Multi-fidelity sensitivity analysis
|
|
- Cuthbert Gallagher
- 5 years ago
- Views:
Transcription
1 Multi-fidelity sensitivity analysis Loic Le Gratiet 1,2, Claire Cannamela 2 & Bertrand Iooss 3 1 Université Denis-Diderot, Paris, France 2 CEA, DAM, DIF, F Arpajon, France 3 EDF R&D, 6 quai Watier, Chatou, France 7th International Conference on Sensitivity Analysis of Model Output July 4th / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
2 Introduction Perform a global sensitivity analysis of a time-consuming numerical model output by calculating variance-based measures of model input parameters. Interest in the so-called Sobol indices. Surrogate the code output by a metamodel for estimating the Sobol indices. Focus on a multi-fidelity cokriging metamodel. Propose a Sobol index estimator taking into account the metamodel error and the sampling error introduced during the estimation. 2 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
3 Outline 1 Multi-fidelity co-kriging model 2 Multi-fidelity sensitivity analysis 3 Applications 3 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
4 Motivations Objective : replace the output of a code, called g 2(x), by a metamodel. g 2(x) : x Q R d R Framework : a coarse version g 1 of g 2 is available. Principle : build a metamodel of g 2(x) which integrates as well observations of the coarse code output. Multi-fidelity co-kriging model 4 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
5 Recursive formulation of the model Multi-fidelity co-kriging model :[Kennedy & O Hagan (2000), Le Gratiet (2012)] j Z2(x) = ρz 1 (x) + δ(x) Z 1 (x) δ(x) where Z 1 (x) [Z 1(x) Z 1 = g 1, β 1, σ 2 1, θ 1], with g 1 = g 1(x), x D 1 and Z 1(x) GP `f t 1(x)β 1, σ 2 1r 1(x, x; θ 1), δ(x) GP `f t δ(x)β δ, σ 2 δr δ (x, x; θ δ ) Parameters estimation : θ 1, θ δ,σ1, 2 σδ 2 : maximum likelihood method βδ β 1, : analytical posterior distribution (Bayesian inference) ρ Finally, Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1] with g 2 = g 2(x), x D 2 We suppose that D 2 D 1 and θ 1, θ δ,σ 2 1, σ 2 δ are known. 5 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
6 Recursive formulation of the model Multi-fidelity co-kriging model :[Kennedy & O Hagan (2000), Le Gratiet (2012)] j Z2(x) = ρz 1 (x) + δ(x) Z 1 (x) δ(x) where Z 1 (x) [Z 1(x) Z 1 = g 1, β 1, σ 2 1, θ 1], with g 1 = g 1(x), x D 1 and Z 1(x) GP `f t 1(x)β 1, σ 2 1r 1(x, x; θ 1), δ(x) GP `f t δ(x)β δ, σ 2 δr δ (x, x; θ δ ) Parameters estimation : θ 1, θ δ,σ1, 2 σδ 2 : maximum likelihood method βδ β 1, : analytical posterior distribution (Bayesian inference) ρ Finally, Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1] with g 2 = g 2(x), x D 2 We suppose that D 2 D 1 and θ 1, θ δ,σ 2 1, σ 2 δ are known. 5 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
7 Recursive formulation of the model Multi-fidelity co-kriging model :[Kennedy & O Hagan (2000), Le Gratiet (2012)] j Z2(x) = ρz 1 (x) + δ(x) Z 1 (x) δ(x) where Z 1 (x) [Z 1(x) Z 1 = g 1, β 1, σ 2 1, θ 1], with g 1 = g 1(x), x D 1 and Z 1(x) GP `f t 1(x)β 1, σ 2 1r 1(x, x; θ 1), δ(x) GP `f t δ(x)β δ, σ 2 δr δ (x, x; θ δ ) Parameters estimation : θ 1, θ δ,σ1, 2 σδ 2 : maximum likelihood method βδ β 1, : analytical posterior distribution (Bayesian inference) ρ Finally, Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1] with g 2 = g 2(x), x D 2 We suppose that D 2 D 1 and θ 1, θ δ,σ 2 1, σ 2 δ are known. 5 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
8 Predictive distribution In Universal Cokriging, the predictive distribution of Z 2 (x) is not Gaussian. The predictive mean and variance can be decomposed as : µ Z2 (x) = E [Z 2(x) Z 2 = g 2, Z 1 = g 1] = ˆρµ Z1 (x) + µ δ (x) σz 2 2 (x) = var (Z 2(x) Z 2 = g 2, Z 1 = g 1]) = ˆρ 2 σz 2 1 (x) + σδ(x) 2 Remarks : in µ Z2 (x) : β and ρ are replaced by their posterior means. in σz 2 2 (x) : we infer from the posterior distributions of β and ρ. 6 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
9 Outline 1 Multi-fidelity co-kriging model 2 Multi-fidelity sensitivity analysis 3 Applications 7 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
10 Sobol index definition Context : The input is a random vector X = (X d 1, X d 2 ) defined on (Ω X, F X, P X), with probability measure µ = µ d 1 µ d 2, d = d 1 + d 2. We replace the code output g 2(x) with a multi-fidelity co-kriging model Z 2 (x) defined on (Ω Z, F Z, P Z). Objective : we are interested in evaluating the closed Sobol indices defined by S d 1 = Vd 1 `EX ˆg2(X) X 1 V = varx d var X (g 2(X)) Classical approach estimation : replace g 2(x) by µ Z 2 (x). + Computationally cheap do not infer from the meta-model error 8 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
11 Sobol index definition Context : The input is a random vector X = (X d 1, X d 2 ) defined on (Ω X, F X, P X), with probability measure µ = µ d 1 µ d 2, d = d 1 + d 2. We replace the code output g 2(x) with a multi-fidelity co-kriging model Z 2 (x) defined on (Ω Z, F Z, P Z). Objective : we are interested in evaluating the closed Sobol indices defined by S d 1 = Vd 1 `EX ˆg2(X) X 1 V = varx d var X (g 2(X)) Classical approach estimation : replace g 2(x) by µ Z 2 (x). + Computationally cheap do not infer from the meta-model error 8 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
12 The meta-model error Another approach :[Oakley & O Hagan (2004), Marrel & al. (2009)] Replace g 2(x) by Z2 (x) and consider the term : `EX ˆZ S d 1 Z2 = varx 2 (X) X 1 d var X (Z2 (X)) (on (Ω Z, F Z, P Z)) They suggest the following quantity : They evaluate its uncertainty with : ˆvarX `EX ˆZ S d 1 Z2 = EZ 2 (X) X 1 d E Z [var X (Z2 (X))] eσ 2 Z 2 + infer from the meta-model error = varz `varx `EX ˆZ 2 (X) X d 1 E Z [var X (Z 2 (X))]2 Computationally expensive (numerical integrations) S d 1 Z 2 is not the expectation of S d 1 Z 2 and e Σ 2 Z 2 is different from the variance of S d 1 Z 2 9 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
13 Sobol index estimation - objective Recall : S d 1 Z 2 is a random variable `EX ˆZ S d 1 Z2 = varx 2 (X) X 1 d var X (Z2 (X)) Objective : We want to build A unbiased Monte Carlo estimator Ŝd 1 Z2,m of Sd 1, where m represents the number of Monte-Carlo particles. Z 2 An estimator of the variance of Ŝd 1 Z 2,m. 10 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
14 Sobol index estimation - sampling Generate a 2m-sample x = (x i, x i) i=1,...,m of the random vector (X, X) [Sobol (1993)] such that X = (X d 1, X d 2 ) X = (X d 1, X d 2 ) X d 2 X d 2 For k = 1,..., N Z : 1. Sample a realization z 2,k(x) of Z 2 (x) at points in x. 2. Compute Ŝd 1 Z 2,m,k,1 from z 2,k(x) with : [Janon & al. (2012)] P 1 m Ŝ d 1 Z2,m,k,1 = m i=1 z 2,k(x i)z2,k( x P 1 i) m 2m i=1 z 2,k(x i) + z2,k( x i) P 1 m P 2 m i=1 z 2,k (xi)2 1 m 2m i=1 z 2,k (xi) + z 2,k ( xi) 3. For l = 2,..., B : Sample with replacements x B l from x Compute Ŝd 1 Z 2,m,k,l from z 2,k(x B l ) / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
15 Sobol index estimation - sampling Generate a 2m-sample x = (x i, x i) i=1,...,m of the random vector (X, X) [Sobol (1993)] such that X = (X d 1, X d 2 ) X = (X d 1, X d 2 ) X d 2 X d 2 For k = 1,..., N Z : 1. Sample a realization z 2,k(x) of Z 2 (x) at points in x. 2. Compute Ŝd 1 Z 2,m,k,1 from z 2,k(x) with : [Janon & al. (2012)] P 1 m Ŝ d 1 Z2,m,k,1 = m i=1 z 2,k(x i)z2,k( x P 1 i) m 2m i=1 z 2,k(x i) + z2,k( x i) P 1 m P 2 m i=1 z 2,k (xi)2 1 m 2m i=1 z 2,k (xi) + z 2,k ( xi) 3. For l = 2,..., B : Sample with replacements x B l from x Compute Ŝd 1 Z 2,m,k,l from z 2,k(x B l ) / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
16 Sobol index estimation - sampling Generate a 2m-sample x = (x i, x i) i=1,...,m of the random vector (X, X) [Sobol (1993)] such that X = (X d 1, X d 2 ) X = (X d 1, X d 2 ) X d 2 X d 2 For k = 1,..., N Z : 1. Sample a realization z 2,k(x) of Z 2 (x) at points in x. 2. Compute Ŝd 1 Z 2,m,k,1 from z 2,k(x) with : [Janon & al. (2012)] P 1 m Ŝ d 1 Z2,m,k,1 = m i=1 z 2,k(x i)z2,k( x P 1 i) m 2m i=1 z 2,k(x i) + z2,k( x i) P 1 m P 2 m i=1 z 2,k (xi)2 1 m 2m i=1 z 2,k (xi) + z 2,k ( xi) 3. For l = 2,..., B : Sample with replacements x B l from x Compute Ŝd 1 Z 2,m,k,l from z 2,k(x B l ) / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
17 Sobol index estimation - sampling Generate a 2m-sample x = (x i, x i) i=1,...,m of the random vector (X, X) [Sobol (1993)] such that X = (X d 1, X d 2 ) X = (X d 1, X d 2 ) X d 2 X d 2 For k = 1,..., N Z : 1. Sample a realization z 2,k(x) of Z 2 (x) at points in x. 2. Compute Ŝd 1 Z 2,m,k,1 from z 2,k(x) with : [Janon & al. (2012)] P 1 m Ŝ d 1 Z2,m,k,1 = m i=1 z 2,k(x i)z2,k( x P 1 i) m 2m i=1 z 2,k(x i) + z2,k( x i) P 1 m P 2 m i=1 z 2,k (xi)2 1 m 2m i=1 z 2,k (xi) + z 2,k ( xi) 3. For l = 2,..., B : Sample with replacements x B l from x Compute Ŝd 1 Z 2,m,k,l from z 2,k(x B l ) / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
18 Sobol index estimation - meta-model and MC errors The estimator of S d 1 Z 2 : Ŝ d 1 Z Ŝd 2,m is defined as the empirical mean of the sample 1 Z2 k=1,...,n,m,k,l : Z l=1,...,b N Ŝ d 1 Z2,m = 1 X Z N Z B BX k=1 l=1 Ŝ d 1 Z 2,m,k,l The variance of the estimator Ŝd 1 Z2,m : ˆΣ 2 is estimated from the empirical variance of 1 Ŝd Ŝ d 1 Z 2 Z,m 2 k=1,...,n,m,k,l : Z l=1,...,b N ˆΣ 2 1 X Z Ŝ d 1 = Z 2 N,m Z B 1 BX k=1 l=1 Ŝd 1 Z 2,m,k,l Ŝd 1 Z 2,m 2 It integrates both meta-modeling error and Monte-Carlo integrations error. 12 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
19 Sobol index estimation - minimal number of MC particles Variance due to the meta-modeling : i var Z E 1 X hŝd Z2,m Z 2 (x) can be estimated by : " bσ 2 Z2 = 1 N BX 1 X Z «# 2 Ŝ d 1 Z B N Z 1 2,m,k,l 1 Ŝd Z2,m,l l=1 k=1 Variance due to Monte-Carlo integrations : var X E 1 Z hŝd Z2,m (Xi, X i i) i=1,...,m can be estimated by : " bσ 2 MC = 1 N X Z 1 BX «# 2 Ŝ d 1 Z N Z B 1 2,m,k,l 1 Ŝd Z2,m,k k=1 Determining the minimal number of Monte-Carlo particles m : l=1 Considering given the cokriging model Z 2 (x), the minimal value m is the one such that b Σ 2 Z 2 (x) b Σ 2 MC 13 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
20 Sobol index estimation - minimal number of MC particles Variance due to the meta-modeling : i var Z E 1 X hŝd Z2,m Z 2 (x) can be estimated by : " bσ 2 Z2 = 1 N BX 1 X Z «# 2 Ŝ d 1 Z B N Z 1 2,m,k,l 1 Ŝd Z2,m,l l=1 k=1 Variance due to Monte-Carlo integrations : var X E 1 Z hŝd Z2,m (Xi, X i i) i=1,...,m can be estimated by : " bσ 2 MC = 1 N X Z 1 BX «# 2 Ŝ d 1 Z N Z B 1 2,m,k,l 1 Ŝd Z2,m,k k=1 Determining the minimal number of Monte-Carlo particles m : l=1 Considering given the cokriging model Z 2 (x), the minimal value m is the one such that b Σ 2 Z 2 (x) b Σ 2 MC 13 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
21 Outline 1 Multi-fidelity co-kriging model 2 Multi-fidelity sensitivity analysis 3 Applications 14 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
22 Ishigami function - kriging case First order Sobol index of the first input parameter when the size n of the learning sample increases. Learning sample : space filling design covariance structure : tensorised 5/2-Matérn kernel m = N Z = 500 B = 300 S % meta model confidence intervals 95% MC+meta model confidence intervals Sensitivity index mean Theoretical Sobol index n 15 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
23 Spherical tank under internal pressure - cokriging case Fine code : g 2(x) is the Von Mises stress at point 1 provided by a finite elements code. x = (P, R int, T shell, T cap, E shell, E cap, σ y,shell, σ y,cap) (d = 8) P : value of the internal pressure. R int : length of the internal radius of the shell. T shell : thickness of the shell. T cap : thickness of the cap. E shell : Young s modulus of the shell material. E cap : Young s modulus of the cap material. σ y,shell : yield stress of the cap material. σ y,cap : yield stress of the cap material. Coarse code : g 1 is the 1D simplification corresponding to a perfectly spherical tank g 1(x) = 3 (R int + T shell ) 3 2 (R int + T shell ) 3 P Rint 3 0 P CAP 1 SHELL Multi-fidelity co-kriging model : built with n 1 = 100 and only n 2 = 20. We estimate the efficiency E ff of the metamodel of g 2(x) by the coefficient of determination calculated on an external set of 7000 points. E ff 86% 0 16 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
24 Spherical tank under internal pressure - results Kriging-based sensitivity analysis of the coarse code output g 1(x) Learning sample : space filling designs n 1 = 100 covariance structure : tensorised 5/2-Matérn kernel m = N Z = 300 B = 150 P 8 i=1 Si 97% S % meta model confidence intervals 95% MC+meta model confidence intervals Sensitivity index mean R int T shell T cap E shell σ y E cap σ y P x 17 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
25 Spherical tank under internal pressure - results Cokriging-based sensitivity analysis of the fine code output g 2(x) Learning sample : space filling design n 1 = 100 n 2 = 20 covariance structures : 2 tensorised 5/2-Matérn kernels m = N Z = 300 B = 150 P 8 i=1 Si 96.7% S % meta model confidence intervals 95% MC+meta model confidence intervals Sensitivity index mean R int T shell T cap E shell σ y E cap σ y P x 17 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
26 Conclusion We have proposed a Sobol index estimator which uses realizations of a predictive process (built from a multi-fidelity cokriging model) at points in a large sample. we use the conditioning kriging [Chilès & Delfiner (1999)] In the computation of its variance, we distinguish among metamodel and Monte-Carlo variance contributions. We can also determine the minimal number of particles m such that the Monte-Carlo variance no longer dominates. The value m is different for each Sobol index estimation. 18 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
27 Bibliography Sobol, I. M. (1993), Sensitivity estimates for non linear mathematical models. Mathematical modeling and Computational Experiments 1, Kennedy, M. C. & O Hagan, A. (2000), Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, Le Gratiet, L. & Garnier, J. (2012), Recursive co-kriging model for Design of Computer experiments with multiple levels of fidelity with an application to hydrodynamic, submitted to International Journal of Uncertainty Quantification. J.E. Oakley & A O Hagan (2004), Probabilistic sensitivity analysis of complex models : A bayesian approach, Journal of the Royal Statistical Society B, 66(3) : A. Marrel, B. Iooss, B. Laurent & O. Roustant (2012), Calculations of Sobol indices for the gaussian process metamodel, Reliability Engineering and System Safety, 94 : A. Janon, T. Klein, A. Lagnoux, M. Nodet & C. Prieur (2013), Asymptotic normality and efficiency of two Sobol index estimators, to appear in ESAIM probability and statistics. J.P. Chilès & P. Delfiner (1999), Geostatistics : modeling spatial uncertainty, Wiley series in probability and statistics (Applied probability and statistics section). L. Le Gratiet,C. Cannamela, & B.Iooss (2013), A Bayesian approach for global sensitivity analysis of (multi-fidelity) computer codes, submitted to SIAM/ASA UQ. R CRAN package : 19 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
28 Sampling from the predictive distribution Two issues : 1. How to sample from Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1]? 2. How to deal with large m? Sampling from Z2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1] : Z2 (x) can be written by : Z2 (x) = ρz1 (x) + δρ,β δ (x) where Z1 (x) and δρ,β δ (x) = [δ(x) ρ, β δ ] are Gaussian processes and (ρ, β δ ) have a known distribution. 1. Sample a realization z1,k (x) of Z 1 (x) 2. Sample a realization (ρ k, β k δ ) of (ρ, β δ ) 3. Sample a realization δk (x) of δ (x) ρ k,β k δ 4. Set z2,k (x) = ρk z 1,k (x) + δ k (x) 20 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
29 Sampling from the predictive distribution Two issues : 1. How to sample from Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1]? 2. How to deal with large m? Sampling from Z2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1] : Z2 (x) can be written by : Z2 (x) = ρz1 (x) + δρ,β δ (x) where Z1 (x) and δρ,β δ (x) = [δ(x) ρ, β δ ] are Gaussian processes and (ρ, β δ ) have a known distribution. 1. Sample a realization z1,k (x) of Z 1 (x) 2. Sample a realization (ρ k, β k δ ) of (ρ, β δ ) 3. Sample a realization δk (x) of δ (x) ρ k,β k δ 4. Set z2,k (x) = ρk z 1,k (x) + δ k (x) 20 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
30 Sampling from the predictive distribution Two issues : 1. How to sample from Z 2 (x) [Z 2(x) Z 2 = g 2, Z 1 = g 1]? 2. How to deal with large m? Sampling with large m : Conditioning kriging result :[Chilès & Delfiner (1999)] We can sample from the conditional distributions of Z 1 (x) and δ (x) by sampling from unconditioned distributions and then by applying a linear transformation on them. We can use efficient algorithms to compute realizations of processes with unconditioned distributions : Stationary kernel : spectral decomposition Tensorised covariance kernel : Karhunen-Loeve decomposition sequentially adding new points to x without re-estimating the decomposition. 20 / 19 L. Le Gratiet, C. Cannamela & B. Iooss Multi-fidelity sensitivity analysis
Multi-fidelity co-kriging models
Application to Sequential design Loic Le Gratiet 12, Claire Cannamela 3 1 EDF R&D, Chatou, France 2 UNS CNRS, 69 Sophia Antipolis, France 3 CEA, DAM, DIF, F-91297 Arpajon, France ANR CHORUS April 3, 214
More informationIntroduction to Gaussian-process based Kriging models for metamodeling and validation of computer codes
Introduction to Gaussian-process based Kriging models for metamodeling and validation of computer codes François Bachoc Department of Statistics and Operations Research, University of Vienna (Former PhD
More informationGaussian Processes for Computer Experiments
Gaussian Processes for Computer Experiments Jeremy Oakley School of Mathematics and Statistics, University of Sheffield www.jeremy-oakley.staff.shef.ac.uk 1 / 43 Computer models Computer model represented
More informationGaussian process regression for Sensitivity analysis
Gaussian process regression for Sensitivity analysis GPSS Workshop on UQ, Sheffield, September 2016 Nicolas Durrande, Mines St-Étienne, durrande@emse.fr GPSS workshop on UQ GPs for sensitivity analysis
More informationIntroduction to emulators - the what, the when, the why
School of Earth and Environment INSTITUTE FOR CLIMATE & ATMOSPHERIC SCIENCE Introduction to emulators - the what, the when, the why Dr Lindsay Lee 1 What is a simulator? A simulator is a computer code
More informationA new estimator for quantile-oriented sensitivity indices
A new estimator for quantile-oriented sensitivity indices Thomas Browne Supervisors : J-C. Fort (Paris 5) & T. Klein (IMT-Toulouse) Advisors : B. Iooss & L. Le Gratiet (EDF R&D-MRI Chatou) EDF R&D-MRI
More informationGaussian Process Optimization with Mutual Information
Gaussian Process Optimization with Mutual Information Emile Contal 1 Vianney Perchet 2 Nicolas Vayatis 1 1 CMLA Ecole Normale Suprieure de Cachan & CNRS, France 2 LPMA Université Paris Diderot & CNRS,
More informationKriging models with Gaussian processes - covariance function estimation and impact of spatial sampling
Kriging models with Gaussian processes - covariance function estimation and impact of spatial sampling François Bachoc former PhD advisor: Josselin Garnier former CEA advisor: Jean-Marc Martinez Department
More informationKriging by Example: Regression of oceanographic data. Paris Perdikaris. Brown University, Division of Applied Mathematics
Kriging by Example: Regression of oceanographic data Paris Perdikaris Brown University, Division of Applied Mathematics! January, 0 Sea Grant College Program Massachusetts Institute of Technology Cambridge,
More informationLectures. Variance-based sensitivity analysis in the presence of correlated input variables. Thomas Most. Source:
Lectures Variance-based sensitivity analysis in the presence of correlated input variables Thomas Most Source: www.dynardo.de/en/library Variance-based sensitivity analysis in the presence of correlated
More informationTotal interaction index: A variance-based sensitivity index for second-order interaction screening
Total interaction index: A variance-based sensitivity index for second-order interaction screening J. Fruth a, O. Roustant b, S. Kuhnt a,c a Faculty of Statistics, TU Dortmund University, Vogelpothsweg
More informationarxiv: v1 [math.st] 14 Oct 2013
ANOVA decomposition of conditional Gaussian processes for sensitivity analysis with dependent inputs Gaelle Chastaing Loic Le Gratiet arxiv:1310.3578v1 [math.st] 14 Oct 2013 Abstract October 15, 2013 Complex
More informationÉtude de classes de noyaux adaptées à la simplification et à l interprétation des modèles d approximation.
Étude de classes de noyaux adaptées à la simplification et à l interprétation des modèles d approximation. Soutenance de thèse de N. Durrande Ecole des Mines de St-Etienne, 9 november 2011 Directeurs Laurent
More informationSURROGATE PREPOSTERIOR ANALYSES FOR PREDICTING AND ENHANCING IDENTIFIABILITY IN MODEL CALIBRATION
International Journal for Uncertainty Quantification, ():xxx xxx, 0 SURROGATE PREPOSTERIOR ANALYSES FOR PREDICTING AND ENHANCING IDENTIFIABILITY IN MODEL CALIBRATION Zhen Jiang, Daniel W. Apley, & Wei
More informationAssessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging. Céline Helbert, Delphine Dupuy and Laurent Carraro
Assessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging., Delphine Dupuy and Laurent Carraro Historical context First introduced in the field of geostatistics (Matheron,
More informationTotal interaction index: A variance-based sensitivity index for interaction screening
Total interaction index: A variance-based sensitivity index for interaction screening Jana Fruth, Olivier Roustant, Sonja Kuhnt To cite this version: Jana Fruth, Olivier Roustant, Sonja Kuhnt. Total interaction
More informationAn introduction to Bayesian statistics and model calibration and a host of related topics
An introduction to Bayesian statistics and model calibration and a host of related topics Derek Bingham Statistics and Actuarial Science Simon Fraser University Cast of thousands have participated in the
More informationDerivative-based global sensitivity measures for interactions
Derivative-based global sensitivity measures for interactions Olivier Roustant École Nationale Supérieure des Mines de Saint-Étienne Joint work with J. Fruth, B. Iooss and S. Kuhnt SAMO 13 Olivier Roustant
More informationUncertainty quantification and calibration of computer models. February 5th, 2014
Uncertainty quantification and calibration of computer models February 5th, 2014 Physical model Physical model is defined by a set of differential equations. Now, they are usually described by computer
More informationIntroduction to Gaussian Processes
Introduction to Gaussian Processes Neil D. Lawrence GPSS 10th June 2013 Book Rasmussen and Williams (2006) Outline The Gaussian Density Covariance from Basis Functions Basis Function Representations Constructing
More informationCross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification
Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification François Bachoc Josselin Garnier Jean-Marc Martinez CEA-Saclay, DEN, DM2S, STMF,
More informationDensity Estimation. Seungjin Choi
Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
Chapter 4 - Fundamentals of spatial processes Lecture notes Geir Storvik January 21, 2013 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites Mostly positive correlation Negative
More informationAn adaptive kriging method for characterizing uncertainty in inverse problems
Int Statistical Inst: Proc 58th World Statistical Congress, 2, Dublin Session STS2) p98 An adaptive kriging method for characterizing uncertainty in inverse problems FU Shuai 2 University Paris-Sud & INRIA,
More informationA comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments
University of Exeter Department of Mathematics A comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments Nathan Edward Owen May 2017 Supervised
More informationUnderstanding the uncertainty in the biospheric carbon flux for England and Wales
Understanding the uncertainty in the biospheric carbon flux for England and Wales John Paul Gosling and Anthony O Hagan 30th November, 2007 Abstract Uncertainty analysis is the evaluation of the distribution
More informationSequential Importance Sampling for Rare Event Estimation with Computer Experiments
Sequential Importance Sampling for Rare Event Estimation with Computer Experiments Brian Williams and Rick Picard LA-UR-12-22467 Statistical Sciences Group, Los Alamos National Laboratory Abstract Importance
More informationKernel-based Approximation. Methods using MATLAB. Gregory Fasshauer. Interdisciplinary Mathematical Sciences. Michael McCourt.
SINGAPORE SHANGHAI Vol TAIPEI - Interdisciplinary Mathematical Sciences 19 Kernel-based Approximation Methods using MATLAB Gregory Fasshauer Illinois Institute of Technology, USA Michael McCourt University
More informationPolynomial chaos expansions for sensitivity analysis
c DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for sensitivity analysis B. Sudret Chair of Risk, Safety & Uncertainty
More informationSequential Monte Carlo Methods for Bayesian Computation
Sequential Monte Carlo Methods for Bayesian Computation A. Doucet Kyoto Sept. 2012 A. Doucet (MLSS Sept. 2012) Sept. 2012 1 / 136 Motivating Example 1: Generic Bayesian Model Let X be a vector parameter
More informationValidating Expensive Simulations with Expensive Experiments: A Bayesian Approach
Validating Expensive Simulations with Expensive Experiments: A Bayesian Approach Dr. Arun Subramaniyan GE Global Research Center Niskayuna, NY 2012 ASME V& V Symposium Team: GE GRC: Liping Wang, Natarajan
More informationPractical unbiased Monte Carlo for Uncertainty Quantification
Practical unbiased Monte Carlo for Uncertainty Quantification Sergios Agapiou Department of Statistics, University of Warwick MiR@W day: Uncertainty in Complex Computer Models, 2nd February 2015, University
More informationGeostatistical Modeling for Large Data Sets: Low-rank methods
Geostatistical Modeling for Large Data Sets: Low-rank methods Whitney Huang, Kelly-Ann Dixon Hamil, and Zizhuang Wu Department of Statistics Purdue University February 22, 2016 Outline Motivation Low-rank
More informationBayesian Quadrature: Model-based Approximate Integration. David Duvenaud University of Cambridge
Bayesian Quadrature: Model-based Approimate Integration David Duvenaud University of Cambridge The Quadrature Problem ˆ We want to estimate an integral Z = f ()p()d ˆ Most computational problems in inference
More informationGeostatistics for Gaussian processes
Introduction Geostatistical Model Covariance structure Cokriging Conclusion Geostatistics for Gaussian processes Hans Wackernagel Geostatistics group MINES ParisTech http://hans.wackernagel.free.fr Kernels
More informationStatistics: Learning models from data
DS-GA 1002 Lecture notes 5 October 19, 2015 Statistics: Learning models from data Learning models from data that are assumed to be generated probabilistically from a certain unknown distribution is a crucial
More informationUncertainty quantification and visualization for functional random variables
Uncertainty quantification and visualization for functional random variables MascotNum Workshop 2014 S. Nanty 1,3 C. Helbert 2 A. Marrel 1 N. Pérot 1 C. Prieur 3 1 CEA, DEN/DER/SESI/LSMR, F-13108, Saint-Paul-lez-Durance,
More informationCovariance function estimation in Gaussian process regression
Covariance function estimation in Gaussian process regression François Bachoc Department of Statistics and Operations Research, University of Vienna WU Research Seminar - May 2015 François Bachoc Gaussian
More informationReview. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Probability and statistics Probability: Framework for dealing with
More informationNonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University
Nonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University this presentation derived from that presented at the Pan-American Advanced
More informationLocal Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs
Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs Sébastien Da Veiga, François Wahl, Fabrice Gamboa To cite this version: Sébastien Da Veiga, François Wahl, Fabrice
More informationSimultaneous Robust Design and Tolerancing of Compressor Blades
Simultaneous Robust Design and Tolerancing of Compressor Blades Eric Dow Aerospace Computational Design Laboratory Department of Aeronautics and Astronautics Massachusetts Institute of Technology GTL Seminar
More informationLecture: Gaussian Process Regression. STAT 6474 Instructor: Hongxiao Zhu
Lecture: Gaussian Process Regression STAT 6474 Instructor: Hongxiao Zhu Motivation Reference: Marc Deisenroth s tutorial on Robot Learning. 2 Fast Learning for Autonomous Robots with Gaussian Processes
More informationMultifidelity Monte Carlo Estimation of Variance and Sensitivity Indices
Estimation of Variance and Sensitivity Indices E. Qian, B. Peherstorfer, D. O Malley, V. V. Vesselinov, and K. Willcox Abstract. Variance-based sensitivity analysis provides a quantitative measure of how
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
TK4150 - Intro 1 Chapter 4 - Fundamentals of spatial processes Lecture notes Odd Kolbjørnsen and Geir Storvik January 30, 2017 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites
More informationRECOMMENDED TOOLS FOR SENSITIVITY ANALYSIS ASSOCIATED TO THE EVALUATION OF MEASUREMENT UNCERTAINTY
Advanced Mathematical and Computational Tools in Metrology and Testing IX Edited by F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill and K. Shirono c 2012 World Scientific Publishing Company (pp.
More informationGLOBAL SENSITIVITY ANALYSIS WITH DEPENDENT INPUTS. Philippe Wiederkehr
GLOBAL SENSITIVITY ANALYSIS WITH DEPENDENT INPUTS Philippe Wiederkehr CHAIR OF RISK, SAFETY AND UNCERTAINTY QUANTIFICATION STEFANO-FRANSCINI-PLATZ 5 CH-8093 ZÜRICH Risk, Safety & Uncertainty Quantification
More informationRobustness criterion for the optimization scheme based on kriging metamodel
Robustness criterion for the optimization scheme based on kriging metamodel Melina Ribaud a,b, Frederic Gillot a, Celine Helbert b, Christophette Blanchet-Scalliet b and Celine Vial c,d a. Laboratoire
More informationPractical Bayesian Optimization of Machine Learning. Learning Algorithms
Practical Bayesian Optimization of Machine Learning Algorithms CS 294 University of California, Berkeley Tuesday, April 20, 2016 Motivation Machine Learning Algorithms (MLA s) have hyperparameters that
More informationDinesh Kumar, Mehrdad Raisee and Chris Lacor
Dinesh Kumar, Mehrdad Raisee and Chris Lacor Fluid Mechanics and Thermodynamics Research Group Vrije Universiteit Brussel, BELGIUM dkumar@vub.ac.be; m_raisee@yahoo.com; chris.lacor@vub.ac.be October, 2014
More informationSensitivity analysis using the Metamodel of Optimal Prognosis. Lectures. Thomas Most & Johannes Will
Lectures Sensitivity analysis using the Metamodel of Optimal Prognosis Thomas Most & Johannes Will presented at the Weimar Optimization and Stochastic Days 2011 Source: www.dynardo.de/en/library Sensitivity
More informationApplications of Randomized Methods for Decomposing and Simulating from Large Covariance Matrices
Applications of Randomized Methods for Decomposing and Simulating from Large Covariance Matrices Vahid Dehdari and Clayton V. Deutsch Geostatistical modeling involves many variables and many locations.
More informationForecasting. This optimal forecast is referred to as the Minimum Mean Square Error Forecast. This optimal forecast is unbiased because
Forecasting 1. Optimal Forecast Criterion - Minimum Mean Square Error Forecast We have now considered how to determine which ARIMA model we should fit to our data, we have also examined how to estimate
More informationFast Update of Conditional Simulation Ensembles
Fast Update of Conditional Simulation Ensembles Clément Chevalier, Xavier Emery, and David Ginsbourger Abstract Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics
More informationKarhunen-Loève decomposition of Gaussian measures on Banach spaces
Karhunen-Loève decomposition of Gaussian measures on Banach spaces Jean-Charles Croix GT APSSE - April 2017, the 13th joint work with Xavier Bay. 1 / 29 Sommaire 1 Preliminaries on Gaussian processes 2
More informationarxiv: v1 [stat.me] 24 May 2010
The role of the nugget term in the Gaussian process method Andrey Pepelyshev arxiv:1005.4385v1 [stat.me] 24 May 2010 Abstract The maximum likelihood estimate of the correlation parameter of a Gaussian
More informationLarge Scale Modeling by Bayesian Updating Techniques
Large Scale Modeling by Bayesian Updating Techniques Weishan Ren Centre for Computational Geostatistics Department of Civil and Environmental Engineering University of Alberta Large scale models are useful
More informationData-driven Kriging models based on FANOVA-decomposition
Data-driven Kriging models based on FANOVA-decomposition Thomas Muehlenstaedt, Olivier Roustant, Laurent Carraro, Sonja Kuhnt To cite this version: Thomas Muehlenstaedt, Olivier Roustant, Laurent Carraro,
More informationAdvanced analysis and modelling tools for spatial environmental data. Case study: indoor radon data in Switzerland
EnviroInfo 2004 (Geneva) Sh@ring EnviroInfo 2004 Advanced analysis and modelling tools for spatial environmental data. Case study: indoor radon data in Switzerland Mikhail Kanevski 1, Michel Maignan 1
More informationBayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling
Bayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling Jon Wakefield Departments of Statistics and Biostatistics University of Washington 1 / 37 Lecture Content Motivation
More informationLecture 5: GPs and Streaming regression
Lecture 5: GPs and Streaming regression Gaussian Processes Information gain Confidence intervals COMP-652 and ECSE-608, Lecture 5 - September 19, 2017 1 Recall: Non-parametric regression Input space X
More informationMonte-Carlo MMD-MA, Université Paris-Dauphine. Xiaolu Tan
Monte-Carlo MMD-MA, Université Paris-Dauphine Xiaolu Tan tan@ceremade.dauphine.fr Septembre 2015 Contents 1 Introduction 1 1.1 The principle.................................. 1 1.2 The error analysis
More informationVariational inference
Simon Leglaive Télécom ParisTech, CNRS LTCI, Université Paris Saclay November 18, 2016, Télécom ParisTech, Paris, France. Outline Introduction Probabilistic model Problem Log-likelihood decomposition EM
More informationADVANCED MACHINE LEARNING ADVANCED MACHINE LEARNING. Non-linear regression techniques Part - II
1 Non-linear regression techniques Part - II Regression Algorithms in this Course Support Vector Machine Relevance Vector Machine Support vector regression Boosting random projections Relevance vector
More informationChallenges In Uncertainty, Calibration, Validation and Predictability of Engineering Analysis Models
Challenges In Uncertainty, Calibration, Validation and Predictability of Engineering Analysis Models Dr. Liping Wang GE Global Research Manager, Probabilistics Lab Niskayuna, NY 2011 UQ Workshop University
More information6.4 Kalman Filter Equations
6.4 Kalman Filter Equations 6.4.1 Recap: Auxiliary variables Recall the definition of the auxiliary random variables x p k) and x m k): Init: x m 0) := x0) S1: x p k) := Ak 1)x m k 1) +uk 1) +vk 1) S2:
More informationBayesian Dynamic Linear Modelling for. Complex Computer Models
Bayesian Dynamic Linear Modelling for Complex Computer Models Fei Liu, Liang Zhang, Mike West Abstract Computer models may have functional outputs. With no loss of generality, we assume that a single computer
More informationarxiv: v2 [stat.ap] 26 Feb 2013
Calibration and improved prediction of computer models by universal Kriging arxiv:1301.4114v2 [stat.ap] 26 Feb 2013 François Bachoc CEA-Saclay, DEN, DM2S, STMF, LGLS, F-91191 Gif-Sur-Yvette, France Laboratoire
More informationLog Gaussian Cox Processes. Chi Group Meeting February 23, 2016
Log Gaussian Cox Processes Chi Group Meeting February 23, 2016 Outline Typical motivating application Introduction to LGCP model Brief overview of inference Applications in my work just getting started
More informationConvergence of Eigenspaces in Kernel Principal Component Analysis
Convergence of Eigenspaces in Kernel Principal Component Analysis Shixin Wang Advanced machine learning April 19, 2016 Shixin Wang Convergence of Eigenspaces April 19, 2016 1 / 18 Outline 1 Motivation
More informationGeostatistics for Seismic Data Integration in Earth Models
2003 Distinguished Instructor Short Course Distinguished Instructor Series, No. 6 sponsored by the Society of Exploration Geophysicists European Association of Geoscientists & Engineers SUB Gottingen 7
More informationSobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis
Sobol-Hoeffding decomposition Application to Global SA Computation of the SI Sobol-Hoeffding Decomposition with Application to Global Sensitivity Analysis Olivier Le Maître with Colleague & Friend Omar
More informationPolynomial chaos expansions for structural reliability analysis
DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for structural reliability analysis B. Sudret & S. Marelli Incl.
More informationModelling Under Risk and Uncertainty
Modelling Under Risk and Uncertainty An Introduction to Statistical, Phenomenological and Computational Methods Etienne de Rocquigny Ecole Centrale Paris, Universite Paris-Saclay, France WILEY A John Wiley
More informationRisk Analysis: Efficient Computation of Failure Probability
Risk Analysis: Efficient Computation of Failure Probability Nassim RAZAALY CWI - INRIA Bordeaux nassim.razaaly@inria.fr 20/09/2017 Nassim RAZAALY (INRIA-CWI) Tail Probability 20/09/2017 1 / 13 Motivation:
More informationLecture Notes 5 Convergence and Limit Theorems. Convergence with Probability 1. Convergence in Mean Square. Convergence in Probability, WLLN
Lecture Notes 5 Convergence and Limit Theorems Motivation Convergence with Probability Convergence in Mean Square Convergence in Probability, WLLN Convergence in Distribution, CLT EE 278: Convergence and
More informationSTA414/2104. Lecture 11: Gaussian Processes. Department of Statistics
STA414/2104 Lecture 11: Gaussian Processes Department of Statistics www.utstat.utoronto.ca Delivered by Mark Ebden with thanks to Russ Salakhutdinov Outline Gaussian Processes Exam review Course evaluations
More informationOn ANOVA decompositions of kernels and Gaussian random field paths
On ANOVA decompositions of kernels and Gaussian random field paths David Ginsbourger, Olivier Roustant, Dominic Schuhmacher, Nicolas Durrande, Nicolas Lenz To cite this version: David Ginsbourger, Olivier
More informationMonte Carlo Integration I [RC] Chapter 3
Aula 3. Monte Carlo Integration I 0 Monte Carlo Integration I [RC] Chapter 3 Anatoli Iambartsev IME-USP Aula 3. Monte Carlo Integration I 1 There is no exact definition of the Monte Carlo methods. In the
More informationLikelihood-Based Methods
Likelihood-Based Methods Handbook of Spatial Statistics, Chapter 4 Susheela Singh September 22, 2016 OVERVIEW INTRODUCTION MAXIMUM LIKELIHOOD ESTIMATION (ML) RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION (REML)
More informationMultiple Speaker Tracking with the Factorial von Mises- Fisher Filter
Multiple Speaker Tracking with the Factorial von Mises- Fisher Filter IEEE International Workshop on Machine Learning for Signal Processing Sept 21-24, 2014 Reims, France Johannes Traa, Paris Smaragdis
More informationREGRESSION WITH SPATIALLY MISALIGNED DATA. Lisa Madsen Oregon State University David Ruppert Cornell University
REGRESSION ITH SPATIALL MISALIGNED DATA Lisa Madsen Oregon State University David Ruppert Cornell University SPATIALL MISALIGNED DATA 10 X X X X X X X X 5 X X X X X 0 X 0 5 10 OUTLINE 1. Introduction 2.
More informationSteps in Uncertainty Quantification
Steps in Uncertainty Quantification Challenge: How do we do uncertainty quantification for computationally expensive models? Example: We have a computational budget of 5 model evaluations. Bayesian inference
More informationThe Laplace driven moving average a non-gaussian stationary process
The Laplace driven moving average a non-gaussian stationary process 1, Krzysztof Podgórski 2, Igor Rychlik 1 1 Mathematical Sciences, Mathematical Statistics, Chalmers 2 Centre for Mathematical Sciences,
More informationComparing Non-informative Priors for Estimation and Prediction in Spatial Models
Environmentrics 00, 1 12 DOI: 10.1002/env.XXXX Comparing Non-informative Priors for Estimation and Prediction in Spatial Models Regina Wu a and Cari G. Kaufman a Summary: Fitting a Bayesian model to spatial
More informationSequential approaches to reliability estimation and optimization based on kriging
Sequential approaches to reliability estimation and optimization based on kriging Rodolphe Le Riche1,2 and Olivier Roustant2 1 CNRS ; 2 Ecole des Mines de Saint-Etienne JSO 2012, ONERA Palaiseau 1 Intro
More informationA stochastic 3-scale method to predict the thermo-elastic behaviors of polycrystalline structures
Computational & Multiscale Mechanics of Materials CM3 www.ltas-cm3.ulg.ac.be A stochastic 3-scale method to predict the thermo-elastic behaviors of polycrystalline structures Wu Ling, Lucas Vincent, Golinval
More informationCalibration and validation of computer models for radiative shock experiment
Calibration and validation of computer models for radiative shock experiment Jean Giorla 1, J. Garnier 2, E. Falize 1,3, B. Loupias 1, C. Busschaert 1,3, M. Koenig 4, A. Ravasio 4, C. Michaut 3 1 CEA/DAM/DIF,
More informationBEST ESTIMATE PLUS UNCERTAINTY SAFETY STUDIES AT THE CONCEPTUAL DESIGN PHASE OF THE ASTRID DEMONSTRATOR
BEST ESTIMATE PLUS UNCERTAINTY SAFETY STUDIES AT THE CONCEPTUAL DESIGN PHASE OF THE ASTRID DEMONSTRATOR M. Marquès CEA, DEN, DER F-13108, Saint-Paul-lez-Durance, France Advanced simulation in support to
More informationReliability Monitoring Using Log Gaussian Process Regression
COPYRIGHT 013, M. Modarres Reliability Monitoring Using Log Gaussian Process Regression Martin Wayne Mohammad Modarres PSA 013 Center for Risk and Reliability University of Maryland Department of Mechanical
More informationProbing the covariance matrix
Probing the covariance matrix Kenneth M. Hanson Los Alamos National Laboratory (ret.) BIE Users Group Meeting, September 24, 2013 This presentation available at http://kmh-lanl.hansonhub.com/ LA-UR-06-5241
More informationParticle Filters. Outline
Particle Filters M. Sami Fadali Professor of EE University of Nevada Outline Monte Carlo integration. Particle filter. Importance sampling. Degeneracy Resampling Example. 1 2 Monte Carlo Integration Numerical
More informationFractional Hot Deck Imputation for Robust Inference Under Item Nonresponse in Survey Sampling
Fractional Hot Deck Imputation for Robust Inference Under Item Nonresponse in Survey Sampling Jae-Kwang Kim 1 Iowa State University June 26, 2013 1 Joint work with Shu Yang Introduction 1 Introduction
More informationPoint spread function reconstruction from the image of a sharp edge
DOE/NV/5946--49 Point spread function reconstruction from the image of a sharp edge John Bardsley, Kevin Joyce, Aaron Luttman The University of Montana National Security Technologies LLC Montana Uncertainty
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationWorkshop Key UQ methodologies and motivating applications
Workshop Key UQ methodologies and motivating applications Uncertainty quantification of numerical experiments: Several issues in industrial applications Bertrand Iooss, EDF R&D Isaac Newton Institute,
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM Repetition of Previous Lecture Sparse grids in Uncertainty Quantification
More informationMonitoring Wafer Geometric Quality using Additive Gaussian Process
Monitoring Wafer Geometric Quality using Additive Gaussian Process Linmiao Zhang 1 Kaibo Wang 2 Nan Chen 1 1 Department of Industrial and Systems Engineering, National University of Singapore 2 Department
More informationSpatial smoothing using Gaussian processes
Spatial smoothing using Gaussian processes Chris Paciorek paciorek@hsph.harvard.edu August 5, 2004 1 OUTLINE Spatial smoothing and Gaussian processes Covariance modelling Nonstationary covariance modelling
More informationEfficient sensitivity analysis for virtual prototyping. Lectures. Thomas Most & Johannes Will
Lectures Efficient sensitivity analysis for virtual prototyping Thomas Most & Johannes Will presented at the ECCOMAS Conference 2012 Source: www.dynardo.de/en/library European Congress on Computational
More information