Global Sensitivity Analysis in Stochastic Systems Olivier Le Matre, - PowerPoint PPT Presentation

Variance-Based Sensitivity Analysis for SODE’s Stochastic Simulators Conclusions Global Sensitivity Analysis in Stochastic Systems Olivier Le Maître, Omar Knio LIMSI CNRS, Orsay, France KAUST, Saudi-Arabia Duke University, Durham, North Carolina CEMRACS - CIRM O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Stochastic Simulators Conclusions Stochastic models Physical systems with Complex small scale dynamics (MD, chemical systems, . . .) Random forcing and source terms (finance, wind-load, . . .) Unresolved scales (turbulence, climate modeling, . . .) are often tackled by means of stochastic modeling where complex / unknown / unresolved phenomenons are accounted for by the introduction of noisy dynamics. In addition to the effect of the noise, the model may involve unknown parameters : e.g. noise level, physical constants and parameters, initial conditions, . . . Our general objective is to propagate / assess the impact of parameters uncertainty within such stochastic models while characterizing the effect of inherent noise : global sensitivity analysis & analysis of the variance O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Stochastic Simulators Conclusions 1 Variance-Based Sensitivity Analysis for SODE’s Variance decomposition PC-Galerkin approximation Examples Stochastic Simulators 2 stochastic simulators Variance Decomposition Examples 3 Conclusions O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Stochastic ODEs We consider a simple systems driven by random noise (Ito equation) : for t ∈ [ 0 , T ] . = T dX ( t ) = C ( X ( t )) dt + D ( X ( t )) dW ( t ) , X ( t = 0 ) = X 0 , where X ( t ) ∈ R is the solution, W ( t ) is the Wiener process, C ( · ) is the drift function, and D ( · ) is the diffusion coefficient. The solution can be computed through MC simulation, solving (e.g.) X i + 1 = X i + C ( X i )∆ t + D ( X i )∆ W i , X i ≈ X ( i ∆ t ) , drawing iid random variables ∆ W i ∼ N ( 0 , ∆ t ) . Sample estimate expectation, moments, quantiles, probability law, . . ., of the stochastic process X ( t ) : M � E { g ( X ( i ∆ t )) } ≈ 1 g ( X l i ) . M l = 1 O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Stochastic ODEs with parametric uncertainty The drift function and diffusion coefficient can involve some uncertain parameters Q : dX ( t ) = C ( X ( t ); Q ) dt + D ( X ( t ); Q ) dW ( t ) , X ( t = 0 ) = X 0 . We consider that : Q random with known probability law, Q and W are assumed independent. The solution can be seen as a functional of W ( t ) and Q : X ( t ) = X ( t , W , Q ) . We shall assume, ∀ t ∈ T , � X 2 � E < ∞ , 1 � E { X | W } 2 � . � � X 2 E = E < ∞ , 2 | W � E { X | Q } 2 � . � � X 2 E = E < ∞ . 3 | Q We want to investigate the respective impact of Q , W on X . O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Classical sensitivity analysis Focusing on the two first moments, global SA for the random parameters Q is based on : 1 approximating the mean and variance of X | Q � � � � = Σ 2 E X | Q = µ X ( Q ) , V X | Q X ( Q ) , perform a GSA of µ X ( Q ) and Σ 2 2 X ( Q ) with respect to the input parameters in Q . In particular, for independent parameters Q , Polynomial Chaos approximations : � � Σ 2 Σ 2 µ X ( Q ) ≈ µ α Ψ α ( Q ) , X ( Q ) ≈ α Ψ α ( Q ) . α α PC expansion coefficients can be computed / estimated by means of Non-Intrusive Spectral Projection, Bayesian identification, . . .. This approach characterizes the dependence of the first moments with respect to the parameters Q . O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Another approach of GSA Here, we exploit the structure of the model to take an alternative approach, inspired from the hierarchical orthogonal Sobol-Hoeffding decomposition of X : ∀ t ∈ T , X ( W , Q ) = X + X W ( W ) + X Q ( Q ) + X W , Q ( W , Q ) , where the functionals in the SH decomposition are mutually orthogonal. In fact, the decomposition is unique and given by X ( t ) . = E { X ( t ) } , X W ( t , W ) . = E { X ( t ) | W } − E { X ( t ) } = X | W ( t ) − X ( t ) , X Q ( t , Q ) . = E { X ( t ) | Q } − E { X ( t ) } = X | Q ( t ) − X ( t ) . Owing to the orthogonality of the SH decomposition, we have � � V { X } = V { X W } + V { X Q } + V X W , Q , from which follow the definitions of the sensitivity indices � � S W = V { X W } S Q = V { X Q } S W , Q = V X W , Q V { X } , V { X } , . V { X } O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Sensitivity indices The sensitivity indices � � S W , Q = V X W , Q S W = V { X W } S Q = V { X Q } V { X } , V { X } , , V { X } then measure the fraction of the variance due to the Wiener noise only, or intrinsic randomness ( S W ), the parameters only, or parametric randomness ( S Q ), the combined effect of intrinsic and parametric randomness ( S W , Q ). In particular, S W measure the part of the variance that cannot be reduced through a better knowledge of the parameters. In addition, � � Σ 2 ( Q ) V Q { µ X ( Q ) } E Q = S Q , but = S W + S W , Q . V { X } V { X } From Σ 2 ( Q ) , one cannot distinguish the intrinsic and mixed randomness effects. O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Polynomial Chaos expansion We express the dependence of X on Q as a PC expansion � X ( t , W , Q ) = X α ( t , W )Ψ α ( Q ) , α where { Ψ α } is a CONS of L 2 ( Q , p Q ) , the expansion coefficients X α are random processes. The random processes X α ( t ) are the solutions of the coupled system of SODEs � �� � � � �� � � dX β ( t ) = F X α ( t )Ψ α ; Q , Ψ β dt + G X α ( t )Ψ α ; Q , Ψ β dW , α α where �· , ·� denotes the inner product in L 2 ( Q , p Q ) . This system can be solved by MC simulation (upon truncation). O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples PC expansion Assuming Ψ 0 = 1, it comes � E { X } = E { X 0 } , X Q ( Q ) = E { X α } Ψ α ( Q ) , X W ( W ) = X 0 ( W ) − E { X 0 } , α � = 0 and � X W , Q ( W , Q ) = ( X α ( W ) − E { X α } ) Ψ α ( Q ) . α � = 0 Finally, the partial variances have for expression : � � � � E { X α } 2 , V { X Q } = V { X W } = V { X 0 } , V X W , Q = V { X α } . α � = 0 α � = 0 Observe : X Q ( Q ) + E { X } = µ X ( Q ) , 1 � � � � � α V { X α } = � − E { X α } 2 = E Q X 2 Σ 2 α E 2 . α X O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Linear additive system Consider SODE with drift and diffusion terms given by : C ( X , Q ) = Q 1 − X D ( X , Q ) = ( ν X + 1 ) Q 2 where Q 1 and Q 2 are independent, uniformly-distributed, random variables with mean µ 1 , 2 and standard deviation σ 1 , 2 . The orthonormal PC basis consists of tensorized Legendre polynomials. We use for initial condition X ( t = 0 ) = 0 almost surely. O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Additive Noise Additive noise model ( ν = 0) with µ 1 = 1, µ 2 = 0 . 1, σ 1 = σ 2 = 0 . 05 : dX ( Q ) = ( Q 1 − X ( Q )) dt + Q 2 dW , a first-order expansion suffices to exactly represent X ( Q ) . +1.5e+00 +7.5e-02 +2.5e-01 [X] k (t) [X] k (t) [X] k (t) +0.0e+00 Mean + 3 σ Mean + 3 σ Mean + 3 σ k=0 k=1 k=2 +0.0e+00 +0.0e+00 -2.5e-01 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 t t t Selected trajectories and variability ranges for [ X k ]( t , W ) . The plots correspond to k = 0 , 1 and 2, arranged from left to right. O. Le Maître Variance-based SA

Variance-Based Sensitivity Analysis for SODE’s Variance decomposition Stochastic Simulators PC-Galerkin approximation Conclusions Examples Multiplicative Noise – I Multiplicative noise : Q 1 ∼ U [ 1 , 0 . 05 ] , Q 2 ∼ U [ 0 . 1 , 0 . 05 ] , ν = 0 . 2 +1.5e+00 [X] k (t) Mean + 3 σ k=0 +0.0e+00 0 2 4 6 8 10 t +7.5e-02 +2.5e-01 [X] k (t) [X] k (t) +0.0e+00 Mean + 3 σ Mean + 3 σ k=1 k=2 +0.0e+00 -2.5e-01 0 2 4 6 8 10 0 2 4 6 8 10 t t Sample trajectories of [ X k ] , 0 ≤ k ≤ 2. Top row : order 0, bottom row : order 1 with and decreasing order in Q 1 from left to right. O. Le Maître Variance-based SA