Stocastic FE

8/10/2019 Stocastic FE

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Roger Ghanem

University of Southern CaliforniaLos Angeles, CaliforniaUSA

SHORT COURSE: UNICAMP, CAMPANIS - BRAZIL- MARCH 2 nd 2007 -

STOCHASTIC FINITE ELEMENTS


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OutlineOutline

MotivationMathematical structure for validation and verification

Packaging of information using this structure:

representing experimental information

Construction of approximation on these structuresNumerical implementation of stochastic predictions (SFEM)

Example applications

Technical challenges and conclusions


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Excerpts from published emails in connection with shuttle Columbias last mission:

MotivationMotivation


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Excerpts from published emails in connection with shuttle Columbias last mission:

simulations showed that landing with 2 flat tires were survivable. Bob and David expressed someskepticism as to the accuracy of the Ames sim in light of other data (Convair 990 testing)


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Problem DefinitionProblem Definition

She says:

I have flipped a coin: if you knowthe outcome then YES

She says:I will flip a coin: if you will knowthe outcome then YES

Not interested !

man has a chance

Consider man-woman:he makes a proposition


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SolutionSolution

Quantity of interest : decision with upper bound on risk Flip coin 100 times calibrated decision tool 52-48

observe initial configuration calibrated decision tool 70-30 observe surface tension calibrated decision tool 80-20

--- --- ---

observe quantum states calibrated decision tool 99-1

Use Model-Based Predictions in lieu of physical experimentsto save time and cost

Ingredients:

calibrate a stochastic plant (you must characterize/parameterize it first !)

evaluate limit on predictability of plant

refine plant if target confidence not achievable

These probabilities are not intrinsic properties of the state. They areconditioned on available information.


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Interaction of model and dataInteraction of model and data


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Error budgetError budget


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WeWe use probabilistic models for uncertaintyuse probabilistic models for uncertainty

A random variable is a measurable mapping on a probability space to

The probability distribution of is the image of under ,defining the probability triple

Set of elementary events

Sigma algebra of all events that make sense

Measure on all elements of

An experiment is defined by a probability triple, (a measurable space)

The probabilistic framework provides a packaging of information that is amenable to a levelof rigor in analysis that permits the quantification of uncertainty.

Although:

unacquainted with problems where wrong results could be attributed to failure to usemeasure theory. (E.T. Jaynes, published 2003)


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What do we mean by things being close ?What do we mean by things being close ?

Modes of Convergence: Let be a sequence of r.v. defined on and let beanother rv on same probability space:

Almost sure Convergence:

Convergence in mean square:

Convergence in distribution: Let and

then converges in distribution to if converges to

Convergence in probability:

Convergence in mean:

Convergence in mean-square permits an analysis of random variables.

First step in a unified perspective in verification and validation.


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A random variable is a functionfunction :

What is a random variableWhat is a random variable

Cameron Martin : Any second-order functional of the Brownian motioncan be expanded as a mean-square convergent series in terms ofinfinite-dimensional Hermite polynomial in gaussian variables.

i.e. : A process that can be modeled as the output of a relatively nicesystem to highly oscillatory input can be represented using the WienerChaos: series of multi-dimensional Hermite polynomials in Gaussianvariables. POLYNOMIAL CHAOS EXPANSION


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CameronCameron --Martin TheoremMartin Theorem


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Another special basis:Another special basis: KarhunenKarhunen --LoeveLoeve expansionexpansion

Reference: J.B. Read (1983)


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KL MODESONE REALIZATION OFFOAM PROPERTIES


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Management of uncertaintyManagement of uncertainty


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Adapted basesAdapted bases

COORDINATES IN THIS SPACE REPRESENT PROBABILISTIC CONTENT


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Adapted basesAdapted bases

COORDINATES IN THIS SPACE REPRESENT PROBABILISTIC CONTENT


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The random quantities are resolved as

These could be, for example:

Parameters in a PDE

Boundaries in a PDE (e.g. Geometry)

Field Variable in a PDE

Multidimensional OrthogonalPolynomials in independentrandom variables

Representing uncertaintyRepresenting uncertainty

Dimension of approximationreflects the stochastic complexityand heterogeneity of the process


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First Step: Model Reduction withFirst Step: Model Reduction with KarhunenKarhunen --LoeveLoeve

Starting with observations of processover a limited subset of indexing set:

Covariance matrix of observations

Reduced order representation: KL:

Where:


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Galerkin Projection

Maximum Likelihood Formulation

Bayesian Inference

Maximum Entropy

Representing uncertaintyRepresenting uncertainty


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The random quantities are resolved as

These could be, for example:

Parameters in a PDE

Boundaries in a PDE (e.g. Geometry)

Field Variable in a PDE

These decompositions provide a resolution (or parameterization) ofthe uncertainty on spatial or temporal scales

Representation of uncertainty:Representation of uncertainty: GalerkinGalerkin projectionprojection

Reference: Sakamoto and Ghanem, PEM and JEM, 2000.


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Marginal probability density function of the stochastic process is specified atevery point in the indexing set, along with the two-point correlation function.

Process is characterized and simulated through its Polynomial Chaoscoordinates.

Simulation ofSimulation of nonnon --gaussiangaussian processesprocesses


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Realizations of the processPDF: 1 st order chaos





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Realizations of the processPDF: 2 nd order chaos





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Realizations of the processPDF: 3 rd order chaos





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Realizations of the processPDF: 4 th order chaos





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Representation of uncertainty: Maximum likelihoodRepresentation of uncertainty: Maximum likelihood

Reference: Descelliers, C., Ghanem, R. and Soize, C. ``Maximum likelihood estimation of stochastic chaos representation from experimental data,'' to appear in International Journal forNumerical Methods in Engineering.


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essentialessential dimensionality of a processdimensionality of a process

0

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z

0 3.25 6.5 9.75 131

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3 x 1010

0 0.5 1 1.5 2 2.50

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0 0.5 1 1.5 2 2.50

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0 0.5 1 1.5 2 2.50

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0 0.5 1 1.5 2 2.50

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(a) (b)

(c) (d)

1 2 3 4 5 6 7 810

4

103

102

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Physical object: LinearElasticity

Stochastic parameters

Convergence as function of dimensionality

Convergence of PDF

Convergence of PDF


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Representation of uncertainty: Bayesian inferenceRepresentation of uncertainty: Bayesian inference

Polynomial Chaos representation of reduced variables:

Constraint on chaos coefficients:

Estimation of stochastic process using estimate of reduced variables:

Objective is to estimate


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Representation of uncertainty: Bayesian inferenceRepresentation of uncertainty: Bayesian inference

Define Cost Function (hats denote estimators):

Then Bayes estimate is:

Bayes rule:

Use kernel density estimation to represent the Likelihood function

Use Markov Chain Monte Carlo to sample from the posterior(metropolis Hastings algorithm) -->BIMH

Reference: Ghanem and Doostan, Journal of Computational Physics, 2006.


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Characterization of Uncertainty:Characterization of Uncertainty:Bayesian InferenceBayesian Inference

Posterior distributions of coefficients inpolynomial Expansion of

Distribution of the recovered process:

Reference: Doostan and Ghanem, , Journal of Computational Physics, 2006.


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Representation of uncertainty:Representation of uncertainty:

MaxEntMaxEnt

and Fisher informationand Fisher information

Moments of observations as constraints:

Maximum Entropy Density Estimation (MEDE) results in joint measure of KL variables:

Fisher Information Matrix:

Then asymptotically (h denotes coefficients in polynomial chaos description of observations):

Observe at N locations and n sets of observations.

Reduce dimensionality using KL.


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Representation of uncertainty:Representation of uncertainty: MaxEntMaxEnt and Fisher informationand Fisher information

Reference: Das, Ghanem, and Spall, SIAM Journal on Scientific Computing, 2006.


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Approximate asymptotic representation:

Representation on the set of observation:

Representation smoothed on the whole domain:

Remark: is formulated by spectral decomposition of .

Remark: Both intrinsic uncertainty and uncertainty due to lack of data are represented.

Uncertainty modeling for system parametersUncertainty modeling for system parameters


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Propagation of uncertainty fromparameters to prediction:

Approximation in the measurespace defined by the parametersand the physics

Stochastic PredictionStochastic Prediction


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A random variable is a functionfunction :

StochasticStochastic PredictionPrediction

Characterize Y, given characterization of


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Assumptions:

Second-order random variables:

Basic random vectors are independent:

Vector does not necessarily involve independent random variables:

Functional representationsFunctional representations


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Given bases of H j,k other bases can be constructed.



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Soize, C., and Ghanem, R. ``Physical Systems with Random Uncertainties: Chaos representations with arbitrary probability measure,'' SIAM Journal of Scientific Computing,Vol. 26, No. 2, pp. 395-410, 2004.


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Some common basesSome common basesInfinite-dimensional case:This is an exercise in stochastic analysis: Hermite polynomials: Gaussian measure (Wiener: Homogeneous Chaos)

Charlier polynomials: Poisson measure (Wiener: Discrete Chaos)

Very few extensions possible: Friedrichs and Shapiro (Integration of functionals) providecharacterization of compatible measures. Segall and Kailath provide an extension to martingales.

Finite-dimensional case: independent variables:This is an exercise in one-dimensional approximation: Askey polynomials:measures from Askey chart (Karniadakis and co-workers)

Legendre polynomials: uniform measure (theoretical results by Babuska and co-workers)

Wavelets: Le-Maitre and co-workers

Arbitrary measures with bounded support: C. Schwab

Finite-dimensional case: dependent variables:This is an exercise in multi-dimensional approximation: Arbitrary measures: Soize and Ghanem.


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Variational Formulation :

Find s.t.:

Where:

should be coercive and continuous

Notice:

Stochastic Finite ElementsStochastic Finite Elements


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Approximation Formulation:

Find s.t.:

Where:

Write:

i th Homogeneous Chaos in

Basis in X Basis in Y

*



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Sources of Error: Spatial Discretization Error:

Random Dimension Discretization Error:

Joint error estimation is possible, for general measures, using nested approximatingspaces (e.g. hierarchical FEM) (Doostan and Ghanem, 2004-2007)

Joint error estimation is possible, for special cases:

infinite-dimesional gaussian measure: Benth et.al, 1998

tensorized uniform measure: Babuska et.al, 2004



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Approximation Formulation:

Find s.t.:

Where:

Write:

i th Homogeneous Chaos in

Basis in X Basis in Y

*



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Typical System Matrix: 1st OrderTypical System Matrix: 1st Order


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Typical System Matrix:Typical System Matrix: 2nd Order2nd Order


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Typical System Matrix: 3rd OrderTypical System Matrix: 3rd Order


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Efficient preEfficient pre

--conditionersconditioners

Reference: Ghanem and Kruger, 1997; Ghanem and Pellissetti, 2003.


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NonNon

--intrusive implementationintrusive implementation

Reference: Ghanem and Ghiocel, 1995; Ghanem and Red-Horse, 1999, 2000, 2001, 2002, 2003; Reagan et.al 2003, 2004, 2005; Soize and Ghanem, 2004.

Stocastic FE

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