i i p88 smit 2013/10/2 page 373 i i Index Symbols High-dimensional model 4D-VAR, 18 representation (HDMR), ANOVA- A Anthropogenic climate forces, 25–28 Acceptance probability, 159 Asymptotic sampling distribution, delayed rejection, 173 139 Acceptance ratio, 170 Atmospheric physics Active subspace method, 113 conservation relations, 13–15 Additive model, 323 phenomenological models, 15 Adjoint Autocorrelation, 170 formal, 349, 351 Automatic differentiation (AD), 145, of unbounded operator, 348–349 305 example, 349–350 Autoregressive models, 89 Adjoint boundary conditions, 314 Adjoint Hilbert space, 348 Adjoint matrix, 348 B Adjoint sensitivity analysis Bayes’ formula, 100 procedure (ASAP) Bayes’ theorem of inverse problems, approach perturbation, 308–309, 156 317–318 Bayesian inference, 100–104 approach variational, 309–311, empirical, 100 316–317 Beta distribution, 74 examples conjugate prior, 103 algebraic problem, 308–309 example, 339–342 boundary value problem, Bilinear form 310–311 ASAP, 313 ODE, 316–318 to construct adjoint, 349–350 functional analysis, 313–314 Binomial distribution, 84, 101–102 Adjoint sensitivity equation Binomial model, 103 algebraic problem, 308 Biological systems, 44–47 Aerosols, 11 HIV model, 47–50, 54–55 Aleatoric uncertainty, 7 uncertainties, 45–50 Analysis of variance (ANOVA), 291 Burn-in, see Metropolis algorithm, ANOVA-HDMR, see convergence 373 i i i i i i p88 smit 2013/10/2 page 374 i i 374 Index C Conservation relations Central limit theorem, 86–87, 139 atmospheric, 13–14 Chebyshev nodes, 252 neutron transport, 40 Chi-squared distribution, 72 subsurface hydrology, 35 Cholesky decomposition, 160 thermal-hydraulic, 41–42 Clenshaw–Curtis Convergence nodes, 242, 254 almost sure, 85 quadrature errors, 243–244 in distribution, 85–86, 139 Climate, 21–33 in probability, 85, 139 aerosol emission, 28 Correlation boundary value problem, 21 Nataf and Rosenblatt deforestration, 28 transformations, 108–109 energy budget, 21 versus identifiability, 125–127 equations of atmospheric Correlation coefficient, 77, 125 physics, 14 Correlation function, 276 greenhouse effect, 25 Covariance, 77 greenhouse gases, 25–28 Covariance matrix uncertainties, 27–28 chain, 172–173 water vapor, 29 definition, 78 ice albedo effects, 29 estimate, 162 segment length curse, 27 in proposal distribution, 160 solar radiation, 23 parameter estimation, 136, 145, uncertainties, 27–28, 30–32 152 volcanic effects, 24 Credible interval, 100 Climate debate, 33 Cubature rules, 250 Climate forces, 22–29 Cumulative distribution function anthropogenic, 25–28 (cdf), 68 feedback mechanisms, 28–29 joint, 76 natural, 23 Cut-HDMR, see High-dimensional Climate models, 21–22, 29–32 model representation Climate questions, 22 (HDMR), cut- Climate scenarios, 30 Climate simulation codes, 29–30 D Collocation method, see Stochastic DAKOTA, see Design Analysis Kit collocation method for Optimization and Conditional pdf, see Probability Terascale Applications density function Data-fit model, see Surrogate model, Confidence interval, 80–82 regression, interpolation for parameters, 139–142, 146, Delayed rejection adaptive 152 Metropolis, see DRAM interpretation, 99 Design Analysis Kit for Optimization versus prediction interval, and Terascale Applications 197–200 (DAKOTA), 236 Conjugacy, 103 GP, kriging models, 299 Conjugate prior, see Prior Detailed balance condition, 96, distribution 168–171 i i i i i i p88 smit 2013/10/2 page 375 i i Index 375 delayed rejection, 174 maximum likelihood, 83–85 DiffeRential Evolution Adaptive OLS, 82, 135 Metropolis, see DREAM point and interval, 79–82 Direct effect, 312 realization of estimator, 80 Discrete projection, see Stochastic Estimator discrete projection confidence interval, 80–82 Distribution, 70 consistent, 86 beta, 74 definition, 80 binomial, 84, 101–102 error variance, 136–137, 142, 146 chi-squared, 72 for parameters, 135, 142, 146, gamma, 73 151 inverse chi-squared, 74 interval, 80 inverse-gamma, 73–74 maximum likelihood, 83–85 conjugate prior, 163 OLS, 82, 135, 142, 146 multivariate normal, 78 unbiased, 80 normal, 70–71 Evolution processes, see Models proposal, 160 Expectation, 70 sampling, 80 Explanatory variables, 132 Student’s t-, 72–73 uniform, 71–72 F DRAM, 172–180 Factors, 331 algorithm, 175–176 Fej´er nodes, 242 examples Fisher information matrix, 164 heat model, 176–179 Forward sensitivity analysis HIV model, 179–180 procedure (FSAP) software, 175 examples DREAM, 181–183 algebraic problem, 306 Dual space, 345 boundary value problem, 310 ODE, 316 E functional analysis, 313 Elementary effect, 125, 331 Fracking, 34 Emulator, see Surrogate model, Fr´echet differential, derivative, regression, interpolation 347–348 Energy budget, 21 Functional, 345 Ensemble forecasts, 19 Functional principal component Epistemic uncertainty, 8 analysis (fPCA), 338 Errors measurement and model, 133 G variance estimator, 136–137, Gˆateaux differential, derivative, 347 142, 146 Gˆateaux variation Bayesian, 163 algebraic problem, 307 Estimate boundary value problem, 310 for covariance matrix, 162 definition, 347 for parameters, 135, 142, 146, response, 310, 312 151 to construct sensitivity maximum a posteriori, 157 equations, 313 i i i i i i p88 smit 2013/10/2 page 376 i i 376 Index Galerkin method, see Stochastic Influential parameters, 114 Galerkin method Initial condition, projection, 281 Gamma distribution, 73 Inner product space, 346 Gauss–Hermite quadrature, see Inputs, 3 Quadrature rule, uncertainties, 6 Gauss–Hermite Interpolation Gaussian process (GP) 1-D, 250–252 as surrogate model, 275–278 Chebyshev nodes, 252 definition, 89 error bound, 252 for model discrepancy, 266 sparse grid, 254 Gelfand triple, 311 tensor product, 253 Generalized Fourier coefficients, 216 Interval estimator, 80–82 Global sensitivity analysis, see Intrusive methods, 214 Sensitivity analysis, global Inverse chi-squared distribution, 74 Greenhouse effect, 25 Inverse-gamma distribution, 73–74, Greenhouse gases, 25–28 163 conjugate prior, 104 H Inverse transform sampling, 76 Hamiltonian, 310 Inverse uncertainty quantification, 6, Hermite basis method, 284 132 Hermite polynomials, 210–211 Irreducible uncertainty, see Aleatoric High-dimensional model uncertainty representation (HDMR), Ishigami function, 329 289–298 ANOVA-, 290–292 J based on cut-, 296–298 Jeffreys prior, 164 cut-, 293–295 Jumping distribution, see Proposal RS-, 292–293 distribution second-order expansion, 323 K Hilbert space, 346 Karhunen–Lo`eve expansion, 109–112 Human immunodeficiency virus relation to POD, 287 (HIV) model, see Models Kernel density estimation (kde), Hyperparameters, 103, 163, 277 75–76 for model discrepancy, 265, 267 Kriging model, 275–278 for model discrepancy, 266 I Kronecker delta, multiple variables, Identifiable parameter subspace 213 definition, 113–114 example, 53, 56 L for model discrepancy, 267 Lagrange basis method, 284 relation to range, 116–117 Lagrange polynomial, 251 versus correlation, 125–127 for cut-HDMR, 295 Importance measures, 324 Law of large numbers, 86, 139 Independent and identically Least squares, see Ordinary least distributed (iid) random squares variables, 79 Lebesgue constant, 252 i i i i i i p88 smit 2013/10/2 page 377 i i Index 377 Legendre polynomials, 211 adaptive, 172–173 Likelihood function convergence, 168–171, 174 Bayesian, 101, 155–156, 161 delayed rejection, 173–174 definition, 83–84 examples surrogate, 298 heat model, 176–179 Linear regression, 134–141 HIV model, 179–180 Local sensitivity analysis, see spring model, 165–167 Sensitivity analysis, local mixing, 161, 173 random walk, 160 M scaled parameters, 176 Marginal pdf, see Probability density using surrogate, 298 function Metropolis–Hastings algorithm, 165 Markov chain, 90–96 Modelcalibration,8,82 definition, 94 Model discrepancy, 133 detailed balance, 96 bias, 9 homogeneous, 91 effects, 261 irreducible, 93 issues, 267–269 parameter density, 159–162 quantification, 265–267 periodic, 94 relation to epistemic stationary distribution, 93 uncertainties, 8 Markov chain Monte Carlo Model errors, see Model discrepancy (MCMC), 159 Models Matrix abstract framework Cholesky decomposition, 160 linear, 63–65 idempotent, 137 nonlinear, 65 null space, 116 algebraic, 62 positive, 94 atmospheric physics, 14 QR factorization, 118 autoregressive (AR), 89 in random algorithm, 119 beam, 58–60 range, 116 Burgers’ equation, 60 row-stochastic, 91 evolution processes, 61–62 SVD, 117–118 exponential processes, 51–52 in random algorithm, 119 groundwater flow, 35–36 trace properties, 137 heat, 55–57 Maximum a posteriori estimate, 157 HIV, 47–50, 54–55 Maximum likelihood estimate neutron, 40, 57–58 (MLE), 84 portfolio, 321, 328–329, 335 Maximum likelihood estimator, simple harmonic oscillator, 83–85 52–54 Mean, 70 SIR, 55 Measurement errors, 133 stationary processes, 62 Meta-model, see Surrogate model, thermal-hydraulic, 41–42 regression, interpolation Morris screening, 331–337 Method of snapshots, 289 elementary effect, 125, 331 Metropolis algorithm, 159–165 scaled, 332 acceptance ratio, 170, 173 factors, 331 i i i i i i p88 smit 2013/10/2 page 378 i i 378 Index for parameter selection, 124–125 Bayesian perspective, 155–158 sampling strategy, 333–335 frequentist perspective, 133–134 sensitivity measures, 332 Parameter selection Multi-index linear problems definition, 212 deterministic, 117–118 sparse grids, 246 random algorithms, 119–122 Multivariate normal distribution, 78 nonlinear problems linearization-based, 123–125 N variance-based, 122–123 Nataf transformation, 109 Parameters Noninfluential parameters, see as random variables,