Equation-Free (EF) Uncertainty Quantification (UQ):Techniques and Applications

Ioannis Kevrekidis and Yu Zou

Princeton University

September 2005

Background for Uncertainty Quantification • Uncertain Phenomena in science and engineering ◊Inherent Uncertainty: Uncertainty Principle of quantum mechanics, Kinetic t

heory of gas, … ◊Uncertainty due to lack of knowledge: randomness of BC, IC and parameter

s in a mathematical model, measurement errors associated with an inaccurate instrument, …

• Scopes of application ◊Estimate and predict propagation of probabilities for occurrences: chemical

reactants, stock and bond values, damage of structural components,… ◊Design and decision making in risk management: optimal selection of rando

m parameters in a manufacturing process, assessment of an investment to achieve maximum profit,...

◊Evaluate and update model predictions via experimental data: validate accuracy of a stochastic model based on experiment, data assimilation for a stochastic contaminant transport, …

• Modeling Techniques ◊Sampling methods (Non-intrusive): Monte Carlo sampling, Markov Chain Mo

nte Carlo, Latin Hypercube Sampling, Quadrature/Cubature rules,… ◊Non-sampling methods (intrusive) : Second-order analysis, higher-order mo

ment analysis, stochastic Galerkin method, …

Dynamical Systems and Their Representation

Model

strong correlation

poor correlation

Pdeterministic system

stochastic system

Fundamentals of Polynomial Chaos (PC)

• The functional of independent random variables

can be used to represent a random variable, a random field or process.

• Spectral expansion aj’s are PC coefficients, Ψj’s are orthogonal polynomial functions with <Ψi,Ψj>=0. if i≠j. The inner product <·, ·> is defined as is a probability measure of .

• Notes ◊ The coefficients aj fully determines . ◊ Selection of Ψj is dependent on the probability measure or distribution of , e.g., if is a Gaussian measure, then Ψj are Hermite polynomials and this leads to the Wiener-Hermite polynomial chaos (Homogeneous Chaos) expansion if is a Lesbeque measure, then Ψj are Legendre polynomials

0

)()(j

jja ξξP

)()()()(),( ξξξξξ dgfgf

))(,),(),((),( 21 nξξP

)(ξ ξ

)(ξPξ

)(ξ

)(ξ

Stochastic Galerkin Method

• Advantages ◊Possibly save large computational resources compared to sampling methods ◊Free of moment closure problems ◊Can establish a strong correlation between input and response.

• Preliminary Formulation

◊ Model: e.g., ODE

◊ Represent the input in terms of expansion of individual r.v.’s (KL, SVD, POD): e.g., parameter ◊ Represent the response in terms of the truncated PC expansion

◊ The solution process involves solving for the PC coefficients fj(t), j=1,2,…,M

n

i

iikKK1

ModelInput: IC, BC, Parameters Response: Solution

M

j

jj tft0

)()(),( ξξf

0);();( KK fgff.

Stochastic Galerkin Method

• Solution technique: Galerkin projection

resulting in coupled ODE’s for fj(t),

where

• Comments

◊ The coupled ODE’s may not be derived explicitly for highly nonlinear systems.

◊ Solution of the ODE’s may be costly even if they are explicitly available.

itf i

M

j

jj

0)(),;)()((0

ξξξ

0))(()(.

tt FGFT

M tftftft ))(),...,(),(()( 10F

Equation-Free Analysis on ODE’s

• Assumption

In a system exhibiting multiple-scale characteristics, the temporal and spatial scales associated with coarse-grained observables and fine-level

observables are well separated and the coarse-grained observables lie in

a manifold that is smooth enough compared with its fine-level counterpart.

• Coarse time-stepper

1kykyLifting Restriction

Micro Simulation

)/()()( 11

.

kkkkk ttyyty Evaluation of temporal derivative

Equation-Free Analysis on ODE’s

• EF techniques

◊ Coarse Projective Integration

◊ Coarse Steady-State Analysis

◊ Coarse Bifurcation, Coarse Dynamic Renormalization, etc.

1kyky

Lifting Restriction

Equation-Free Galerkin-Free Uncertainty Quantification

• Principle Assuming smoothness of Polynomial Chaos coefficients in a large tempo

ral scale, we use the PC coefficients of random solutions as coarse-grained observables and individual realizations of random solutions as fine-level observables. The microsimulator is the Monte Carlo simulation using any sampling method.

• Coarse time-stepper ◊ Lifting (standard Monte Carlo sampling, quadrature/cubature points sampl

ing,…):

◊ Microsimulation:

◊ Restriction:

For standard Monte Carlo sampling, For quadrature/cubature points sampling, is the weight associated with each sampling point.

e

M

j

kjj

k Nktft ,,2,1,)()(),(0

00

ξξf

ekkk NkKtt ,,2,1,0))();,((),( ξξξ fgf

.

Mjttf jjjj ,,1,0,)(),(/)(),,()( ξξξξf

)(),()(),,(1

kj

N

k

kkj

e

tt ξξξξ

ff

ek N/1k

Equation-Free Galerkin-Free Uncertainty Quantification

• Domain of applications corresponding to EF analysis on ODE’s

EF analysis on ODE’s EF GF UQ

◊ Coarse Projective Integration ◊ Expedited evolution of a stochastic system

◊ Coarse steady-state computation ◊ Random steady-state computation

◊ Coarse limit-cycle computation ◊ Random limit-cycle computation

◊ Coarse bifurcation analysis ◊ Random bifurcation analysis

Example 1: Continuous Stirred-tank Chemical Reactor

)(/1

exp)1(

/1exp)1(

222

212

2

2

211

1

ca

a

xxx

xxDBx

dt

dx

x

xxDx

dt

dx

x1: the conversionx2: the dimensionless temperatureDa: Damkoehler numberB: heat of reactionβ: heat transfer coefficient γ: activation energyx2c: coolant temperature

Sampling technique in lifting: standard Monte Carlo

Evolution of x1, B=22, Da=0.07,

β=3( 1+0.1ξ)

Evolution of x1, B=22,β=3

Da varies

Example 1: Continuous Stirred-tank Chemical Reactor

Random steady states of x1,

B=22,β=3(1+0.1ξ), Da variesRandom steady states of x1, B=22,β=3,Da=<

Da>(1+0.1ξ), <Da> varies

Example 1: Continuous Stirred-tank Chemical Reactor

Evolution of x1, B=22,β=3

Da variesB=22,β=3, Da ~(0.083,0.084)

Example 2: CO oxidation in a Pt surface

BArB

BArAA

kdt

d

kdt

d

2*

*

2

Chemical reaction: A(CO)+1/2B2(O2) -> AB (CO2)

Model: θ A: mean coverage of Aθ B: mean coverage of Bθ *: mean coverage of vacant sitesα=1.6; γ=0.04; kr= 4; β=6+0.25ξ

Sampling method 1: standard Monte Carlo (Ne=40,000)

Example 2: CO oxidation in a Pt surface

Sampling method 2: Gauss-Legendre quadrature points (Ne=200)

Example 2: CO oxidation in a Pt surface

Standard Monte Carlo (Ne=40,000)β=< β>(1+0.05ξ)

G-L quadrature (Ne=200)β=< β>(1+0.05ξ)