty Quantification......Page 646
  1 Introduction......Page 647
  2 One-Dimensional Multiresolution System......Page 648
   2.1 Multiresolution Analysis and Multiresolution Space......Page 649
   2.2 Stochastic Element Basis......Page 650
    2.3.1 Detail Spaces......Page 652
    2.3.2 Construction of the MW Mother Functions......Page 654
   3.1 Binary-Tree Representation......Page 655
   3.2 Multidimensional Extension......Page 657
    3.3.1 Restriction Operator......Page 661
  4 Adaptivity......Page 662
    4.1.2 Coarsening Procedure......Page 663
   4.2 Anisotropic Enrichment......Page 664
    4.2.1 Multidimensional Enrichment Criterion......Page 665
    4.2.2 Directional Enrichment Criterion......Page 667
   5.1 Simple ODE Problem......Page 668
    5.2.1 Test Problem......Page 671
    5.2.2 Adaptive Computations......Page 674
    5.2.3 Convergence and Computational Time Analysis......Page 676
  6 Conclusions......Page 679
  References......Page 680
  Contents......Page 682
  1 Introduction......Page 683
  2 Surrogate Modeling for Forward Propagation......Page 684
   3.1 Input PC Specification......Page 686
   3.2 PC Surrogate Construction......Page 688
    3.2.1 Bayesian Regression......Page 690
    3.3.1 Model Selection and Validation......Page 692
    3.3.2 High Dimensionality......Page 693
   3.4 Moment Evaluation......Page 697
   3.5 Global Sensitivity Analysis......Page 699
  4 Conclusions......Page 702
  References......Page 704
  Contents......Page 708
  1 Introduction......Page 709
  2 Definition of Stochastic Collocation......Page 710
   3.1 Formulation......Page 711
    3.2.1 Tensor Nodes......Page 713
    3.2.2 Sparse Grids......Page 715
   3.3 Interpolation on Unstructured Samples......Page 717
   4.1 Over-sampled Case: Least Squares......Page 718
   4.2 Under-sampled Case: Sparse Approximations......Page 719
  5 Stochastic Collocation via Pseudo Projection......Page 721
  References......Page 723
 21 Sparse Collocation Methods for Stochastic Interpolation and Quadrature......Page 726
  1 Introduction......Page 727
  2 Problem Setting......Page 729
  3 Stochastic Finite Element Method......Page 732
   3.2 Stochastic Fully Discrete Approximation......Page 733
    3.3.1 Standard Sparse Global Polynomial Subspaces......Page 734
    3.3.2 Best M-Term Polynomial Subspaces......Page 735
    3.3.3 Quasi-optimal Polynomial Subspaces......Page 738
    3.3.4 Local Piecewise Polynomial Subspaces......Page 739
   4.1 Lagrange Global Polynomial Interpolation in Parameter Space......Page 740
   4.2 Generalized Sparse Grid Construction......Page 741
    4.2.1 Clenshaw-Curtis Points on Bounded Hypercubes......Page 744
    4.2.3 Selection of the Anisotropic Weights for Example 3......Page 747
    4.2.4 Sparse Grid gSCM Error Estimates for Example 3......Page 748
   4.3 Nonintrusive Sparse Interpolation in Quasi-optimal Subspaces......Page 753
    5.1.1 One-Dimensional Piecewise Linear Hierarchical Interpolation......Page 755
    5.1.2 Multidimensional Hierarchical Sparse Grid Interpolation......Page 757
    5.1.3 Hierarchical Sparse Grid Stochastic Collocation......Page 761
   5.2 Adaptive Hierarchical Stochastic Collocation Methods......Page 762
    5.2.1 Other Choices of Hierarchical Basis......Page 764
  6 Conclusion......Page 767
  References......Page 768
  Contents......Page 772
   1.1 Randomness in Mathematical Models......Page 773
   1.3 Uncertainty Quantification in PDEs with Random Coefficients......Page 774
  2 Method of Distributions......Page 775
   2.1 PDF Methods......Page 776
   2.2 CDF Methods......Page 779
  3 Distribution Methods for PDEs......Page 780
   3.1 Weakly Nonlinear PDEs Subject to Random Initial Conditions......Page 781
   3.2 Weakly Nonlinear PDEs with Random Coefficients......Page 782
   3.3 Nonlinear PDEs with Shocks......Page 783
   3.4 Systems of PDEs......Page 785
  4 Conclusions......Page 789
  Appendix......Page 790
  References......Page 791
 23 Sampling via Measure Transport: An Introduction......Page 793
  1 Introduction......Page 794
  2 Transport Maps and Optimal Transport......Page 797
   3.1 Preliminaries......Page 798
   3.2 Optimization Problems......Page 799
   3.3 Convergence, Bias, and Approximate Maps......Page 802
   4.1 Optimization Problem......Page 805
   4.2 Convexity and Separability of the Optimization Problem......Page 807
   4.3 Computing the Inverse Map......Page 809
   5.1 Polynomial Representations......Page 811
   5.3 Monotonicity Constraints and Monotone Parameterizations......Page 813
  6 Related Work......Page 814
  7 Conditional Sampling......Page 818
  8 Example: Biochemical Oxygen Demand Model......Page 821
   8.1 Inverse Transport: Map from Samples......Page 822
   8.2 Direct Transport: Map from Densities......Page 825
  9 Conclusions and Outlook......Page 827
  References......Page 830
  Contents......Page 834
   1.1 Problem Formulation......Page 835
   1.2 The Askey Scheme......Page 837
   1.3 Polynomial Order......Page 838
   1.4 Sparsity in PCE......Page 839
  2 Solution Computation......Page 840
   2.1 Basis Pursuit Denoising (BPDN)......Page 841
   3.1 Phase Transition Diagrams......Page 842
   3.2 Restricted Isometry Constant (RIC)......Page 843
   3.3 Coherence of Bounded Orthonormal Systems......Page 844
   3.4 Mutual Coherence......Page 845
   4.1 Sampling Distribution......Page 846
    4.1.1 Sampling for Minimizing (24.19)......Page 847
   4.2 Column Weighting......Page 849
   4.3 Incorporating Derivative Evaluations......Page 851
  5 Thermally Driven Cavity Flow Example......Page 853
  References......Page 858
  Contents......Page 863
  1 Introduction......Page 864
   2.1 Best Rank-m Approximation and Optimal Subspaces......Page 866
   2.3 Singular Value Decomposition......Page 867
    3.1.1 Interpolation......Page 868
    3.1.2 Galerkin Projections......Page 869
    3.2.1 From Samples of the Function......Page 871
    3.2.2 From Approximations of the Correlation Operator......Page 872
    3.2.3 From the Model Equations......Page 873
  4 Low-Rank Approximation of Multivariate Functions......Page 875
   4.1 Tensor Ranks and Corresponding Low-Rank Formats......Page 876
   4.2 Relation with Other Structured Approximations......Page 877
   4.3 Properties of Low-Rank Formats......Page 878
   5.1 Least Squares......Page 879
   5.2 Interpolation/Projection......Page 880
   6.1 Tensor-Structured Equations......Page 881
   6.2 Iterative Solvers and Low-Rank Truncations......Page 882
   6.3 Optimization on Low-Rank Manifolds......Page 883
  References......Page 885
 26 Random Vectors and Random Fields in High Dimension: Parametric Model-Based Representation, Identification from Data, and Inverse Problems......Page 889
  Contents......Page 890
   2.1 What Is a Random Vector or a Random Field with a High Stochastic Dimension?......Page 892
   2.2 What Is a Parametric Model-Based Representation for the Statistical Identification of a Random Model Parameter from Experimental Data?......Page 893
   3.1 Classical Methods for Statistical Inverse Problems......Page 894
   3.3 Case for Which the Model Parameter Is a Non-Gaussian Second-Order Random Field......Page 895
   3.5 Parameterization of the Non-Gaussian Second-Order Random Vector η......Page 896
   3.6 Statistical Inverse Problem for Identifying a Non-Gaussian Random Field as a Model Parameter of a BVP, Using Polynomial Chaos Expansion......Page 897
   3.7 Algebraic Prior Stochastic Models of the Model Parameters of BVP......Page 898
  4 Overview......Page 899
   5.4 Norms and Usual Operators......Page 900
   6.1 Stochastic Elliptic Operator and Boundary Value Problem......Page 901
   6.2 Stochastic Finite Element Approximation of the Stochastic Boundary Value Problem......Page 903
   6.4 Statistical Inverse Problem to be Solved......Page 904
   7.1 Introduction a Class of Lower-Bounded Random Fields for [K] and Normalization......Page 905
   7.2 Construction of the Nonlinear Transformation G......Page 906
   7.3 Truncated Reduced Representation of Second-Order Random Field [G] and Its Polynomial Chaos Expansion......Page 908
   7.5 Parameterized Representation for Non-Gaussian Random Field [K]......Page 910
   8.1 Step 1: Introduction of a Family {[KAPSM(x;s)],x  Ω} of Algebraic Prior Stochastic Models (APSM) for Non-Gaussian Random Field [K]......Page 911
   8.2 Step 2: Identification of an Optimal Algebraic Prior Stochastic Model (OAPSM) for Non-Gaussian Random Field [K]......Page 912
   8.4 Step 4: Construction of a Truncated Reduced Representation of Second-Order Random Field [GOAPSM]......Page 913
   8.5 Step 5: Construction of a Truncated Polynomial Chaos Expansion of ηOAPSM and Representation of Random Field [KOAPSM]......Page 914
   8.6 Step 6: Identification of the Prior Stochastic Model [Kprior] of [K] in the General Class of the Non-Gaussian Random Fields......Page 918
   8.7 Step 7: Identification of a Posterior Stochastic Model [Kpost] of [K]......Page 919
  9 Construction of a Family of Algebraic Prior Stochastic Models......Page 920
    9.2.1 Introduction of an Adapted Representation......Page 921
    9.2.2 Construction of Random Field [G0] and Its Generator of Realizations......Page 922
    9.3.1 Positive-Definite Matrices Belonging to a Symmetry Class......Page 925
    9.3.2 Introduction of the Matrices [C], [S], and [A] Related to the Mean Value of the Matrix-Valued Random Field......Page 927
    9.3.3 Introduction of an Adapted Representation for the Random Field......Page 928
    9.3.4 Remarks Concerning the Control of the Statistical Fluctuations and the Limit Cases......Page 929
    9.3.5 Parameterization of Random Field {[A(x)],x Rd}......Page 930
    9.3.6 Construction of the pdf for Random Vector Y(x) Using the MaxEnt Principle......Page 931
    9.3.7 Constructing a Spatial-Correlation Structure for Random Field {Y(x),xRd} and Its Generator......Page 932
    9.3.8 Discretization Scheme of the Family of ISDE......Page 935
    9.3.9 Definition of the Hyperparameter s......Page 936
  11 Conclusions......Page 937
  References......Page 938
 27 Model Order Reduction Methods in Computational Uncertainty Quantification......Page 942
  1 Introduction......Page 943
   2.1 A Class of Forward Problems with Uncertain Input Data......Page 947
   2.2 Uncertainty Parametrization......Page 951
   2.3 Parameter Sparsity in Forward UQ......Page 956
    2.3.1 (b,p)-Holomorphy......Page 957
    2.3.2 Sparse Polynomial Approximation......Page 958
    2.3.3 Sparse Grid Interpolation......Page 959
  3 Model Order Reduction......Page 960
   3.1 High-Fidelity Approximation......Page 961
   3.2 Reduced Basis Compression......Page 963
    3.3.1 Proper Orthogonal Decomposition......Page 965
    3.3.2 Greedy Algorithm......Page 966
    3.4.1 High-Fidelity Approximation......Page 968
    3.4.2 Reduced Basis Compression......Page 969
    3.4.3 Tight A Posteriori Error Bound......Page 970
    3.4.4 Stable RB-PG Compression......Page 971
    3.5.1 High-Fidelity Approximation......Page 973
    3.5.2 Reduced Basis Compression......Page 974
    3.5.3 Empirical Interpolation......Page 975
    3.5.4 Computable a Posterior Error Indicator......Page 978
    3.5.5 RB-EI-PG Compression......Page 979
   3.6 Sparse Grid RB Construction......Page 980
   4.1 Bayes