Uncertainty Quantification in Multiscale...

Intro Analysis-Reduction Coupling Conclusions

Uncertainty Quantification in Multiscale Models

H.N. Najm

Sandia National LaboratoriesLivermore, CA

Workshop onStochastic Multiscale Methods: Mathematical Analysis and Algorithms

Univ. Southern California, Aug 10–11, 2009

SNL Najm Uncertainty Quantification in Multiscale Models 1 / 29


Uncertainty Quantification in Multiscale Models

Complex physical systems involve wide ranges of lengthand time scalesUncertainties abound in

specification of the full detailed modelidentification of reduced submodels over ranges of scalescoupling between submodelsdata assimilationcalibration and validation of coupled multiscale models

UQ is needed in the analysis, construction, and utilizationof multiscale models



Outline

1 Introduction

2 Analysis and Reduction of Multiscale Models

3 Coarse-Fine Coupling in Multiscale Models

4 Conclusions



Analysis and Reduction of Multiscale Models

There are many methods for analysis of multiscale physicalsystems, enabling reduction of their size, complexity, and/ordimensionality.

A deterministic reduction requirement:

Given specified observables, desired error thresholds, and thedetailed model, identify the smallest acceptable model

A potential reduction requirement under uncertainty:

Given specified observables, desired error thresholds, and anuncertain detailed model, identify the smallest acceptable(deterministic or uncertain) model

Homescu, Petzold, & Serban, SIAM Review, 2007



Multiscale Coupling

There are manyavailable methods forcoupling multiscalemodels across a rangeof application domains

Ma Mb

Coupling techniques generally pursue accuracy incommunication of relevant physical quantities acrossheterogeneous numerical representations

Communication of uncertainty across model interfaces isimportant from a general UQ perspective, but also morespecifically to account for multiscale coupling errors



Physical Models Spanning a Large Range of Scales

The degree of coupling among length/time scales affectsthe extent to which a model can be reduced, arriving at asimplified coupled system of submodelsThere are many analysis methods for examining thiscoupling and identifying low order reduced models

PDE systemsProper Orthogonal DecompositionPrincipal Component Analysis

Elliptic systemsFast Multipole Methods

Stochastic systemsKarhunen-Loeve Expansion

ODE systemsEigenvalue and SVD methodsComputational Singular Perturbation (CSP)



Structure of Stiff ODE Models

Macroscale chemical models typically involve largestrongly-nonlinear stiff ODE systems

Stiffness is associated with the presence of a large rangeof time scalesStiff ODE system dynamics typically exhibit lowdimensional 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 subspacesprovides means of model analysis, and reduction

– CSP (Lam & Goussis, 1988)



CSP Highlights

dydt

= g(y), y(t = 0) = y0

g = a1f 1 + · · · + aNf N, f i = bi · g, τi < τi+1, i = 1, . . . , N

g = a1f 1 + · · · + aMf M

︸︷︷︸

gfast≈0

+ aM+1f M+1 + · · · + aNf N

︸︷︷︸

gslow

Manifold:

f i(y) ≈ 0, i = 1, · · · , M

Slow Evolution:

gslow =

N∑

s=M+1

asfs =

(

I −M∑

r=1

arbr

)

g = Pg



Model Reduction based on CSP Analysis

Employ CSP slow/fast Importance Indices I ik

– Measure importance of process k to the slow/fastevolution 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 modelValorani et al., 2006

Algorithm maintains controlled accuracy in representing boththe low dimensional manifold and time evolution along it



CSP reduction of nHeptane-air model

Detailed model:560 sp, 2538 rn

A posteriori erroranalysis forhomogeneousignition of annHeptane-airmixture:

# species

Err

or%

τ ign

100 150 200 250 300 350 400

10-1

100

101

102

T0 = 700 KT0 = 850 KT0 = 1100 K

Error in ignition time versus the no. ofspecies in the reduced model

(Valorani et al., 2007)



Uncertainty in ODE model

Consider uncertainty in g(y) (structure/parameters).

dydt

= gω(y)

Depending on the uncertain elements of gω

, this can implyuncertainty in the fast and/or slow subspaces

– with consequences to model reduction

Stochastic eigenvalue problem

– Galerkin versus collocation formulation

... work in progress



Coupled Multiscale Models

Accuracy of multiscale models based on coupled, possiblyheterogeneous, submodels is function of:

accuracy of individual submodelsaccuracy of coupling at model interfaces

Need to communicate uncertainty at model interfaces for:Forward propagation of model/parametric uncertainty inotherwise deterministic coupled multiscale constructionsAccounting for information mismatch at the interface

– Coarse-graining, upscaling, lifting, restrictionAccounting for errors due to finite sample size in stochasticsubmodels, e.g.

– SDE coupled with a deterministic model– MD/BD-continuum coupling



Model/Parametric Uncertainty in Multiscale Models

Ma(ωa) Mb(ωb)

Uncertainties in {Ma, ωa,Mb, ωb} need to becommunicated in both directions between the two models

Sampling/collocation based UQ methods compute Ndeterministic instances of the coupled multiscale system

– no algorithmic consequences for the interfaceIntrusive Polynomial Chaos (PC) methods

Galerkin projection of the original governing equationsSpectral representations of random variable or fields

– Spectral representations of uncertain quantitiesneed to be communicated at the interface



Coarse Graining

ExamplesBrownian dynamicscomputations employinglumped atom models andrelevant force-fieldsDSMCHomogenization-upscaling

VILLIN-1QQV

Fitting rich information from a complex model, or from data,using a lower fidelity model with a smaller number ofdegrees of freedom

Identify effective force fields, material properties, or otherparameters of the low fidelity coarse model, by calibratingwith respect to the complex model and/or data



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 tocalibrate the uncertain effective properties

Arnst & Ghanem, 2008

The density of the uncertain model outputs provides thebasis for the likelihood function in a Bayesian inferenceprocedure to similarly solve the calibration problem

Ghanem & Doostan, 2006; Liu et al., 2009

In the presence of measurement data, the associatednoise convolves with the uncertain model output density toform the likelihood function



Coarse-Fine Model CouplingA Multiscale Combustion system Model

Detailed 2D reactingflow model in smallcombustor region

Non-reacting 1Dcompressible flow modelin long exhaust zone

l L >> l

Averaging of a 2D upstream flow/mixture structure at theinterface, to that of a 1D model, at each coarse time-stepAccounting for uncertainty:

Discrepancy between the two models can be representedusing a statistical modelThis provides the uncertain BC for the 1D domain



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 onthe availability of datawhether or not the model is directly observable

– model costthe intended use



Limited Information at Interface

Information availablefrom/to microscale modelor data assimilation atselect spatial locations inthe macroscale model

Ma

Mb/D Mb/D Mb/D

Boundary condition for Ma can be formulated as astatistical model, expressing uncertainty in the field alongthe interface away from the available points

Parameters of the statistical model can be determinedusing Bayesian inference or least-squares methods

Measurement uncertainty resulting from noise in the datacan be directly incorporated in the inference procedure



Statistical Model Correlation Structure

With limited information, the random process or statisticalmodel representing the uncertain field at the interface isnot sufficiently specified

Uncorrelated fine-scale/data inputsNo information on spatial correlation structure along themacro dimension

Yet there are physical constraints on the range ofcorrelation lengths

The structure of material-property variability along theinterfaceThe length scales inherent in the macroscale forcing

These need to be used to constrain the correlationstructure



Operator-Upscaling, Variational Multiscale Method

Using the known detailed material properties in afull-domain solution is prohibitively expensive.

Solve the fine-scale problem independently within eachcoarse-scale element

Solve the coarse-scale problem, with the same physicalfine-scale material properties but an upscaled operator

Arbogast, 1998; Hughes, 1995

Stochastic version employing stochastic variationalmultiscale methods with Polynomial Chaos andKarhunen-Loeve Expansions

Accounting for inherent randomness of material properties

Asokan and Zabaras, 2006

Ganapathysubramanian and Zabaras, 2007, 2009



Projective Time Integration

Detailed full-scale solution time integration over shortbursts of time, possibly at select macroscale locations

Use resulting statistics to inform large-∆t coarse-level timeintegration

Lifting and Restriction operationsKevrekidis, 2000-

Used for modeling diffusion with uncertain diffusivityXiu and Kevrekidis, 2005

MC Sampled fine-scale computationsEquation-free coarse-scale time integration of uncertainPolynomial Chaos representation



Stochastic Coupling

Averaged informationfrom stochastic modelcommunicated todeterministic model

Finite sample sizeslead to uncertainty inunconverged averages

Ma

Deterministic

Mb

StochasticSDE/MD/BD

A wealth of deterministic MD/Continuum coupling methods. . . Kevrekidis 2003, Prudhomme 2009, . . .

What about coupling while accounting for averaginguncertainty?



Multiscale Modeling of Desalination Membranes

Electrochemicalacqueous transportof Na+ and Cl− innanoporousmembranes

Adalsteinsson, Debusschere, Long, and Najm, 2008

Continuum: Poisson-Nernst-Planck – finite element

Particle: Brownian Dynamics



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



Residual Averaging Noise in BD Fluxes

Multiple replica simulationsat each iteration step

Standard deviation ofreplica simulation fluxes iscomparable with theiteration-to-iteration noisein the mean

0 2 4 6 8 10iteration step

0.0e+00

5.0e-17

1.0e-16

1.5e-16

flux

[m

ol/s

]

JNa

, 150-50 mMJ

Cl, 150-50 mM

JNa

, 150-100 mMJ

Cl, 150-100 mM

Residual variability could be communicated to thecontinuum model as an uncertainty in the flux BC

– work in progress



Summary

Numerous methods for analyzing, reducing, and couplingmultiscale models

Uncertainties abound in the inherent formulation andpractical implementation of these systems

Methods have been developed to handle and communicatethese uncertaintiesMany opportunities remain

Coupling multiple/all-relevant sources of uncertaintyBalancing error budgets and model reduction underuncertaintyHandling systems with strong coupling in length/time scales



Acknowledgements

B.J. Debusschere,H. Adalsteinsson . . . Sandia National Labs, Livermore, CA

O.M. Knio . . . Johns Hopkins Univ., Baltimore, MDR.G. Ghanem . . . Univ. Southern California, Los Angeles, CAY.M. Marzouk . . . Massachusetts Inst. Tech., Cambridge, MAM. Valorani . . . La Sapienza Univ. Rome, ItalyD. Goussis . . . National Tech. Univ. Athens, Greece

This work was supported by the US Department of Energy (DOE), Office of Science (SC), Office of Basic EnergySciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences, the SciDAC ComputationalChemistry Program, and the Office of Advanced Scientific Computing Research (ASCR). Sandia NationalLaboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for theUnited States Department of Energy under contract DE-AC04-94-AL85000.



Relevant Papers – 1

H. Adalsteinsson, B.J. Debusschere, K.R. Long, and H.N. Najm, “Components for atomistic-to-continuummultiscale modeling of flow in micro- and nanofluidic systems”, Scientific Programming, 16(4):297-313,2008.

T. Arbogast, S.E. Minkoff, and P.T. Keenan, “An operator-based approach to upscaling the pressureequation”, Computational Methods in Water Resources XII, Vol. 1: Computational Methods inContamination and Remediation of Water Resources, V.N. Burganos et al. Eds., Computational MechanicsPublications, Southampton, U.K., pp. 405-412, 1998.

M. Arnst and R. Ghanem, “Probabilistic equivalence and stochastic model reduction in multiscale analysis”,Comput. Methods Appl. Mech. Engrg., 197:3584-35923, 2008.

B.V. Asokan and N. Zabaras, “A stochastic variational multiscale method for diffusion in heterogeneousrandom media”, J. Comp. Phys., 218:654-676, 2006.

P.T. Bauman, J.T. Oden, and S. Prudhomme, “Adaptive multiscale modeling of polymeric materials withArlequin coupling and Goals algorithms”, Comput. Methods Appl. Mech. Engrg., 198:799-818, 2009.

B. Ganapathysubramanian and N. Zabaras, “Modeling diffusion in random heterogeneous media:Data-driven models, stochastic collocation and the variational multiscale method”, J. Comp. Phys.,226:326-353, 2007.

B. Ganapathysubramanian and N. Zabaras, “A stochastic multiscale framework for modeling flow throughrandom heterogeneous porous media”, J. Comp. Phys., 228:591-618, 2009.

R. Ghanem and A. Doostan, “On the construction and analysis of stochastic models: Characterization andpropagation of the errors associated with limited data”, J. Comp. Phys., 217:63-81, 2006.

T.J.R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgridscale models, bubbles and the origins of stabilized methods”, Comput. Methods Appl. Mech. Engrg.,127:387-501, 1995.



Relevant Papers – 2

V. Kolobov, R. Arslanbekov, ad A. Vasenkov, “Coupling Atomistic and Continuum Models for Multi-scaleSimulations of Gas Flows”, Y.Shi et al. (Eds.): ICCS 2007, Part I, LNCS 4487, pp. 858-865, Springer-VerlagBerlin Heidelberg 2007.

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 stochasticcomputations”, Int. J. Numer. Meth. Engng., submitted, 2009.

S. Prudhomme, H.B. Dhia, P.T. Bauman, N. Elkhodja, and J.T. Oden, “Computational analysis of modelingerror for the coupling of particle and continuum models by the Arlequin method”, Comput. Methods Appl.Mech. Engrg., 197:3399-3409, 2008.

S. Prudhomme, L. Chamoin, H.B. Dhia, and B.T. Bauman, “An adaptive strategy for the control of modelingerror in two-dimensional atomic-to-continuum coupling simulations”, Comput. Methods Appl. Mech. Engrg.,198:1887-1901, 2009.

M. Valorani, F. Creta, D.A. Goussis, J.C. Lee, and H.N. Najm, “Chemical Kinetics Mechanism Simplificationvia CSP”, Combustion and Flame, 146:29-51, 2006.

M. Valorani, F. Creta, F. Donato, H.N. Najm, and D.A. Goussis, “Skeletal Mechanism Generation andAnalysis for n-heptane with CSP”, Proc. Comb. Inst., 31:483-490, 2007.

D. Xiu, I.G. Kevrekidis, and R. Ghanem, “An Equation-Free Multiscale Approach to UncertaintyQuantification”, Computing in Science & Engineering, pp.18-23, May/June 2005.

D. Xiu and I.G. Kevrekidis, “Equation-free, multiscale computation for unsteady random diffusion”, MultiscaleModel. Simul., 4(3):915-935, 2005.


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