Surrogate models and reduction methods for UQ and inference in - PowerPoint PPT Presentation

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Surrogate models and reduction methods for UQ and inference in large-scale models Olivier Le Maître Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur Joint work with Loïc Giraldi, Omar M. Knio and Ibrahim Hoteit (KAUST) Guoto Li (Duke) WTDMF 2017 O. Le Maître (LIMSI) WTDMF 2017 1 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Table of contents 1. Motivation 2. Global Sensitivity Analysis via Surrogates 3. Bayesian Inference 4. Conclusions and outlooks O. Le Maître (LIMSI) WTDMF 2017 2 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Many physical systems are characterized by: the presence of broad range of scales (turbulence, porous flow, . . . ) subjected to highly varying / unknown forcing and source terms (finance, wind-load, gust, waves, earthquake, . . . ) details at all scales are often not relevant for the prediction (turbulence flows, climate modeling, . . . ) Tempering the computational complexity calls for physical models restricted at the scales of interest (upscaling, homogenization, sub-scale modeling, . . . ) modeling the -known- unknown (probabilistic approaches) These procedures introduce parameters: they usually must be calibrated (inferred, learned, identified,. . . ) from experimental observations typically parameters can not be determined exactly (limited observations, because of imperfect models and setups or becauses processes are inherently uncertain Our final goal is to infer model parameters in the most objective manner and propagate them to assess the predictive uncertainty. global sensitivity analysis & Bayesian inference O. Le Maître (LIMSI) WTDMF 2017 3 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Table of contents 1. Motivation 2. Global Sensitivity Analysis via Surrogates 3. Bayesian Inference 4. Conclusions and outlooks O. Le Maître (LIMSI) WTDMF 2017 4 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Input Uncertainty Sources of data uncertainty Inherent variability ( e.g. industrial processes). Epistemic uncertainty ( e.g. model constants). May not be fully reducible, even theoretically . O. Le Maître (LIMSI) WTDMF 2017 5 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Input Uncertainty Sources of data uncertainty Inherent variability ( e.g. industrial processes). Epistemic uncertainty ( e.g. model constants). May not be fully reducible, even theoretically . Probabilistic framework Define an abstract probability space (Ω , A , d µ ). Consider model parameter Q as random quantity: Q ( ω ) , ω ∈ Ω. Simulation output S is random and on (Ω , A , d µ ). O. Le Maître (LIMSI) WTDMF 2017 5 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Input Uncertainty Sources of data uncertainty Inherent variability ( e.g. industrial processes). Epistemic uncertainty ( e.g. model constants). May not be fully reducible, even theoretically . Probabilistic framework Define an abstract probability space (Ω , A , d µ ). Consider model parameter Q as random quantity: Q ( ω ) , ω ∈ Ω. Simulation output S is random and on (Ω , A , d µ ). Parameter Q and simulation output S are dependent random quantities (through the mathematical model M ): M ( S ( ω ) , Q ( ω )) = 0 , ∀ ω ∈ Ω . O. Le Maître (LIMSI) WTDMF 2017 5 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Uncertainty Propagation Propagation of data uncertainty Parameter density Solution density M ( S , Q ) = 0 O. Le Maître (LIMSI) WTDMF 2017 6 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification Uncertainty Propagation Propagation of data uncertainty Parameter density Solution density M ( S , Q ) = 0 Variability in model output: numerical error bars. Assessment of predictability . Support decision making process . What type of information (abstract quantities, confidence intervals, density estimations, structure of dependencies, . . . ) one needs? O. Le Maître (LIMSI) WTDMF 2017 6 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification UQ Methods Deterministic methods Sensitivity analysis (adjoint based, AD, . . . ): local. Perturbation techniques : limited to low order and simple data uncertainty. Neumann expansions : limited to low expansion order. Moments method: closure problem (non-Gaussian / non-linear problems). Simulation techniques Monte-Carlo Spectral Methods O. Le Maître (LIMSI) WTDMF 2017 7 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification UQ Methods Deterministic methods Simulation techniques Monte-Carlo Generate a sample set of parameter realizations and compute the corresponding sample set of model ouput . Use sample set based random estimates of abstract characterizations (moments, correlations, . . . ). Plus: Very robust and re-use deterministic codes: (parallelization, complex data uncertainty). Minus: slow convergence of the random estimates with the sample set dimension. Spectral Methods O. Le Maître (LIMSI) WTDMF 2017 7 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Uncertainty Quantification UQ Methods Deterministic methods Simulation techniques Monte-Carlo Spectral Methods Parameterization with random variables (RVs). ⊥ projection of solution on the ( L 2 ) space spanned by the RVs. Plus: arbitrary level of uncertainty, deterministic approach, convergence rate, information contained. Minus: parameterizations (limited # of RVs), adaptation of simulation tools (legacy codes), robustness (non-linear problems, non-smooth output, . . . ). Not suited for model uncertainty O. Le Maître (LIMSI) WTDMF 2017 7 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Spectral methods Spectral methods Input parametrization Parametrization of Q using N < ∞ independent RVs with prescribed distribution p ( ξ ξ ξ ): Q ( ω ) ≈ Q ( ξ ξ ξ ( ω )) , ξ ξ ξ = ( ξ 1 , . . . , ξ N ) ∈ Ξ . Iso-probabilistic transformations of RVs, Karhunen-Loève expansion : Q ( x x x , ω ) stochastic field/process, Indentification ( e.g. Bayesian). Model Solution expansion O. Le Maître (LIMSI) WTDMF 2017 8 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Spectral methods Spectral methods Input parametrization Model We assume that for a.e. ξ ξ ∈ Ξ, the problem M ( S , Q ( ξ ξ ξ ξ )) = 0 is well-posed, 1 has a unique solution 2 and that ξ ) ∈ L 2 (Ξ , p ξ ) the random solution S ( ξ ξ : � � E � S 2 � S 2 ( ξ S 2 ( ξ ξ ξ ξ ξ ξ < + ∞ . = ξ ( ω )) d µ ( ω ) = ξ ) p ( ξ ξ ) d ξ Ω Ξ Solution expansion O. Le Maître (LIMSI) WTDMF 2017 8 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Spectral methods Spectral methods Input parametrization Model Solution expansion Let { Ψ 0 , Ψ 1 , . . . } be a basis of L 2 (Ξ , p ξ ) then � S ( ξ ξ ξ ) = s k Ψ k ( ξ ξ ) . ξ k Knowledge of the spectral coefficients s k fully determine the random solution. Makes explicit the dependence between Q ( ξ ξ ξ ) and S ( ξ ξ ξ ). Need to select the basis and efficient procedure(s) to compute the s k . O. Le Maître (LIMSI) WTDMF 2017 8 / 51

Motivation Global Sensitivity Analysis via Surrogates Bayesian Inference Conclusions and outlooks Spectral methods Polynomial Chaos expansions [Wiener, 1938] Any well behaved RV U ( ω ) ( e.g. 2nd-order one) defined on (Ω , A , d µ ) has a convergent expansion of the form: ∞ ∞ i 1 � � � U ( ω ) = u 0 Γ 0 + u i 1 Γ 1 ( ξ i 1 ( ω )) + u i 1 , i 2 Γ 2 ( ξ i 1 ( ω ) , ξ i 2 ( ω )) i 1 =1 i 1 =1 i 2 =1 ∞ i 1 i 2 � � � + u i 1 , i 2 , i 3 Γ 3 ( ξ i 1 ( ω ) , ξ i 2 ( ω ) , ξ i 3 ( ω )) + . . . i 1 =1 i 2 =1 i 3 =1 { ξ 1 , ξ 2 , . . . } : independent normalized Gaussian RVs. Γ p polynomials with degree p , orthogonal to Γ q , ∀ q < p . Convergence in the mean square sense [Cameron & Martin, 1947] . O. Le Maître (LIMSI) WTDMF 2017 9 / 51