PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS

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1 PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS M. Navarro, O.P. Le Maître,2, O.M. Knio,3 CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, KSA. 2 ion Logo Center Lock-up for Uncertainty LIMSI-CNRS, UPR-325, Orsay, France. Logo Lock-up 3 Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA. SAMO 26 VIII International Conference on Sensitivity Analysis of Model Output, 3 Nov 3 Dec 26, Le Tampon, Réunion, France.

2 OUTLINE NON-INTRUSIVE PSP FOR SDES WITH PARAMETRIC UNCERTAINTY NUMERICAL EXAMPLES CONCLUSIONS ion Logo Center Lock-up for Uncertainty Logo Lock-up

3 NON-INTRUSIVE PSP FOR SDES WITH PARAMETRIC UNCERTAINTY NUMERICAL EXAMPLES CONCLUSIONS ion Logo Center Lock-up for Uncertainty Logo Lock-up

4 MODEL DEFINITION Let (Ω, Σ, P) be a probability space in which we consider the following SDE: dx(t, ω) = C(X(t, ω))dt + D(X(t, ω))dw (t, ω) X(t =, ω) = X () () where: X : (t, ω) T Ω R is a stochastic process defined in the time interval T = [, Tf ] with T f > W (t, ω) is a Wiener process C : R R is the drift coefficient and D : R R is the diffusion coefficient Note that there is only one source of randomness in SDE (), the Wiener process. We consider uncertain drift and diffusion coefficients through the introduction of random parameters ξ(ω) ion Logo Center Lock-up for dx(t, Uncertainty ω) = C(X(t, ω), ξ(ω))dt + D(X(t, ω), ξ(ω))dw Logo (t, Lock-up ω) X(t =, ξ(ω)) = X () (ξ(ω)) (2) Note that there are two sources of randomness in SDE (2), the Wiener process and the uncertain parameters.

5 VARIANCE DECOMPOSITION The variance-based GSA of X relies on its Sobol-Hoeffding (SH) decomposition: X = X + X par(ξ) + X noise (W ) + X mix (ξ, W ), (3) where: X = E {X} Xpar(ξ) = E {X ξ} X Xnoise (W ) = E {X W } X Xmix (ξ, W ) = X X E {X ξ} E {X W } Since the SH functions on the rhs of (3) are orthogonal, V {X} can be decomposed as: V {X} = V par + V noise + V mix, where:. V par is the variance due to the parametric uncertainty 2. ion Logo Center VLock-up noise is the variance for Uncertainty due to the Wiener noise Logo Lock-up 3. V mix is the variance due their interactions Finally the sensitivity indices (SI) are computed: S par = Vpar V {X}, S noise = V noise V {X}, S mix = V mix V {X}.

6 POLYNOMIAL CHAOS EXPANSION (PCE) Assumptions:. The Wiener noise and the uncertain parameters are considered as independent random variables 2. The solution of SDE (2), X(t, W, ξ), is a second order random variable for almost any trajectory of W (t) X(t, W, ξ) admits a truncated PC expansion of the form []: X(t, W, ξ) X(t, W, ξ) = α A [X α](t, W )Ψ α(ξ) where: ξ(ω) ξ = {ξ,, ξ N } where ξ i is a real-valued independent random variable (rv) with pdf p i (ξ i ) α = (α,..., α N ) is a multi-index Ψα(ξ) are multivariate orthonormal polynomials defined through, Ψ. α(ξ) = N i= ψα i (ξ i i ), where ion Logo Center Lock-up for Uncertainty Logo Lock-up {ψ i α, α N } is a complete orthonormal set with respect to p i (ξ i ) The random processes [Xα] are the modes of the expansion.

7 NON-INTRUSIVE PSEUDO SPECTRAL PROJECTION (PSP) To compute the modes, [X α](t, W ) for α A, we rely on non-intrusive PSP [X α](t, W ) = obtained through: ion Logo Center Lock-up for Uncertainty Logo Lock-up N Q j= P PSP αj X(t, W, ξ j ), where: P PSP is a non-intrusive operator that uses sparse tensorization of one-dimensional projection operators at different levels NQ is the number of sparse grids points ξj are the quadrature points In particular, one SDE at each N Q is solved for a particular trajectory W (i) = W (ω i ). The expansion coefficients of the corresponding trajectory of X(t) are finally [X α] (i) (t) = N Q j= P PSP αj X (i) (t, ξ j )

8 PC-BASED EXPRESSIONS FOR SENSITIVITY INDICES PC expansion of X: X(t, W, ξ) X(t, W, ξ) = α A [X α](t, W )Ψ α(ξ) The approximated expressions of the statistic of the SDE solution can be obtained using the PC coefficients: X α A E {[Xα](W )} E {Ψα} = E {[X ](W )} V {X} = E { X 2} X 2 α A E {[X α] 2} E {[X ]} 2 S par = Vpar V{X} and Vpar = α A \ E {[Xα]}2 S noise = V noise V{X} and V noise = V {[X ]} ion Logo Center Lock-up for Uncertainty Logo Lock-up S mix = V mix V{X} and V mix = α A \ V {[Xα]} For a detailed proof see [2]

9 NON-INTRUSIVE PSP FOR SDES WITH PARAMETRIC UNCERTAINTY NUMERICAL EXAMPLES CONCLUSIONS ion Logo Center Lock-up for Uncertainty Logo Lock-up

10 TEST PROBLEM Let us consider the process governed by the following SDE dx(w, ξ) = (ξ X(W, ξ))dt + (νx(w, ξ) + )ξ 2 dw X(t = ) = almost surely (4) where: ξ and ξ 2 are independent and uniformly distributed random variables ξ U ([.95,.5]) and ξ 2 U ([.2,.22]) ν =.2 so the problem has multiplicative noise Note that for ν =, X is the Ornstein-Uhlenbeck (OU) process ion Logo Center Lock-up for Uncertainty Logo Lock-up

11 TEST PROBLEM X(t,W,ξ) X(t,W,ξ) t (A) Sampling W t (B) Sampling ξ ion Logo Center Lock-up for Uncertainty Logo Lock-up FIGURE: Samples trajectories of X computed using the PC expansion. (A) Trajectories for samples of W at a fixed value of the parameters ξ =.5 and ξ 2 =.2. (B) Trajectories for samples of ξ and a fixed realization of W. ξ U [.95,.5] and ξ 2 U [.2,.22].

12 CASE OF SMOOTH QOI Assuming that we are not interested in X(t, W, ξ), but in some derived quantity such as the integral of X over t [, ]: A(W, ξ) =. X(t, W, ξ)dt..4 X(t,W (i),ξ).6 A(W (i),ξ) ion Logo Center 2 Lock-up for 4Uncertainty 6 8 Logo.5 Lock-up t ξ (A) Trajectories of X(W (i), ξ) (B) Response surface of A(W (i), ξ) FIGURE: (A) Different trajectories of X, for a fixed noise W (i) and different realizations of ξ. (b) A (i) (ξ) versus ξ and ξ 2 for fixed W (i).

13 CONVERGENCE OF THE DIRECT NI PROJECTION A Â A Â A Â ξ.5.5 ξ.5.5 ξ.5 (A) Approximation error in Â. L2-error -5 - L2-error -5 - ion Logo Center Lock-up for Uncertainty Logo Lock-up Level log(nq) (B) L 2 -norm of approximation error. FIGURE: (A) A Â versus ξ for fixed W (i). Plotted are 2D surfaces for l =, 2 and 3 arranged from left to right. (B) Normalized L 2 error versus l (left) and log(n Q ) (right).

14 CONVERGENCE OF THE SENSITIVITY INDICES Spar Snoise.5 Smix PSP l= PSP l=2 MC PSP l= PSP l=2 MC NW NW NW FIGURE: S par, S noise and S mix versus N W for different PSP levels. Also shown for comparison are the estimators for a direct MC approach (without PSP approximation, labeled MC). The solid lines correspond to the settings ion Logo Center of the Test problem, Lock-up for Uncertainty whereas dashed lines correspond to results obtained using Logo PSP with Lock-up a reduced diffusion coefficient ξ 2 U [.2,.3].

15 ERROR OF THE SENSITIVITY INDICES SE(Spar) -2-3 SE(Snoise) -2-3 SE(Smix) PSP l= -6 MC PSP l= MC NW NW NW (A) SE (bootstrapping) versus N W SE(Spar) -2-3 SE(Snoise) -2-3 SE(Smix) -4-4 MC ion Logo Center Lock-up -5 for costuncertainty Logo Lock-up cost cost reduced diffusion coefficient ξ 2 U [.2,.3] PSP l= MC PSP l= (B) SE versus computational cost which is estimated as N Q N W. FIGURE: Standard Errors in the sensitivity indices of A, using the direct PSP with N Q = 5 and standard MC methods. The solid lines correspond to the settings of Test problem, whereas the dashed lines correspond to the

16 CASE OF NON-SMOOTH QOI We continue to consider the solution, X, of the SDE in (4), but focus on the variance decomposition of the exit time, T, corresponding to the exit boundary X = c: T (W, ξ) = min{x(t, W, ξ) > c} we set c =. t>.8 6 X(t,W (i),ξ) ion Logo Center Lock-up 5 for Uncertainty t ξ Logo Lock-up (A) Trajectories of X(W (i), ξ) T(W, ξ) (B) Response surface of T (W (i), ξ) FIGURE: (A) Different trajectories of X, for a fixed noise W (i) and different realizations of ξ. (B) T (i) (ξ) versus ξ and ξ 2 for fixed W (i).

17 DIRECT NI PROJECTION OF NON-SMOOTH QOI Exact (l = ) l = l = T(W,ξ) 4 ˆT(W,ξ) 2 ˆT(W,ξ) ξ.5.5 ξ.5.5 ξ.5 l = 3 l = 4 l = ˆT(W,ξ) 3 ˆT(W,ξ) 4 ˆT(W,ξ) ion Logo Center ξ Lock-up for Uncertainty ξ Logo ξ Lock-up FIGURE: Direct PSP approximation of T (W (i), ξ) for different levels l as indicated. Also shown as circles are the sparse grid points used in the PSP constructions.

18 INDIRECT NON-INTRUSIVE PSP PROJECTION KEY IDEA: the exact exit time T (or its direct non-intrusive approximation) is substituted with the surrogate T = T ( ˆX), indirectly constructed through T (W, ξ) T (W, ξ) = min t> { ˆX(t, W, ξ) > c = }, where: ˆX(t, W, ξ) = α A [Xα](t, W )Ψα(ξ) ion Logo Center Lock-up for Uncertainty Logo Lock-up

19 CONVERGENCE OF THE INDIRECT NON-INTRUSIVE PSP PROJECTION l = l = 2 l = T(W,ξ) 4 T(W,ξ) 4 T(W,ξ) ξ.5.5 ξ.5.5 ξ.5 l = l = 2 l = T T.5.5 T T ion Logo Center Lock-up for Uncertainty Logo.5 Lock-up.5 ξ ξ ξ T T FIGURE: Indirect approximation T of the exit time (top row) and absolute indirect approximation error T T (bottom row). The plots correspond to a fixed trajectory W of the noise and different PSP levels l as indicated.

20 CONVERGENCE OF THE INDIRECT NON-INTRUSIVE PSP PROJECTION L2-error -5 - T ˆT L2-error -5 - T ˆT Level -5 2 log(nq) FIGURE: L 2 error norms of the direct and indirect PSP approximations of T. ion Logo Center Lock-up for Uncertainty Logo Lock-up

21 CONVERGENCE OF THE SENSITIVITY INDICES High noise Spar NW Snoise NW Smix PSP l= PSP l=2.28 PSP l=3 MC NW Low noise PSP l= MC Spar ion Logo Center.46 Lock-up for Uncertainty.5.4 Logo Lock-up Snoise NW NW NW Smix FIGURE: Sensitivity indices S par, S noise and S mix versus the number, N W, of noise samples for different PSP levels. Also shown for comparison are the standard MC estimators (labeled MC). High noise: ξ U [.95,.5], ξ 2 U [.2,.22] and ν =.2. Low noise: ξ U [.95,.5], ξ 2 U [.2,.3] and ν =.

22 NON-INTRUSIVE PSP FOR SDES WITH PARAMETRIC UNCERTAINTY NUMERICAL EXAMPLES CONCLUSIONS ion Logo Center Lock-up for Uncertainty Logo Lock-up

23 CONCLUSIONS Very efficient for large S par or smooth QoIs with respect to ξ It is trivially implemented in parallel (see [4]) It can be generalized to any sensitivity indices Extension to complex problems and acceleration to MLMC Extension to stochastic simulators (see [3]) ion Logo Center Lock-up for Uncertainty Logo Lock-up

24 BIBLIOGRAPHY R. H. Cameron and W. T. Martin. The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Annals of Mathematics, 48: , 947. O. P. Le Maître and O. M. Knio. Pc analysis of stochastic differential equations driven by wiener noise. Reliability Engineering & System Safety, 35:7 24, 25. M. Navarro, O. P. Le Maître, and O. M. Knio. Global sensitivity analysis in stochastic simulators of uncertain reaction networks. The Journal of Chemical Physics (accepted), 26. M. Navarro, O. P. Le Maître, and O. M. Knio. Non-intrusive polynomial chaos expansions for sensitivity analysis in stochastic differential equations. SIAM/ASA Journal on Uncertainty (JUQ)-(Feb 26 First submission, Minor revision August 26 submitted revised version), 26. ion Logo Center Lock-up for Uncertainty Logo Lock-up