Uncertainty Quantification in Multiscale Models

Transcription

1 Uncertainty Quantification in Multiscale Models H.N. Najm Sandia National Laboratories Livermore, CA Workshop on Stochastic Multiscale Methods: Mathematical Analysis and Algorithms Univ. Southern California, Aug 10 11, 2009 SNL Najm Uncertainty Quantification in Multiscale Models 1 / 29

2 Uncertainty Quantification in Multiscale Models Complex physical systems involve wide ranges of length and time scales Uncertainties abound in specification of the full detailed model identification of reduced submodels over ranges of scales coupling between submodels data assimilation calibration and validation of coupled multiscale models UQ is needed in the analysis, construction, and utilization of multiscale models SNL Najm Uncertainty Quantification in Multiscale Models 2 / 29

3 Outline 1 Introduction 2 Analysis and Reduction of Multiscale Models 3 Coarse-Fine Coupling in Multiscale Models 4 Conclusions SNL Najm Uncertainty Quantification in Multiscale Models 3 / 29

4 Analysis and Reduction of Multiscale Models There are many methods for analysis of multiscale physical systems, enabling reduction of their size, complexity, and/or dimensionality. A deterministic reduction requirement: Given specified observables, desired error thresholds, and the detailed model, identify the smallest acceptable model A potential reduction requirement under uncertainty: Given specified observables, desired error thresholds, and an uncertain detailed model, identify the smallest acceptable (deterministic or uncertain) model Homescu, Petzold, & Serban, SIAM Review, 2007 SNL Najm Uncertainty Quantification in Multiscale Models 4 / 29

5 Multiscale Coupling There are many available methods for coupling multiscale models across a range of application domains M a M b Coupling techniques generally pursue accuracy in communication of relevant physical quantities across heterogeneous numerical representations Communication of uncertainty across model interfaces is important from a general UQ perspective, but also more specifically to account for multiscale coupling errors SNL Najm Uncertainty Quantification in Multiscale Models 5 / 29

6 Physical Models Spanning a Large Range of Scales The degree of coupling among length/time scales affects the extent to which a model can be reduced, arriving at a simplified coupled system of submodels There are many analysis methods for examining this coupling and identifying low order reduced models PDE systems Proper Orthogonal Decomposition Principal Component Analysis Elliptic systems Fast Multipole Methods Stochastic systems Karhunen-Loève Expansion ODE systems Eigenvalue and SVD methods Computational Singular Perturbation (CSP) SNL Najm Uncertainty Quantification in Multiscale Models 6 / 29

7 Structure of Stiff ODE Models Macroscale chemical models typically involve large strongly-nonlinear stiff ODE systems Stiffness is associated with the presence of a large range of time scales Stiff ODE system dynamics typically exhibit low dimensional algebraic manifolds The solution exhibits fast attraction towards the manifold the fast subspace The system evolves slowly along the manifold the slow subspace Identifying and decoupling the fast and slow subspaces provides means of model analysis, and reduction CSP (Lam & Goussis, 1988) SNL Najm Uncertainty Quantification in Multiscale Models 7 / 29

8 CSP Highlights dy dt = g(y), y(t = 0) = y 0 g = a 1 f a N f N, f i = b i g, τ i < τ i+1, i = 1,..., N g Manifold: = a 1 f a M f M }{{} g fast 0 f i (y) 0, + a M+1 f M a N f N }{{} g slow i = 1,, M Slow Evolution: ( ) N M g slow = a s f s = I a r b r g = Pg s=m+1 r=1 SNL Najm Uncertainty Quantification in Multiscale Models 8 / 29

9 Model Reduction based on CSP Analysis Employ CSP slow/fast Importance Indices I i k Measure importance of process k to the slow/fast evolution of species i Use a given database of solutions of the detailed model, along with a specification of observables of interest u y thresholds on importance indices arrive at reduced ODE model Valorani et al., 2006 Algorithm maintains controlled accuracy in representing both the low dimensional manifold and time evolution along it SNL Najm Uncertainty Quantification in Multiscale Models 9 / 29

10 CSP reduction of nheptane-air model Detailed model: 560 sp, 2538 rn A posteriori error analysis for homogeneous ignition of an nheptane-air mixture: Error % τ ign 10 2 T 0 = 700 K T 0 = 850 K T 0 = 1100 K # species Error in ignition time versus the no. of species in the reduced model (Valorani et al., 2007) SNL Najm Uncertainty Quantification in Multiscale Models 10 / 29

11 Uncertainty in ODE model Consider uncertainty in g(y) (structure/parameters). dy dt = g ω (y) Depending on the uncertain elements of g ω, this can imply uncertainty in the fast and/or slow subspaces with consequences to model reduction Stochastic eigenvalue problem Galerkin versus collocation formulation... work in progress SNL Najm Uncertainty Quantification in Multiscale Models 11 / 29

12 Coupled Multiscale Models Accuracy of multiscale models based on coupled, possibly heterogeneous, submodels is function of: accuracy of individual submodels accuracy of coupling at model interfaces Need to communicate uncertainty at model interfaces for: Forward propagation of model/parametric uncertainty in otherwise deterministic coupled multiscale constructions Accounting for information mismatch at the interface Coarse-graining, upscaling, lifting, restriction Accounting for errors due to finite sample size in stochastic submodels, e.g. SDE coupled with a deterministic model MD/BD-continuum coupling SNL Najm Uncertainty Quantification in Multiscale Models 12 / 29

13 Model/Parametric Uncertainty in Multiscale Models M a (ω a ) M b (ω b ) Uncertainties in {M a,ω a,m b,ω b } need to be communicated in both directions between the two models Sampling/collocation based UQ methods compute N deterministic instances of the coupled multiscale system no algorithmic consequences for the interface Intrusive Polynomial Chaos (PC) methods Galerkin projection of the original governing equations Spectral representations of random variable or fields Spectral representations of uncertain quantities need to be communicated at the interface SNL Najm Uncertainty Quantification in Multiscale Models 13 / 29

14 Coarse Graining Examples Brownian dynamics computations employing lumped atom models and relevant force-fields DSMC Homogenization-upscaling VILLIN-1QQV Fitting rich information from a complex model, or from data, using a lower fidelity model with a smaller number of degrees of freedom Identify effective force fields, material properties, or other parameters of the low fidelity coarse model, by calibrating with respect to the complex model and/or data SNL Najm Uncertainty Quantification in Multiscale Models 14 / 29

15 Upscaling with Uncertainty Quantificaton Define stochastic uncertain effective upscaled properties Forward UQ provides uncertain coarse-model prediction Least-squares regression and optimization can be used to calibrate the uncertain effective properties Arnst & Ghanem, 2008 The density of the uncertain model outputs provides the basis for the likelihood function in a Bayesian inference procedure to similarly solve the calibration problem Ghanem & Doostan, 2006; Liu et al., 2009 In the presence of measurement data, the associated noise convolves with the uncertain model output density to form the likelihood function SNL Najm Uncertainty Quantification in Multiscale Models 15 / 29

16 Coarse-Fine Model Coupling A Multiscale Combustion system Model Detailed 2D reacting flow model in small combustor region Non-reacting 1D compressible flow model in long exhaust zone l L >> l Averaging of a 2D upstream flow/mixture structure at the interface, to that of a 1D model, at each coarse time-step Accounting for uncertainty: Discrepancy between the two models can be represented using a statistical model This provides the uncertain BC for the 1D domain SNL Najm Uncertainty Quantification in Multiscale Models 16 / 29

17 Statistical Model Construction A statistical model can be built based on: Probability Distribution. Parametric/non-parametric Gaussian process model e.g. Kennedy & O Hagan, 2000 Gaussian mixture models Kernel density estimation Moments Spectral expansion for RV/Random Process Optimal choice depends on the availability of data whether or not the model is directly observable model cost the intended use SNL Najm Uncertainty Quantification in Multiscale Models 17 / 29

18 Limited Information at Interface Information available from/to microscale model or data assimilation at select spatial locations in the macroscale model M a M b /D M b /D M b /D Boundary condition for M a can be formulated as a statistical model, expressing uncertainty in the field along the interface away from the available points Parameters of the statistical model can be determined using Bayesian inference or least-squares methods Measurement uncertainty resulting from noise in the data can be directly incorporated in the inference procedure SNL Najm Uncertainty Quantification in Multiscale Models 18 / 29

19 Statistical Model Correlation Structure With limited information, the random process or statistical model representing the uncertain field at the interface is not sufficiently specified Uncorrelated fine-scale/data inputs No information on spatial correlation structure along the macro dimension Yet there are physical constraints on the range of correlation lengths The structure of material-property variability along the interface The length scales inherent in the macroscale forcing These need to be used to constrain the correlation structure SNL Najm Uncertainty Quantification in Multiscale Models 19 / 29

20 Operator-Upscaling, Variational Multiscale Method Using the known detailed material properties in a full-domain solution is prohibitively expensive. Solve the fine-scale problem independently within each coarse-scale element Solve the coarse-scale problem, with the same physical fine-scale material properties but an upscaled operator Arbogast, 1998; Hughes, 1995 Stochastic version employing stochastic variational multiscale methods with Polynomial Chaos and Karhunen-Loève Expansions Accounting for inherent randomness of material properties Asokan and Zabaras, 2006 Ganapathysubramanian and Zabaras, 2007, 2009 SNL Najm Uncertainty Quantification in Multiscale Models 20 / 29

21 Projective Time Integration Detailed full-scale solution time integration over short bursts of time, possibly at select macroscale locations Use resulting statistics to inform large- t coarse-level time integration Lifting and Restriction operations Kevrekidis, Used for modeling diffusion with uncertain diffusivity Xiu and Kevrekidis, 2005 MC Sampled fine-scale computations Equation-free coarse-scale time integration of uncertain Polynomial Chaos representation SNL Najm Uncertainty Quantification in Multiscale Models 21 / 29

22 Stochastic Coupling Averaged information from stochastic model communicated to deterministic model Finite sample sizes lead to uncertainty in unconverged averages M a Deterministic M b Stochastic SDE/MD/BD A wealth of deterministic MD/Continuum coupling methods... Kevrekidis 2003, Prudhomme 2009,... What about coupling while accounting for averaging uncertainty? SNL Najm Uncertainty Quantification in Multiscale Models 22 / 29

23 Multiscale Modeling of Desalination Membranes Electrochemical acqueous transport of Na + and Cl in nanoporous membranes Adalsteinsson, Debusschere, Long, and Najm, 2008 Continuum: Particle: Poisson-Nernst-Planck finite element Brownian Dynamics SNL Najm Uncertainty Quantification in Multiscale Models 23 / 29

24 Multiscale Coupling Subdomains solved in an alternating fashion Schwarz alternating method Hadjiconstantinou, JCP 1999; Schwarz, 1896 Electrostatic field computation continuum: ignore particles inside the nanopore nanopore: superpose continuum field and local particle Coulomb interactions Handshake zone coupling region Continuum solution provides concentration BC for the particle domain BD solution provides flux BC for the continuum domain SNL Najm Uncertainty Quantification in Multiscale Models 24 / 29

25 Residual Averaging Noise in BD Fluxes Multiple replica simulations at each iteration step Standard deviation of replica simulation fluxes is comparable with the iteration-to-iteration noise in the mean flux [mol/s] 1.5e e e-17 J Na, mm J Cl, mm J Na, mm J Cl, mm 0.0e iteration step Residual variability could be communicated to the continuum model as an uncertainty in the flux BC work in progress SNL Najm Uncertainty Quantification in Multiscale Models 25 / 29

26 Summary Numerous methods for analyzing, reducing, and coupling multiscale models Uncertainties abound in the inherent formulation and practical implementation of these systems Methods have been developed to handle and communicate these uncertainties Many opportunities remain Coupling multiple/all-relevant sources of uncertainty Balancing error budgets and model reduction under uncertainty Handling systems with strong coupling in length/time scales SNL Najm Uncertainty Quantification in Multiscale Models 26 / 29

27 Acknowledgements B.J. Debusschere, H. Adalsteinsson... Sandia National Labs, Livermore, CA O.M. Knio... Johns Hopkins Univ., Baltimore, MD R.G. Ghanem... Univ. Southern California, Los Angeles, CA Y.M. Marzouk... Massachusetts Inst. Tech., Cambridge, MA M. Valorani... La Sapienza Univ. Rome, Italy D. Goussis... National Tech. Univ. Athens, Greece This work was supported by the US Department of Energy (DOE), Office of Science (SC), Office of Basic Energy Sciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences, the SciDAC Computational Chemistry Program, and the Office of Advanced Scientific Computing Research (ASCR). Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL SNL Najm Uncertainty Quantification in Multiscale Models 27 / 29

28 Relevant Papers 1 H. Adalsteinsson, B.J. Debusschere, K.R. Long, and H.N. Najm, Components for atomistic-to-continuum multiscale modeling of flow in micro- and nanofluidic systems, Scientific Programming, 16(4): , T. Arbogast, S.E. Minkoff, and P.T. Keenan, An operator-based approach to upscaling the pressure equation, Computational Methods in Water Resources XII, Vol. 1: Computational Methods in Contamination and Remediation of Water Resources, V.N. Burganos et al. Eds., Computational Mechanics Publications, Southampton, U.K., pp , M. Arnst and R. Ghanem, Probabilistic equivalence and stochastic model reduction in multiscale analysis, Comput. Methods Appl. Mech. Engrg., 197: , B.V. Asokan and N. Zabaras, A stochastic variational multiscale method for diffusion in heterogeneous random media, J. Comp. Phys., 218: , P.T. Bauman, J.T. Oden, and S. Prudhomme, Adaptive multiscale modeling of polymeric materials with Arlequin coupling and Goals algorithms, Comput. Methods Appl. Mech. Engrg., 198: , B. Ganapathysubramanianand N. Zabaras, Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method, J. Comp. Phys., 226: , B. Ganapathysubramanianand N. Zabaras, A stochastic multiscale framework for modeling flow through random heterogeneous porous media, J. Comp. Phys., 228: , R. Ghanem and A. Doostan, On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data, J. Comp. Phys., 217:63-81, T.J.R. Hughes, Multiscale phenomena: Green s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127: , SNL Najm Uncertainty Quantification in Multiscale Models 28 / 29

29 Relevant Papers 2 V. Kolobov, R. Arslanbekov, ad A. Vasenkov, Coupling Atomistic and Continuum Models for Multi-scale Simulations of Gas Flows, Y.Shi et al. (Eds.): ICCS 2007, Part I, LNCS 4487, pp , Springer-Verlag Berlin Heidelberg W.K. Liu, L. Siad, R. Tian, S. Lee, D. Lee, X. Yin, W. Chen, S. Chan, G.B. Olson, L.-E. Lingden, M.F. Horstmeyer, Y.-S. Chang, J.-B. Choi and Y.J. Kim, Complexity science of multiscale materials via stochastic computations, Int. J. Numer. Meth. Engng., submitted, S. Prudhomme, H.B. Dhia, P.T. Bauman, N. Elkhodja, and J.T. Oden, Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method, Comput. Methods Appl. Mech. Engrg., 197: , S. Prudhomme, L. Chamoin, H.B. Dhia, and B.T. Bauman, An adaptive strategy for the control of modeling error in two-dimensional atomic-to-continuum coupling simulations, Comput. Methods Appl. Mech. Engrg., 198: , M. Valorani, F. Creta, D.A. Goussis, J.C. Lee, and H.N. Najm, Chemical Kinetics Mechanism Simplification via CSP, Combustion and Flame, 146:29-51, M. Valorani, F. Creta, F. Donato, H.N. Najm, and D.A. Goussis, Skeletal Mechanism Generation and Analysis for n-heptane with CSP, Proc. Comb. Inst., 31: , D. Xiu, I.G. Kevrekidis, and R. Ghanem, An Equation-Free Multiscale Approach to Uncertainty Quantification, Computing in Science & Engineering, pp.18-23, May/June D. Xiu and I.G. Kevrekidis, Equation-free, multiscale computation for unsteady random diffusion, Multiscale Model. Simul., 4(3): , SNL Najm Uncertainty Quantification in Multiscale Models 29 / 29