Multi-fidelity sensitivity analysis

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quai Watier, 78401 Chatou, France 7th International Conference on Sensitivity Analysis of Model

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

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