Challenges in Uncertainty Quantification in Computational Models

Transcription

1 Introduction UQ Basics Challenges Closure Challenges in Uncertaint Quantification in Computational Models Habib N. Najm Sandia National Laboratories, Livermore, CA Computational and Theoretical Challenges in Interdisciplinar Predictive Modeling Over Random Fields 12th Annual Red Raider Mini-Smposium Department of Mathematics and Statistics Teas Tech Universit, Lubbock, TX October 26, 2012 SNL Najm Challenges in UQ 1 / 52

2 Introduction UQ Basics Challenges Closure Acknowledgement B.J. Debusschere, R.D. Berr, K. Sargsan, C. Safta, K. Chowdhar Sandia National Laboratories, CA R.G. Ghanem U. South. California, Los Angeles, CA O.M. Knio Duke Univ., Durham, NC O.P. Le Maître CNRS, Paris, France Y.M. Marzouk Mass. Inst. of Tech., Cambridge, MA This work was supported b: DOE Office of Basic Energ Sciences, Div. of Chem. Sci., Geosci., & Biosci. DOE Office of Advanced Scientific Computing Research (ASCR), Scientific Discover through Advanced Computing (SciDAC) DOE ASCR Applied Mathematics program American Recover and Reinvesment Act. DOE Office of Biological and Environmental Research Sandia National Laboratories is a multiprogram laborator operated b Sandia Corporation, a Lockheed Martin Compan, for the United States Department of Energ under contract DE-AC04-94-AL SNL Najm Challenges in UQ 2 / 52

3 Introduction UQ Basics Challenges Closure Outline 1 Introduction 2 Uncertaint Quantification Basics Forward Uncertaint Propagation Statistical Inverse Problems 3 Uncertaint Quantification Challenges a Selection Characterization of Uncertain Inputs High-dimensionalit Discontinuities Oscillator Dnamics 4 Closure SNL Najm Challenges in UQ 3 / 52

4 Introduction UQ Basics Challenges Closure The Case for Uncertaint Quantification (UQ) UQ is needed in: Assessment of confidence in computational predictions Validation and comparison of scientific/engineering models Design optimization Use of computational predictions for decision-support Assimilation of observational data and model construction Multiscale and multiphsics model coupling SNL Najm Challenges in UQ 4 / 52

5 Introduction UQ Basics Challenges Closure Overview of UQ Methods Estimation of model/parametric uncertaint Epert opinion, data collection Regression analsis, fitting, parameter estimation Baesian inference of uncertain models/parameters Forward propagation of uncertaint in models Local sensitivit analsis (SA) and error propagation Fuzz logic; Evidence theor interval math Probabilistic framework Global SA / stochastic UQ Random sampling, statistical methods Galerkin methods Polnomial Chaos (PC) intrusive/non-intrusive Collocation, interpolants, regression, fitting... PC/other SNL Najm Challenges in UQ 5 / 52

6 Introduction UQ Basics Challenges Closure Forward Inverse Probabilistic Forward UQ & Polnomial Chaos Representation of Random Variables With = f(), a random variable, estimate the RV Can describe a RV in terms of its densit, moments, characteristic function, or most fundamentall as a function on a probabilit space Constraining the analsis to RVs with finite variance, enables the representation of a RV as a spectral epansion in terms of orthogonal functions of standard RVs. Polnomial Chaos Enables the use of available functional analsis methods for forward UQ SNL Najm Challenges in UQ 6 / 52

7 Introduction UQ Basics Challenges Closure Forward Inverse Polnomial Chaos Epansion (PCE) Model uncertain quantities as random variables (RVs) Given a germ ξ(ω) = {ξ 1,,ξ n } a set of i.i.d. RVs where p(ξ) is uniquel determined b its moments An RV in L 2 (Ω,S(ξ), P) can be written as a PCE: u(, t,ω) = f(, t,ξ) P u k (, t)ψ k (ξ(ω)) k=0 u k (, t) are mode strengths Ψ k () are functions orthogonal w.r.t. p(ξ) With dimension n and order p: P+1 = (n+p)! n!p! SNL Najm Challenges in UQ 7 / 52

8 Introduction UQ Basics Challenges Closure Forward Inverse Orthogonalit B construction, the functions Ψ k () are orthogonal with respect to the densit of ξ u k (, t) = uψ k Ψ 2 k = 1 Ψ 2 k u(, t;λ(ξ))ψ k (ξ)p ξ (ξ)dξ Eamples: Hermite polnomials with Gaussian basis Legendre polnomials with Uniform basis,... Global versus Local PC methods Adaptive domain decomposition of the support of ξ SNL Najm Challenges in UQ 8 / 52

9 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV Lognormal; WH PC order = 1 Wiener-Hermite PCE constructed for a Lognormal RV PCE-sampled PDF superposed on true PDF Order = P u = u k Ψ k (ξ) k=0 = u 0 + u 1 ξ SNL Najm Challenges in UQ 9 / 52

10 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV Lognormal; WH PC order = 2 Wiener-Hermite PCE constructed for a Lognormal RV PCE-sampled PDF superposed on true PDF Order = P u = u k Ψ k (ξ) k=0 = u 0 + u 1 ξ + u 2 (ξ 2 1) SNL Najm Challenges in UQ 9 / 52

11 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV Lognormal; WH PC order = 3 Wiener-Hermite PCE constructed for a Lognormal RV PCE-sampled PDF superposed on true PDF Order = P u = u k Ψ k (ξ) k=0 = u 0 + u 1 ξ + u 2 (ξ 2 1)+u 3 (ξ 3 3ξ) SNL Najm Challenges in UQ 9 / 52

12 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV Lognormal; WH PC order = 4 Wiener-Hermite PCE constructed for a Lognormal RV PCE-sampled PDF superposed on true PDF Order = P u = u k Ψ k (ξ) k= = u 0 + u 1 ξ + u 2 (ξ 2 1)+u 3 (ξ 3 3ξ)+u 4 (ξ 4 6ξ 2 + 3) SNL Najm Challenges in UQ 9 / 52

13 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV Lognormal; WH PC order = 5 Wiener-Hermite PCE constructed for a Lognormal RV PCE-sampled PDF superposed on true PDF Order = P u = u k Ψ k (ξ) k= = u 0 + u 1 ξ + u 2 (ξ 2 1)+u 3 (ξ 3 3ξ)+u 4 (ξ 4 6ξ 2 + 3) + u 5 (ξ 5 10ξ ξ) SNL Najm Challenges in UQ 9 / 52

14 Introduction UQ Basics Challenges Closure Forward Inverse PC Illustration: WH PCE for a Lognormal RV 1.4 Lognormal RV WH PC order WH PC order Lognormal RV PC mode amplitudes u 0 u PC function u(ξ) Order 1 5 Fifth-order Wiener-Hermite PCE represents the given Lognormal well Higher order terms are negligible SNL Najm Challenges in UQ 10 / 52

15 Introduction UQ Basics Challenges Closure Forward Inverse Random Fields KLE Smooth random fields can be repesented with a small no. of stochastic degrees of freedom Karhunen-Loeve Epansion (KLE) for a RF with a continuous covariance function M(,ω) = µ()+ λi η i (ω)φ i () µ() is the mean of M(,ω) at λ i and φ i () are the eigenvalues and eigenfunctions of the covariance C( 1, 2 ) = [M( 1,ω) µ( 1 )][M( 2,ω) µ( 2 )] The η i are uncorrelated zero-mean unit-variance RVs i=1 KLE representation of random fields using PC SNL Najm Challenges in UQ 11 / 52

16 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: KL of 2D Gaussian Process δ = 0.1 δ = δ = 0.5 2D Gaussian Process with covariance: Cov( 1, 2 ) = ep( /δ 2 ) Realizations smoother as covariance length δ increases SNL Najm Challenges in UQ 12 / 52

17 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - Modes for δ = 0.1 λ1 φ 1 λ2 φ 2 λ3 φ 3 λ4 φ 4 λ5 φ 5 λ6 φ 6 λ7 φ 7 λ8 φ 8 SNL Najm Challenges in UQ 13 / 52

18 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - Modes for δ = λ1 φ 1 λ2 φ 2 λ3 φ 3 λ4 φ 4 λ5 φ 5 λ6 φ 6 λ7 φ 7 λ8 φ 8 SNL Najm Challenges in UQ 14 / 52

19 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - Modes for δ = 0.5 λ1 φ 1 λ2 φ 2 λ3 φ 3 λ4 φ 4 λ5 φ 5 λ6 φ 6 λ7 φ 7 λ8 φ 8 SNL Najm Challenges in UQ 15 / 52

20 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - eigenvalue spectrum δ = terms 16 terms Eigenvalue Magnitude δ =0.1 δ = δ = Eigenvalue # 32 terms 64 terms SNL Najm Challenges in UQ 16 / 52

21 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - eigenvalue spectrum δ = 4 terms 16 terms Eigenvalue Magnitude δ =0.1 δ = δ = Eigenvalue # 32 terms 64 terms SNL Najm Challenges in UQ 17 / 52

22 Introduction UQ Basics Challenges Closure Forward Inverse RF Illustration: 2D KL - eigenvalue spectrum δ = terms 16 terms Eigenvalue Magnitude δ =0.1 δ = δ = Eigenvalue # 32 terms 64 terms SNL Najm Challenges in UQ 18 / 52

23 Introduction UQ Basics Challenges Closure Forward Inverse Essential Use of PC in UQ Strateg: Represent model parameters/solution as random variables Construct PCEs for uncertain parameters Evaluate PCEs for model outputs Advantages: Computational efficienc Sensitivit information Requirement: Random variables in L 2, i.e. with finite variance SNL Najm Challenges in UQ 19 / 52

24 Introduction UQ Basics Challenges Closure Forward Inverse Intrusive PC UQ: A direct non-sampling method Given model equations: M(u(, t);λ) = 0 Epress uncertain parameters/variables using PCEs P P u = u k Ψ k ; λ = λ k Ψ k k=0 k=0 Substitute in model equations; appl Galerkin projection G(U(, t),λ) = 0 New set of equations: with U = [u 0,...,u P ] T, Λ = [λ 0,...,λ P ] T Solving this sstem once provides the full specification of uncertain model ouputs SNL Najm Challenges in UQ 20 / 52

25 Introduction UQ Basics Challenges Closure Forward Inverse Laminar 2D Channel Flow with Uncertain Viscosit Incompressible flow Viscosit PCE ν = ν 0 +ν 1 ξ Streamwise velocit P v = v i Ψ i i=0 v 0 : mean v i : i-th order mode P σ 2 = v 2 v0 v1 v2 v3 sd i Ψ 2 i v 0 v 1 v 2 v 3 σ i=1 SNL Najm Challenges in UQ 21 / 52

26 Introduction UQ Basics Challenges Closure Forward Inverse Non-intrusive Spectral Projection (NISP) PC UQ Sampling-based Relies on black-bo utilization of the computational model Evaluate projection integrals numericall For an quantit of interest φ(, t;λ) = P k=0 φ k(, t)ψ k (ξ) φ k (, t) = 1 φ(, t;λ(ξ))ψ Ψ 2 k (ξ)p ξ (ξ)dξ, k = 0,..., P k Integrals can be evaluated using A variet of (Quasi) Monte Carlo methods Slow convergence; indep. of dimensionalit Quadrature/Sparse-Quadrature methods Fast convergence; depends on dimensionalit SNL Najm Challenges in UQ 22 / 52

27 Introduction UQ Basics Challenges Closure Forward Inverse 1D H 2 -O 2 SCWO Flame NISP UQ/Chemkin-Premi σ σ 20 Rns 1,6 All modes Rns 5,8,1,6 Y OH 5 σ (OH) 10 Rns 8,1,6 All 1st order (cm) (cm) Fast growth in OH uncertaint in the primar reaction zone Constant uncertaint and mean of OH in post-flame region Uncertaint in pre-eponential of Rn.5 (H 2O 2+OH=H 2O+HO 2) has largest contribution to uncertaint in predicted OH SNL Najm Challenges in UQ 23 / 52

28 Introduction UQ Basics Challenges Closure Forward Inverse Other non-intrusive methods Response surface emploing PC or other functional basis Collocation: Fit interpolant to samples Oscillation concern Regression: Estimate best-fit response surface Least-squares Baesian inference Useful when quadrature methods are infeasible, e.g. when Can t choose sample locations; samples given a priori Can t take enough samples Forward model is nois SNL Najm Challenges in UQ 24 / 52

29 Introduction UQ Basics Challenges Closure Forward Inverse Inverse UQ Estimation of Uncertain Inputs Forward UQ requires specification of uncertain inputs Probabilistic setting Require joint PDF on input space Baesian setting PDF on uncertain inputs can be found given data Probabilistic inference an inverse problem Uncertaint in computational predictions can depend strongl on detailed structure of the parametric PDF SNL Najm Challenges in UQ 25 / 52

30 Introduction UQ Basics Challenges Closure Forward Inverse The strong role of detailed input PDF structure Simple nonlinear algebraic model (u, v) = ( 2 2, 2) Two input PDFs, p(, ) same nominals/bounds different correlation structure Drasticall different output PDFs different nominals and bounds SNL Najm Challenges in UQ 26 / 52

31 Introduction UQ Basics Challenges Closure Forward Inverse Baes formula for Parameter Inference Data Model (fit model + noise model): Baes Formula: p(λ, ) = p(λ )p() = p( λ)p(λ) p(λ ) Posterior = Likelihood p( λ) Prior: knowledge of λ prior to data p() Evidence Prior p(λ) = f(λ) g(ǫ) Likelihood: forward model and measurement noise Posterior: combines information from prior and data Evidence: normalizing constant for present contet SNL Najm Challenges in UQ 27 / 52

32 Introduction UQ Basics Challenges Closure Forward Inverse Eploring the Posterior Given an sample λ, the un-normalized posterior probabilit can be easil computed p(λ ) p( λ)p(λ) Eplore posterior w/ Markov Chain Monte Carlo (MCMC) Metropolis-Hastings algorithm: Random walk with proposal PDF & rejection rules Computationall intensive, O(10 5 ) samples Each sample: evaluation of the forward model Surrogate models Evaluate moments/marginals from the MCMC statistics SNL Najm Challenges in UQ 28 / 52

33 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc UQ Challenges - Characterization of Uncertain Inputs Computational Model M(u, λ) = 0 Uncertain input parameter λ Eperimental Measurement F(, λ) = 0 Uncertain model inputs can be estimated from data on Regression Baesian inference Quite frequentl, we have partial data/information Partial missing data, e.g. failed measurements Full data loss No data, but have summar information, e.g. moments and/or quantiles on data processed data products fitted parameters SNL Najm Challenges in UQ 29 / 52

34 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Dealing with Partial Data Imputation Baesian Multiple Imputation (Rubin, 1987) Use the posterior predictive conditioned on observed data to generate replicates of the missing data z t mis, t = 1,..., m For each full data set, (z obs, z t mis ) appl Baesian inference to get a posterior densit on the model parameters Marginalize over the missing data to get the observed data posterior p(β z obs ) p(β z obs ) = q(β z mis, z obs )f(z mis z obs )dz mis SNL Najm Challenges in UQ 30 / 52

35 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Parameter Estimation in the Absence of Data Frequentl: we know summar statistics about data or parameters from previous work the raw data used to arrive at these statistics is not available How can we construct a joint PDF on the parameters? In the absence of data, the structure of the fit model, combined with the summar statistics, implicitl inform the joint PDF on the parameters Goal: Make available information eplicit in the joint PDF Data Free Inference (DFI) Discover a consensus joint PDF on the parameters consistent with given information in the absence of data SNL Najm Challenges in UQ 31 / 52

36 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Generate ignition "data" using a detailed model+noise Ignition using a detailed chemical model for methane-air chemistr Ignition time versus Initial Temperature Multiplicative noise error model 11 data points: Ignition time (sec) GRI+noise GRI d i = tig,i GRI (1+σǫ i ) ǫ N(0, 1) Initial Temperature (K) SNL Najm Challenges in UQ 32 / 52

37 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Fitting with a simple chemical model Fit a global single-step irreversible chemical model CH 4 + 2O 2 CO 2 + 2H 2 O R = [CH 4 ][O 2 ]k f k f = A ep( E/R o T) Infer 3-D parameter vector (ln A, ln E, lnσ) Use an adaptive MCMC procedure Start at the maimum likelihood estimate SNL Najm Challenges in UQ 33 / 52

38 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Baesian Inference Posterior and Nominal Prediction Ignition time (sec) GRI+noise GRI GRI GRI+noise Fit Model Marginal joint posterior on (ln A, ln E) ehibits strong correlation Initial Temperature (K) Nominal fit model is consistent with the true model SNL Najm Challenges in UQ 34 / 52

39 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc DFI Challenge Chemical Model Problem Discarding initial data, reconstruct marginal (ln A, ln E) posterior using the following information Form of fit model Range of initial temperature Nominal fit parameter values of ln A and ln E Marginal 5% and 95% quantiles on ln A and ln E Further, for now, presume Multiplicative Gaussian errors N = 8 data points SNL Najm Challenges in UQ 35 / 52

40 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc DFI Algorithm Structure Basic idea: Eplore the space of hpothetical data sets MCMC chain on the data Each state defines a data set For each data set: MCMC chain on the parameters Evaluate statistics on resulting posterior Accept data set if posterior is consistent with given information Evaluate pooled posterior from all acceptable posteriors Logarithmic pooling: p(λ ) = [ K ] 1/K p(λ i ) i=1 SNL Najm Challenges in UQ 36 / 52

41 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc 2D Marginal Pooled Posteriors vs. Reference Posterior ln(e) ln(e) ln(e) ln(a) ln(a) ln(a) ln(e) ln(e) ln(e) ln(a) ln(a) ln(a) SNL Najm Challenges in UQ 37 / 52

42 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Empirical Convergence of 3D Pooled Posteriors Kullback-Leibler divergence between posteriors of successivel increased data volumes Combinatorial choices of chains pooled at each stage statistical scatter of KLdiv KLdiv Overall convergence evident 1/N c 2-4c 4-8c 8-16c 16-32c SNL Najm Challenges in UQ 38 / 52

43 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Challenges in PC UQ High-Dimensionalit Dimensionalit n of the PC basis: ξ = {ξ 1,...,ξ n } number of degrees of freedom P+1 = (n+p)!/n!p! grows fast with n Impacts: Size of intrusive sstem # non-intrusive (sparse) quadrature samples Generall n number of uncertain parameters Reduction of n: Sensitivit analsis Dependencies/correlations among parameters Dominant eigenmodes of random fields Manifold learning: Isomap, Diffusion maps Sparsification: Compressed Sensing, LASSO SNL Najm Challenges in UQ 39 / 52

44 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc PC Quadrature in hid Full quadrature: N = (N 1D ) n Sparse Quadrature Wide range of methods Nested & hierarchical Clenshaw-Curtis: N = O(n p ) Adaptive greed algorithms Number of points can still be ecessive in hi-d Large no. of terms Reduction/sparsit Number 1e+07 1e+06 1e D Surrogate No. of Sparse Quadrature Points No. of PC Terms PC Order SNL Najm Challenges in UQ 40 / 52

45 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc PC coefficients via sparse regression PCE: K 1 = f() c k Ψ k () k=0 with R n, Ψ k ma order p, and K = (p+n)!/p!/n! N samples ( 1, 1 ),...,( N, N ) Estimate K terms c 0,..., c K 1, s.t. min Ac 2 2 where R N, c R K, A ik = Ψ k ( i ), A R N K With N << K under-determined Need some form of regularization SNL Najm Challenges in UQ 41 / 52

46 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Regularization Compressive Sensing (CS) l 2 -norm Tikhonov regularization; Ridge regression: min { Ac c 2 2} l 1 -norm Compressive Sensing; LASSO; basis pursuit min { Ac c 1} min { Ac 2 2 } subject to c 1 ǫ min { c 1 } subject to Ac 2 2 ǫ discover of sparse signals SNL Najm Challenges in UQ 42 / 52

47 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Baesian Regression Baes formula p(c D) p(d c)π(c) Baesian regression: prior as a regularizer, e.g. Log Likelihood Ac 2 2 Log Prior c p p Laplace sparsit priors π(c k α) = 1 2α e c k /α LASSO (Tibshirani 1996)... formall: min { Ac 2 2 +λ c 1 } Solution the posterior mode of c in the Baesian model N(Ac, I N ), c k 1 2α e c k /α Baesian LASSO (Park & Casella 2008) SNL Najm Challenges in UQ 43 / 52

48 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Baesian Compressive Sensing (BCS) BCS (Ji 2008; Babacan 2010) hierarchical priors: Gaussian priors N(0,σ 2 k ) on the c k Gamma priors on the σ 2 k Laplace sparsit priors on the c k Evidence maimization establishes ML estimates of the σ k man of which are found 0 c k 0 iterativel include terms that lead to the largest increase in the evidence iterative BCS (ibcs) (Sargsan 2012): adaptive iterative order growth BCS on order-p Legendre-Uniform PC repeat with order-p+1 terms added to surviving p-th order terms SNL Najm Challenges in UQ 44 / 52

49 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc CS and BCS Oscillator Genz function f() = cos(2πr + n i=1 a i i ); a i 1/i 2 ; r = 0 Legendre-Uniform PC, 10 th -order in 5d; (5, 6) th -order in 10d Baesian CS l 1 minimization BCS 5 th order CS 5 th order BCS 6 th order L 2 error 1 L 2 error Number of measurements 5d Number of measurements 10d SNL Najm Challenges in UQ 45 / 52

50 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Oscillator function BCS number of terms (n,p)=(10,6) (n,p)={(5,10),(10,5)} 3003 Number of Terms dim=5, ord=10 dim=10, ord=5 dim=10, ord= Number of measurements SNL Najm Challenges in UQ 46 / 52

51 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Challenges in PC UQ Non-Linearit Bifurcative response at critical parameter values Raleigh-Bénard convection Transition to turbulence Chemical ignition Discontinuous u(λ(ξ)) Failure of global PCEs in terms of smooth Ψ k () failure of Fourier series in representing a step function Local PC methods Subdivide support of λ(ξ) into regions of smooth u λ(ξ) Emplo PC with compact support basis on each region A spectral-element vs. spectral construction Domain mapping SNL Najm Challenges in UQ 47 / 52

52 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Multi-Block Multiwavelet PC UQ in Ignition 2.0e 12 OH, MRA, full F_1 F_7 Mean +/ 1σ envelope OH Concentration (mole/cm 3 sec) 1.5e 12 e e 13 e time (sec) H 2 -O 2 supercritical water oidation model Empiricall-based uncertaint in all 7 reactions Adaptive refinement of MW block decomposition (Le Maître, 2004, 2007) Challenges in hi-dimensional contets SNL Najm Challenges in UQ 48 / 52

53 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Uncertaint in Discontinuous Climate Response Climate sensitivit [K] Recover No Recover Rate of CO 2 increase [%] Atlantic meridional ocean circulation (AMOC) Predicted response to increasing CO 2 (Webster, 2007) Circulation ON/OFF response over parameter space Rate of CO 2 increase Climate sensitivit SNL Najm Challenges in UQ 49 / 52

54 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Domain Mapping for Discontinuous Response (c) Input λ Computational Model PC Epansion Input λ MOC ON pdf(z) 4 MOC OFF η η 1 η η 1 Initial set of computational samples z Discover uncertain discontinuit with Baesian inference Map sub-domains to unit hpercubes; Rosenblatt transform PC quadrature in mapped domains; map back Marginalize over uncertain curve (Sargsan, 2012) SNL Najm Challenges in UQ 50 / 52

55 Introduction UQ Basics Challenges Closure Characterization hi-d Discont Osc Challenges in PC UQ Time Dnamics Sstems with limit-ccle or chaotic dnamics Large amplification of phase errors over long time horizon PC order needs to be increased in time to retain accurac Time shifting/scaling remedies Futile to attempt representation of detailed turbulent velocit field v(, t;λ(ξ)) as a PCE Fast loss of correlation due to energ cascade Problem studied in 60 s and 70 s Focus on flow statistics, e.g. Mean/RMS quantities Well behaved Argues for non-intrusive methods with DNS/LES of turbulent flow SNL Najm Challenges in UQ 51 / 52

56 Introduction UQ Basics Challenges Closure Closure Probabilistic UQ framework Forward UQ Polnomial Chaos representation of random variables Intrusive and non-intrusive forward PC UQ methods Inverse UQ Baesian methods Highlighted UQ Challenges Missing data Imputation / DFI High dimensionalit Non-linearit Long term oscillator dnamics SNL Najm Challenges in UQ 52 / 52