sign in sign up
<<
Home , Conjugate prior

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 , 100–104 approach variational, 309–311, empirical, 100 316–317 , 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 , 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 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 , 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 , 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 density estimation (kde), , 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 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 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 , 78 nonlinear problems linearization-based, 123–125 N variance-based, 122–123 Nataf transformation, 109 Parameters Noninfluential parameters, see as random variables, 107–112 Parameters correlated, 108 Noninformative prior, 100 finite-dimensional Nonintrusive methods, 214 representation, 109–112 Normal distribution, 70–71 mutually independent, 107 Nuclear reactor confidence interval, 139–142, CASL, 37 146, 152 design, 36–39 estimator and estimate, light water reactors, 37–39 135–136, 142, 146, 151 models, 39–42 identifiability, 113–114 neutron, 39–41, 57–58 Bayesian algorithms, 171–172 simulation packages, 41, 42 relation to range, 116–117 thermal-hydraulic, 41–42 versus correlation, 125–127 QoI, 43 in Markov chain, 159 uncertainties, 42–43 influential, 114 Nugget, in Kriging model, 277 polynomial representation Null space, see Matrix multiple variables, 212–214 Numerical errors and uncertainties, 7 single variable, 209 Numerical weather prediction relation to inputs, 3 (NWP), see Weather, sampling distribution, 138–139, models 145, 151 Perron–Frobenius theorem, 94 O POD, see Proper orthogonal Optimization routines, 143 decomposition Ordinary least squares (OLS) Point estimate, 79 estimate, scaled, 143 Polynomial chaos (PC), 208 estimator, 82, 135, 142, 146, 151 Posterior density, 100–101, 156 functional, 135, 142 based on conjugate, 163 scaled, 143 Prediction interval, 197–203 Orthogonal complement, 113 definition, 199 Orthogonal polynomials extrapolation, 199–200 Hermite polynomials, 210–211 for uncertainty quantification, Legendre polynomials, 211 201–202 versus confidence interval, P 197–200 Parameter estimation, 6, 132 Predictive estimation, 8–10

i i

i i i i p88 smit 2013/10/2 page 379 i i

Index 379

Predictive science, 1 ODE, 215 Prior distribution stationary problems, 218 conjugate, 103–104, 163 weather models, 3, 19, 21 Jeffreys, 164 noninformative, 100 R parameter estimation, 156 Radial basis functions, 278 Probabilistic risk assessment, 44 Random differential equation, 96 Probability density function (pdf), Random field, 89 69–70 Random process, 87–90 conditional, 78–79 correlated and uncorrelated, 111 marginal, 78 definition, 88 Probability mass function, 70 Gaussian, 89 Probability space, 67 in Kriging model, 277 image space, 107, 209 polynomial expansion, 208–209 Propagation of moments, 194 second-order, 88 Proper orthogonal decomposition spectral representation, 207–208 (POD), 285–289 state space, 90 relation to SVD, 288 stationary, 89 Proposal distribution, 160 Random range algorithm, 119 delayed rejection, 173 definition, 68 Q iid, 79 QR factorization, 118 independent, 77 in random algorithm, 119 multiple, 76 Quadrature rule normal, representation, 212 Gauss–Hermite, 211, 241 polynomial representation Gauss–Legendre, 241 mean and variance, 210 nested, 241–243 multiple variables, 212–214 Clenshaw–Curtis, 242 single variable, 209 composite trapezoid, 242 S-valued, 90 error bound, 243 uncorrelated, 77 sparse grid, 244–250 uniform, representation, 212 adaptive, 249 Random vector, 76 Clenshaw–Curtis, 246 Random walk Metropolis, see error, 248 Metropolis algorithm multi-index, 246 Realization, 68 nodal set, 246 Reduced-order model, see Surrogate stochastic, 239–240 model, projection-based tensor product, 243–244 Regression, see Linear regression Clenshaw–Curtis error, 244 Regressor variables, 132 Quantile-quantile (Q-Q) plot, 74 Response surface model, see Quantity of interest (QoI), 4 Surrogate model, climate models, 22, 31 regression, interpolation evolutionary PDE, 222 Riesz representation theorem, 346 HIV model, 50 for ASAP, 311 nuclear reactor models, 43 Rosenblatt transformation, 109

i i

i i i i p88 smit 2013/10/2 page 380 i i

380 Index

RS-HDMR, see High-dimensional Sensitivity indices model representation local, 322 (HDMR), RS- Morris, 332 Runge function, 251 Sobol, 324 computational algorithm, S 330–331 Sample mean, 80, 173 general densities, 327 Sample variance, 80 Sensitivity matrix Sampling distribution, 80 parameter estimation, 144, 152 asymptotic, 139 Singular value decomposition (SVD), parameter, 138–139, 145, 151 117–118 Sandwich relation, 193 in random algorithm, 119 Screening techniques, see Morris relation to POD, 288 screening Snapshot set, 284 Sensitivity analysis Sobol function, 335 examples Sobol indices, 324 Ishigami function, 329 computational algorithm, neutron diffusion, 306–311, 330–331 314–315 for parameter selection, 123 portfolio model, 321–323, general densities, 327 328–329, 335 statistical properties, 325 SIR disease model, 339–342 total sensitivity indices, 324 Sobol function, 335–337 Sobol representation, 289, 323 spring model, 304–305, general densities, 326–329 315–318 Sparse grid global interpolation, 254 definition, 304 quadrature, 244–250 for parameter selection, 123 adaptive, 249 Morris screening, 331–337 Clenshaw–Curtis, 246 Sobol indices, 324–329 error, 248 time- or space-dependent, multi-index, 246 338–342 nodal set, 246 local Standard deviation, 70 ASAP, 308–311, 313 Standard error, 141, 152 definition, 303 State space, 90 derivative relation, 322 Stationary distribution, 93, 168–171 for parameter selection, Stationary processes, see Models 123–125 Statistic, 80 for uncertainty quantification, Statistical inference 192 Bayesian, 100–104 FSAP, 306, 310, 313 frequentist vs. Bayesian, 98–100 sigma-normalized, 322 goals, 98 Sensitivity equations Statistical model, 133, 142 algebraic problem, 307 Statistical uncertainty, see Aleatoric boundary value problem, 310 uncertainty general model, 313 Stochastic collocation method

i i

i i i i p88 smit 2013/10/2 page 381 i i

Index 381

as surrogate model, 279–280 Systematic uncertainty, see attributes, 224–225 Epistemic uncertainty evolutionary PDE, 223 ODE, 217 T stationary problems, 220–221 Taylor basis method, 284 Stochastic differential equation U (SDE), 97 Unbounded operator, adjoint, Stochastic discrete projection 348–349 attributes, 226 Uncertainties evolutionary PDE, 223 aleatoric, 7 examples, 234–235 epistemic, 8 ODE, 218 sources stationary problems, 221 experimental, 5 Stochastic Galerkin method models and inputs, 5–7 attributes, 223–224 numerical, 7 evolutionary PDE, 222 Uncertainty propagation examples, 226–234 linear models, 188–191 ODE, 216 perturbation methods, 192–197 stationary problems, 220 sampling methods, 191–192 Stochastic polynomial packages, 235 Unidentifiable parameter subspace, Stochastic process, see Random see Identifiable parameter process subspace Stochastic weak model formulation, Uniform distribution, 71–72 216, 219, 233 Student’s t-distribution, 72–73 V Subset selection, 113 Validation, 4 Subsurface hydrology, 33–36 Validation regime, 8 hydraulic fracturing, 33 Vandemonde system, 217 models, 35–36 Variance uncertainties, 35–36 definition, 70 partial and total, 324 Yucca mountain, 33 for parameter selection, 123 Sum of squares error, 156 general densities, 327 with surrogate, 298 statistical properties, 325 Surrogate model Verification, 4 for model calibration, 298–299 projection-based, 280–282 W eigenfunction, 283–284 Weak model formulation HDMR, 289–298 deterministic, 218 POD, 284–289 stochastic, 216, 219, 233 regression, interpolation, Weather, 2–3, 11–20 273–280 ensemble forecasts, 19 kriging, GP, 275–278 equations of atmospheric quadratic, 275 physics, 14 radial basis function, 278–279 Katrina, 19 stochastic collocation, 279–280 models

i i

i i i i p88 smit 2013/10/2 page 382 i i

382 Index

4D-VAR, 18 numerical models, 15–16 closure relations, 15 ECMWF, UK Met, 16, 18 conservation relations, 13–15 uncertainties, 18–19 data assimilation, 16–18 primitive equations, 14 Wiener PC, 209

i i

i i