Preface ........................................................ ix
Notation ..................................................... xiii
Acronyms and Initialisms ..................................... xvii
1 Introduction ................................................. 1
1.1 Nature of Uncertainties and Errors ...................... 4
1.2 Predictive Estimation ................................... 8
2 Large-Scale Applications .................................... 11
2.1 Weather Models ......................................... 11
2.2 Climate Models ......................................... 21
2.3 Subsurface Hydrology and Geology ....................... 33
2.4 Nuclear Reactor Design ................................. 36
2.5 Biological Models ...................................... 44
3 Prototypical Models ......................................... 51
3.1 Models ................................................. 51
3.2 Evolution, Stationary, and Algebraic Models ............ 61
3.3 Abstract Modeling Framework ............................ 63
3.4 Notation for Parameters and Inputs ..................... 65
3.5 Exercises .............................................. 66
4 Fundamentals of Probability, Random Processes, and
Statistics .................................................. 67
4.1 Random Variables, Distributions, and Densities ......... 67
4.2 Estimators, Estimates, and Sampling Distributions ...... 79
4.3 Ordinary Least Squares and Maximum Likelihood
Estimators ............................................. 82
4.4 Modes of Convergence and Limit Theorems ................ 85
4.5 Random Processes ....................................... 87
4.6 Markov Chains .......................................... 90
4.7 Random versus Stochastic Differential Equations ........ 96
4.8 Statistical Inference .................................. 98
4.9 Notes and References .................................. 104
4.10 Exercises ............................................. 105
5 Representation of Random Inputs ............................ 107
5.1 Mutually Independent Random Parameters ................ 107
5.2 Correlated Random Parameters .......................... 108
5.3 Finite-Dimensional Representation of Random
Coefficients .......................................... 109
5.4 Exercises ............................................. 112
6 Parameter Selection Techniques ............................. 113
6.1 Linearly Parameterized Problems ....................... 115
6.2 Nonlinearly Parameterized Problems .................... 122
6.3 Parameter Correlation versus Identifiability .......... 125
6.4 Notes and References .................................. 127
6.5 Exercises ............................................. 128
7 Frequentist Techniques for Parameter Estimation ............ 131
7.1 Parameter Estimation from a Frequentist Perspective ... 133
7.2 Linear Regression ..................................... 134
7.3 Nonlinear Parameter Estimation Problem ................ 141
7.4 Notes and References .................................. 152
7.5 Exercises ............................................. 153
8 Bayesian Techniques for Parameter Estimation ............... 155
8.1 Parameter Estimation from a Bayesian Perspective ...... 155
8.2 Markov Chain Monte Carlo (MCMC) Techniques ............ 159
8.3 Metropolis and Metropolis-Hastings Algorithms ......... 159
8.4 Stationary Distribution and Convergence Criteria ...... 168
8.5 Parameter Identifiability ............................. 171
8.6 Delayed Rejection Adaptive Metropolis (DRAM) .......... 172
8.7 DiffeRential Evolution Adaptive Metropolis (DREAM) .... 181
8.8 Notes and References .................................. 184
8.9 Exercises ............................................. 184
9 Uncertainty Propagation in Models .......................... 187
9.1 Direct Evaluation for Linear Models ................... 188
9.2 Sampling Methods ...................................... 191
9.3 Perturbation Methods .................................. 192
9.4 Prediction Intervals .................................. 197
9.5 Notes and References .................................. 203
9.6 Exercises ............................................. 204
10 Stochastic Spectral Methods ................................ 207
10.1 Spectral Representation of Random Processes ........... 207
10.2 Galerkin, Collocation, and Discrete Projection
Frameworks ............................................ 214
10.3 Stochastic Galerkin Method—Examples ................... 226
10.4 Discrete Projection Method—Example .................... 234
10.5 Stochastic Polynomial Packages ........................ 235
10.6 Exercises ............................................. 236
11 Sparse Grid Quadrature and Interpolation Techniques ........ 239
11.1 Quadrature Techniques ................................. 239
11.2 Interpolating Polynomials for Collocation ............. 250
11.3 Sparse Grid Software .................................. 254
11.4 Exercises ............................................. 255
12 Prediction in the Presence of Model Discrepancy ............ 257
12.1 Effects of Unaccommodated Model Discrepancy ........... 261
12.2 Incorporation of Missing Physical Mechanisms .......... 263
12.3 Techniques to Quantify Model Errors ................... 265
12.4 Issues Pertaining to Model Discrepancy
Representations ....................................... 267
12.5 Notes and References .................................. 269
12.6 Exercises ............................................. 269
13 Surrogate Models ........................................... 271
13.1 Regression or Interpolation-Based Models .............. 273
13.2 Projection-Based Models ............................... 280
13.3 Eigenfunction or Modal Expansions ..................... 283
13.4 Snapshot-Based Methods including POD .................. 284
13.5 High-Dimensional Model Representation (HDMR)
Techniques ............................................ 289
13.6 Surrogate-Based Bayesian Model Calibration ............ 298
13.7 Notes and References .................................. 299
13.8 Exercises ............................................. 300
14 Local Sensitivity Analysis ................................. 303
14.1 Motivating Examples—Neutron Diffusion ................. 306
14.2 Functional Analytic Framework for FSAP and ASAP ....... 312
14.3 Notes and References .................................. 318
14.4 Exercises ............................................. 319
15 Global Sensitivity Analysis ................................ 321
15.1 Variance-Based Methods ................................ 323
15.2 Morris Screening ...................................... 331
15.3 Time- or Space-Dependent Responses .................... 337
15.4 Notes and References .................................. 343
15.5 Exercises ............................................. 344
A Concepts from Functional Analysis .......................... 345
A.l Exercises ............................................. 351
Bibliography .................................................. 353
Index ......................................................... 373
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