Polynomial chaos expansions for sensitivity analysis

Transcription

1 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 Quantification Ecole ASPEN May 9th, 2014 Les Houches

2 Global framework for uncertainty quantification Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Distribution Mean, std. deviation Probability of failure Step C B.S., Uncertainty propagation and sensitivity analysis in mechanical models, Habilitation thesis, B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

3 Global framework for uncertainty quantification Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Distribution Mean, std. deviation Probability of failure Step C B.S., Uncertainty propagation and sensitivity analysis in mechanical models, Habilitation thesis, B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

4 Global framework for uncertainty quantification Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Distribution Mean, std. deviation Probability of failure Step C B.S., Uncertainty propagation and sensitivity analysis in mechanical models, Habilitation thesis, B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

5 Global framework for uncertainty quantification Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Distribution Mean, std. deviation Probability of failure Step C B.S., Uncertainty propagation and sensitivity analysis in mechanical models, Habilitation thesis, B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

6 Outline Polynomial chaos expansions 1 Polynomial chaos expansions Computation of the expansion coefficients Error estimation and validation 2 Application to sensitivity analysis Sobol indices PC-based Sobol indices 3 Ishigami function Morris function Marelli function B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

7 Computing the coefficients Error estimation and validation Spectral approach Xiu & Karniadakis (2002) Soize & Ghanem (2004) The input parameters are modelled by a random vector X over a probabilistic space (Ω, F, P) such that P (dx) = f X (x) dx The response random vector Y = M(X) is considered as an element of L 2 (Ω, F, P) A basis of multivariate orthogonal polynomials is built up with respect to the input PDF (assuming independent components) The response random vector Y is completely determined by its coordinates in this basis B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

8 Computing the coefficients Error estimation and validation Spectral approach Xiu & Karniadakis (2002) Soize & Ghanem (2004) The input parameters are modelled by a random vector X over a probabilistic space (Ω, F, P) such that P (dx) = f X (x) dx The response random vector Y = M(X) is considered as an element of L 2 (Ω, F, P) A basis of multivariate orthogonal polynomials is built up with respect to the input PDF (assuming independent components) The response random vector Y is completely determined by its coordinates in this basis B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

9 Computing the coefficients Error estimation and validation Spectral approach Xiu & Karniadakis (2002) Soize & Ghanem (2004) The input parameters are modelled by a random vector X over a probabilistic space (Ω, F, P) such that P (dx) = f X (x) dx The response random vector Y = M(X) is considered as an element of L 2 (Ω, F, P) A basis of multivariate orthogonal polynomials is built up with respect to the input PDF (assuming independent components) The response random vector Y is completely determined by its coordinates in this basis where : Y = α N M y α Ψ α(x) y α : coefficients to be computed (coordinates) Ψ α(x) : basis B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

10 Computing the coefficients Error estimation and validation Univariate orthogonal polynomials For each marginal distribution f Xi (x i) one can define a functional inner product: φ 1, φ 2 i = D i φ 1(x) φ 2(x) f Xi (x i) dx i and a family of orthogonal polynomials {P (i) k, k N} such that:, P (i) = (x) P (i) (x) f X i (x) dx = aj i δ jk P (i) j k P (i) j k Classical families Xiu & Karniadakis (2002) Type of variable Weight function Orthogonal polynomials Hilbertian basis ψ k (x) Uniform 1 ] 1,1[ (x)/2 Legendre P k (x) P k (x)/ 1 2k+1 Gaussian 1 e x2 /2 Hermite He (x) 2π k He (x)/ k! k Gamma x a e x Γ(k+a+1) 1 R + (x) Laguerre La k (x) La k (x)/ k! Beta 1 ] 1,1[ (x) (1 x)a (1+x) b Jacobi J a,b (x) J a,b (x)/j B(a) B(b) k k a,b,k J 2 a,b,k = 2k+a+b+1 2a+b+1 Γ(k+a+1)Γ(k+b+1) Γ(k+a+b+1)Γ(k+1) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

11 Computing the coefficients Error estimation and validation Univariate orthogonal polynomials For each marginal distribution f Xi (x i) one can define a functional inner product: φ 1, φ 2 i = D i φ 1(x) φ 2(x) f Xi (x i) dx i and a family of orthogonal polynomials {P (i) k, k N} such that:, P (i) = (x) P (i) (x) f X i (x) dx = aj i δ jk P (i) j k P (i) j k Classical families Xiu & Karniadakis (2002) Type of variable Weight function Orthogonal polynomials Hilbertian basis ψ k (x) Uniform 1 ] 1,1[ (x)/2 Legendre P k (x) P k (x)/ 1 2k+1 Gaussian 1 e x2 /2 Hermite He (x) 2π k He (x)/ k! k Gamma x a e x Γ(k+a+1) 1 R + (x) Laguerre La k (x) La k (x)/ k! Beta 1 ] 1,1[ (x) (1 x)a (1+x) b Jacobi J a,b (x) J a,b (x)/j B(a) B(b) k k a,b,k J 2 a,b,k = 2k+a+b+1 2a+b+1 Γ(k+a+1)Γ(k+b+1) Γ(k+a+b+1)Γ(k+1) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

12 Computing the coefficients Error estimation and validation Multivariate polynomials Let us define the multi-indices (tuples) α = {α 1,..., α M }, of degree M α = α i. The associated multivariate polynomial reads: i=1 M Ψ α(x) = P α (i) i (x i) Orthonormality of the basis E [Ψ α(x)ψ β (X)] = δ αβ (Kronecker symbol: δ αβ = 1 if α = β, 0 otherwise) i=1 The set of multivariate polynomials {Ψ α, α N M } forms a basis of the space of second order variables: Y = y α Ψ α(x) α N M B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

13 Practical implementation Computing the coefficients Error estimation and validation The input random variables are first transformed into reduced variables (e.g. standard normal variables N (0, 1), uniform variables on [-1,1], etc.): X = T (ξ) dim ξ = M (isoprobabilistic transform) The model response is cast as a function of the reduced variables and expanded: Y = M(X) = M T (ξ) = y α Ψ α(ξ) α N M A truncation scheme is selected and the associated finite set of multi-indices is generated, e.g. : ( ) A M,p = {α N M : α p} card A M,p M + p P = p B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

14 Application example (1) Computing the coefficients Error estimation and validation Computational model Y = M(X 1, X 2) Probabilistic model X i N (µ i, σ i) Isoprobabilistic transform X i = µ i + σ i ξ i Hermite polynomials Recurrence: H 1(x) = H 0(x) = 1 H n+1(x) = x H n(x) n H n 1(x) where H n 2 = n! First (normalized) polynomials: Hen (x) x P 0(x) = 1 P 1(x) = x P 2(x) = (x 2 1)/ 2 P 3(x) = (x 3 3x)/ 6 He1 He2 He3 He4 He5 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

15 Application example (3) Computing the coefficients Error estimation and validation Truncature scheme Consider all multivariate polynomials of total degree α = M than or equal to p. i=1 αi less α 3 α 4 α 5 α 6 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

16 Application example (4) Computing the coefficients Error estimation and validation Third order truncature p = 3 All the polynomials of ξ 1, ξ 2 that are product of univariate Hermite polynomials and whose total degree is less than 3 are considered j α Ψ α Ψ j 0 [0, 0] Ψ 0 = 1 1 [1, 0] Ψ 1 = ξ 1 2 [0, 1] Ψ 2 = ξ 2 3 [2, 0] Ψ 3 = (ξ1 2 1)/ 2 4 [1, 1] Ψ 4 = ξ 1 ξ 2 5 [0, 2] Ψ 5 = (ξ2 2 1)/ 2 6 [3, 0] Ψ 6 = (ξ1 3 3ξ 1)/ 6 7 [2, 1] Ψ 7 = (ξ1 2 1)ξ 2/ 2 8 [1, 2] Ψ 8 = (ξ2 2 1)ξ 1/ 2 9 [0, 3] Ψ 9 = (ξ2 3 3ξ 2)/ 6 Ỹ M PC (ξ 1, ξ 2 ) = y 0 + y 1 ξ 1 + y 2 ξ 2 + y 3 (ξ1 2 1)/ 2 + y 4 ξ 1 ξ 2 + y 5 (ξ2 2 1)/ 2 + y 6 (ξ1 3 3ξ 1)/ 6 + y 7 (ξ1 2 1)ξ 2/ 2 + y 8 (ξ2 2 1)ξ 1/ 2 + y 9 (ξ2 3 3ξ 2)/ 6 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

17 Outline Polynomial chaos expansions Computing the coefficients Error estimation and validation 1 Polynomial chaos expansions Computation of the expansion coefficients Error estimation and validation 2 Application to sensitivity analysis 3 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

18 Computing the coefficients Error estimation and validation Various methods for computing the coefficients Intrusive approaches Historical approaches: projection of the equations residuals in the Galerkin sense Ghanem & Spanos, 1991, 2003 Proper generalized decompositions Nouy, Non intrusive approaches Non intrusive methods consider the computational model M as a black box They rely upon a design of numerical experiments, i.e. a n-sample X = {x (i) D X, i = 1,..., n} of the input parameters Different classes of methods are available: Projection : by simulation or quadrature Stochastic collocation Least-square minimization B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

19 Computing the coefficients Error estimation and validation Statistical approach: least-square minimization Berveiller et al. (2006) Principle The exact (infinite) series expansion is considered as the sum of a truncated series and a residual: P 1 Y = M(X) = y j Ψ j(x) + ε P Y T Ψ(X) + ε P j=0 where : Y = {y 0,..., y P 1 } Ψ(x) = {Ψ 0(x),..., Ψ P 1 (x)} B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

20 Computing the coefficients Error estimation and validation Least-Square Minimization: continuous solution Least-square minimization The unknown coefficients are gathered into a vector Ŷ = {yα, α A}, and computed by minimizing the mean square error: [ (Y Ŷ = arg min E T Ψ(X) M(X) ) ] 2 Analytical solution (continuous case) The least-square minimization problem may be solved analytically: ŷ α = E [M(X) Ψ α(x)] α A The solution is identical to the projection solution due to the orthogonality of the regressors B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

21 Computing the coefficients Error estimation and validation Least-Square Minimization: discretized solution Resolution An estimate of the mean square error (sample average) is minimized: [ (Y Ŷ L.S = arg min Ê T Ψ(X) M(X) ) ] 2 = arg min 1 n n ( Y T Ψ(x (i) ) M(x (i) ) ) 2 i=1 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

22 Computing the coefficients Error estimation and validation Least-Square Minimization: procedure Select an experimental design X = { x (1),..., x (n)} T that covers at best the domain of variation of the parameters Evaluate the model response for each sample (exactly as in Monte carlo simulation) M = { M(x (1) ),..., M(x (n) ) } T Compute the experimental matrix A ij = Ψ j ( x (i) ) i = 1,..., n ; j = 0,..., P 1 Solve the least-square minimization problem Ŷ = (A T A) 1 A T M B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

23 Choice of the experimental design Random designs Computing the coefficients Error estimation and validation Monte Carlo samples obtained by standard random generators Latin Hypercube designs that are both random and space-filling Quasi-random sequences (e.g. Sobol sequence) Monte Carlo Latin Hypercube Sampling Sobol sequence Size of the ED The size n of the experimental design shall be scaled with the number of unknown coefficients, e.g. P = ( ) M+p p The thumb rule n = k P where k = 2 3 is used B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

24 Choice of the experimental design Random designs Computing the coefficients Error estimation and validation Monte Carlo samples obtained by standard random generators Latin Hypercube designs that are both random and space-filling Quasi-random sequences (e.g. Sobol sequence) Monte Carlo Latin Hypercube Sampling Sobol sequence Size of the ED The size n of the experimental design shall be scaled with the number of unknown coefficients, e.g. P = ( ) M+p p The thumb rule n = k P where k = 2 3 is used B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

25 Outline Polynomial chaos expansions Computing the coefficients Error estimation and validation 1 Polynomial chaos expansions Computation of the expansion coefficients Error estimation and validation 2 Application to sensitivity analysis 3 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

26 Validation of the surrogate model Computing the coefficients Error estimation and validation The truncated series expansions are convergent in the mean square sense. However one does not know in advance where to truncate (problem-dependent) Most people truncate the series according to the total maximal degree of the polynomials, say (p=2,3,4, etc.). Several values of p are tested until some kind of convergence is empirically observed The recent research deals with the development of error estimates: adaptive integration in the projection approach cross validation in the least-square minimization approach B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

27 Error estimators Coefficient of determination Polynomial chaos expansions Computing the coefficients Error estimation and validation The least-squares technique is based on the minimization of the mean square error. The generalization error is defined as: [ (M(X) E gen = E M PC (X) ) ] 2 M PC (X) = y α Ψ α(x) α A It may be estimated by the empirical error using the already computed response quantities: E emp = 1 n ( M(x (i) ) M PC (x (i) ) ) 2 n i=1 The coefficient of determination R 2 is often used as an error estimator: R 2 = 1 Eemp Var [Y] Var [Y] = 1 n (M(x(i) ) Ȳ)2 This error estimator leads to overfitting B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

28 Error estimators Leave-one-out cross validation Polynomial chaos expansions Computing the coefficients Error estimation and validation Principle In statistical learning theory, cross validation consists in splitting the experimental design Y into two parts, namely a training set (which is used to build the model) and a validation set The leave-one-out technique consists in using each point of the experimental design as a single validation point for the meta-model built from the remaining n 1 points n different meta-models are built and the error made on the remaining point is computed, then mean-square averaged B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

29 Cross validation Implementation Polynomial chaos expansions Computing the coefficients Error estimation and validation For each x (i), a polynomial chaos expansion is built using the following experimental design: X \x (i) = {x (j), j = 1,..., n, j i}, denoted by M PC\i (.) The predicted residual is computed in point x (i) : i = M(x (i) ) M PC\i (x (i) ) The PRESS coefficient (predicted residual sum of squares) is evaluated: PRESS = The leave-one-out error and related Q 2 error estimator are computed: n i=1 2 i E LOO = 1 n n i=1 2 i Q 2 = 1 E LOO Var [Y] B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

30 Cross validation Implementation Polynomial chaos expansions Computing the coefficients Error estimation and validation In practice one does not need to explicitly derive the n different models M PC\i (.) In contrast, a single least-square analysis using the full ED is carried out. The predicted residual reads: i = M(x (i) ) M PC\i (x (i) ) = M(x(i) ) M PC (x (i) ) 1 h i where h i is the i-th diagonal term of matrix A(A T A) 1 A T, where: A ij = Ψ j(x (i) ) Thus: E LOO = 1 n n i=1 ( ) M(x (i) ) M PC (x (i) 2 ) 1 h i B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

31 Computing the coefficients Error estimation and validation Sparse PC expansions Blatman & S., J. Comp. Phys. (2011) Blatman & S., RESS (2010) The LOO error allows one to assess a posteriori the accuracy of a given expansion Adaptive algorithms may be used based on a target accuracy Other truncation schemes (e.g. hyperbolic) may be used to decrease the computational cost The Least Angle regression algorithm allows one to compute sparse expansion within a candidate set of polynomials Sparse expansions allows one to address problems of dimension M = B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

32 Outline Polynomial chaos expansions Sobol indices PC-based Sobol indices 1 Polynomial chaos expansions 2 Application to sensitivity analysis Sobol indices PC-based Sobol indices 3 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

33 Sobol indices PC-based Sobol indices Sobol (1993) Saltelli et al. (2000) Sobol decomposition aims at quantifying what are the input parameters (or combinations thereof) that influence the most the response variability. Global sensitivity analysis relies on so-called variance decomposition techniques. Consider a model M : x [0, 1] M M(x) R. The Hoeffding-Sobol decomposition reads : M M(x) = M 0 + M i(x i) + M ij(x i, x j) + + M 12...M (x) where : i=1 1 i<j M M 0 is the mean value of the function M i(x i) are univariate functions M ij(x i, x j) are bivariate functions etc. B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

34 Sobol decomposition Properties Polynomial chaos expansions Sobol indices PC-based Sobol indices The functional decomposition is unique if the orthogonality of the terms (with respect to the uniform measure) is imposed [0, 1] M, i.e. {i 1,..., i s} = {j 1,..., j t}: [0,1] M M i1...i s (x i1,..., x is )M j1...j t (x j1,..., x jt ) dx = 0 A construction definition of the terms is obtained by recurrence: M i(x i) = M ij(x i, x j) = M(x) dx i M 0 [0,1] M 1 [0,1] M 2 M(x) dx {ij} M i(x i) M j(x j) + M 0 where [0,1] M 1 (.)dx i denoted the integration over all variables except for the i-th one. B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

35 Sobol indices Polynomial chaos expansions Sobol indices PC-based Sobol indices Variance decomposition Assume X i U(0, 1), i = 1,..., M (possibly after some isoprobabilistic transform) Due to the orthogonality of the decomposition: D Var [M(X)] = E [ (M(X) M 2] 0) = E = {i 1,...,i s} {1,...,M} {i 1,...,i s} {1,...,M} 2 M i1...i s (X i1,..., X is ) E [ M 2 i 1...i s (X i1,..., X is ) ] B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

36 Sobol indices Polynomial chaos expansions Sobol indices PC-based Sobol indices Partial variance Consider: D i1...i s = M 2 i 1...i s (x i1,..., x is ) dx i1... dx is [0,1] s Then: M D Var [Y ] = D i + D ij D 12...M i=1 1 i<j M The Sobol indices are obtained by normalization : S i1...i s = Di 1...i s D They represent the fraction of the total variance Var [Y ] that can be attributed to each input variable i (S i) or combinations of variables {i 1... i s} B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

37 Sobol indices PC-based Sobol indices First order and total Sobol indices Saltelli et al. (1997) First order Sobol indices S i = Di D D i = Var Xi [M i(x)] = Var Xi [E [M(X) X i = x i]] They quantify the (additive) effect of each input parameter separately, i.e. the reduction of variance to expect if X i is set equal to x i Total Sobol indices S T i def = S i1...i s i {i 1...i s} They quantify the total effect of X i including the first order effect and the interactions with the other variables. B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

38 Outline Polynomial chaos expansions Sobol indices PC-based Sobol indices 1 Polynomial chaos expansions 2 Application to sensitivity analysis Sobol indices PC-based Sobol indices 3 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

39 Sobol indices PC-based Sobol indices Polynomial chaos expansions: moments Consider Y = M(X) where X f X with independent components: Y = y α Ψ α(x) α N M Due to the orthogonality properties of the polynomial chaos basis, one gets: E [Ψ α(x)] = 0 E [Ψ α(x) Ψ β (X)] = δ αβ Thus the mean and variance: M 0 = E [M(X)] = y 0 D = Var [M(X)] = α N M α 0 y 2 α B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

40 Sobol indices PC-based Sobol indices Sobol decomposition from PC expansions Sudret ( ) ; Blatman & S. (2010) Interaction sets Let A u be the set of multi-indices depending exactly on the subset of variables u = def {i 1,..., i s}: A u = { α N M : k u α k 0 } A u = N M u {1,...,M} Sobol decomposition def By unicity of the Sobol decomposition one gets (x u = {x i1,..., x is }): M(x) = M 0 + M u(x u) u {1,...,M} where: M u(x u) def = α A u y α Ψ α(x) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

41 Partial variances Polynomial chaos expansions Sobol indices PC-based Sobol indices The partial variances D u def = D i1...i s = Var [M u(x)] are obtained by summing up the square of selected PC coefficients First order contribution D i = yα 2 α A i A i = { α N M : α i > 0, α j i = 0 } Higher order contribution D u = yα 2 α A u A i1...i s = { α N M : k u α k > 0 } The Sobol indices come after normalization: S u = Du D B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

42 Sobol indices PC-based Sobol indices Sobol decomposition from PC expansions Sudret ( ) ; Blatman & S. (2010) First order indices S i = yα/d 2 α A i A i = { α N M : α i > 0, α j i = 0 } Higher order indices S i1,...,i s = yα/d 2 A i1,...,i s = { α N M : k {i 1,..., i s} α j 0 } α A i1,...,is Total indices S T i = y 2 α/d A T i = { α N M : α i > 0 } α A T i B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

43 Sobol indices PC-based Sobol indices Example (M = 2) Computational model Y = M(X 1, X 2) Probabilistic model X i N (µ i, σ i) Isoprobabilistic transform Chaos degree X i = µ i + σ i ξ i p = 3, i.e. P = 10 terms j α Ψα Ψ j 0 [0, 0] Ψ 0 = 1 1 [1, 0] Ψ 1 = ξ 1 2 [0, 1] Ψ 2 = ξ 2 3 [2, 0] Ψ 3 = (ξ 1 2 1)/ 2 5 [0, 2] Ψ 5 = (ξ 2 2 1)/ 2 4 [1, 1] Ψ 4 = ξ 1 ξ 2 6 [3, 0] Ψ 6 = (ξ 3 1 3ξ 1 )/ 6 7 [2, 1] Ψ 7 = (ξ 2 1 1)ξ 2 / 2 8 [1, 2] Ψ 8 = (ξ 2 2 1)ξ 1 / 2 9 [0, 3] Ψ 9 = (ξ 3 2 3ξ 2 )/ 6 Variance D = Sobol indices 9 j=1 y 2 j S 1 = ( y1 2 + y3 2 + y6) 2 /D S 2 = ( y2 2 + y5 2 + y9) 2 /D S 12 = ( y4 2 + y7 2 + y8) 2 /D B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

44 Outline Polynomial chaos expansions Ishigami function Morris function Marelli function 1 Polynomial chaos expansions 2 Application to sensitivity analysis 3 Ishigami function Morris function Marelli function B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

45 Ishigami function Polynomial chaos expansions Ishigami function Morris function Marelli function M = 3 variables Definition Y = sin X 1 + a sin 2 X 2 + b X 4 3 sin X 1 a = 7, b = 0.1 where X i U[ π, π] are independent uniform random variables Analytical solution D = a2 8 + b π4 5 + b2 π D 1 = b π4 5 + b2 π , D 2 = a2 8, D 3 = 0 D 12 = D 23 = 0, D 13 = 8 b2 π 8 225, D 123 = 0 B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

46 First order Sobol indices Ishigami function Morris function Marelli function Si S 1 (MCS) S 1 (PC) S 2 (MCS) S 2 (PC) S 3 (MCS) S 3 (PC) Index Value S S S 3 0 S 12 0 S S Sample size N B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

47 First order Sobol indices Ishigami function Morris function Marelli function Si S (ref) i S 1 (MCS) S 1 (PC) S 2 (MCS) S 2 (PC) S 13 (MCS) S 13 (PC) Sample size N B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

48 Total Sobol indices Polynomial chaos expansions Ishigami function Morris function Marelli function S T i S T 1 (MCS) S T 1 (PC) S T 2 (MCS) S T 2 (PC) S T 3 (MCS) S T 3 (PC) Sample size N B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

49 Ishigami function Morris function Marelli function Total Sobol indices small design and replication S T 1 (MCS) S T 1 (PC) S T 2 (MCS) S T 2 (PC) S T i 0.4 S T i Sample size N Sample size N S T 3 (MCS) S T 3 (PC) S T i Sample size N B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

50 Morris function Definition Polynomial chaos expansions Ishigami function Morris function Marelli function M = 20 variables Y = β β iw i + 20 β ijw iw j + 20 β ijl w iw jw l + 20 i=1 i<j i<j<l i<j<l<s β ijls w iw jw l w s where: { 2 (1.1Xi/(X i + 0.1) 0.5) if i = 3, 5, 7 w i = 2(X i 0.5) otherwise and: X i U(0, 1) β i = 20 for i = 1,..., 10 and β i = ( 1) i otherwise β ij = 15 for i = 1,..., 6 and β ij = ( 1) i+j otherwise β ijl = 10 for i = 1,..., 5 and β ijl = 0 otherwise β ijls = 5 for i = 1,..., 4 and β ijls = 0 otherwise B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

51 Ishigami function Morris function Marelli function Morris function Sensitivity results Blatman & Sudret, RESS (2010) Reference : N = 440, 000 Monte Carlo simulations + bootstrap Adaptive sparse PC: Q 2 tgt = 0.9 (resp. 0.99) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

52 Morris function Sparsity of the model Polynomial chaos expansions Ishigami function Morris function Marelli function Full chaos: card A 20,11 = ( ) = 84, 672, 315 Hyperbolic truncature: card A 20, = 4, 234 LAR: 339 terms (IS2 = 8%) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

53 High dimensional example Ishigami function Morris function Marelli function Definition Y =3 5 M M kx k + 1 M k=1 ( M kxk 3 + ln 1 3M k=1 M = 86 variables M ( ) ) k Xk 2 + X k 4 k=1 + X 1 X 2 2 X 5X 3 + X 2 X 4 + X 50 + X 50 X 2 54 X U([1, 2]) M, with X 20 U([1, 3]) Strongly non-linear Continuous and smooth function Well-located peaks in the sensitivity indices (X 2, X 20, X 50, X 54) Interactions: (X 1, X 2), (X 50, X 54) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

54 Reference results Polynomial chaos expansions Ishigami function Morris function Marelli function Model response Total Sobol indices Second Order Sobol indices B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

55 Reference results Polynomial chaos expansions Ishigami function Morris function Marelli function Model response Total Sobol indices Second Order Sobol indices B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

56 Reference results Polynomial chaos expansions Ishigami function Morris function Marelli function Model response Total Sobol indices Second Order Sobol indices B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

57 Ishigami function Morris function Marelli function PC-based vs. MCS-based Sobol indices Total Sobol indices PCE-based (1200 model runs) MCS-based (1,760,000 model runs) B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

58 Conclusions Polynomial chaos expansions Ishigami function Morris function Marelli function Polynomial chaos expansions are a powerful technique for building surrogate models used in uncertainty quantification They can be post-processed for moment- and sensitivity analysis straightforwardly. Sobol indices are obtained as a sum of squares of expansions coefficients Sparse expansions can be obtained at an affordable cost for high-dimensional problems and used for screening purpose Extensions for sensitivity problems with correlated inputs and to derivative-based sensitivity measures have been proposed Thank you very much for your attention! B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49

59 Conclusions Polynomial chaos expansions Ishigami function Morris function Marelli function Polynomial chaos expansions are a powerful technique for building surrogate models used in uncertainty quantification They can be post-processed for moment- and sensitivity analysis straightforwardly. Sobol indices are obtained as a sum of squares of expansions coefficients Sparse expansions can be obtained at an affordable cost for high-dimensional problems and used for screening purpose Extensions for sensitivity problems with correlated inputs and to derivative-based sensitivity measures have been proposed Thank you very much for your attention! UQLab The Uncertainty Quantification Toolbox in Matlab B. Sudret (Chair of Risk & Safety) PC expansions Ecole ASPEN May 9th, / 49