equation-free uncertainty quantification on stochastic chemical reactions

13
Princeton University Department of Chemical Engineering and PACM Department of Chemical Engineering and PACM Equation-Free Uncertainty Quantification on Stochastic Chemical Reactions SIAM Annual Meeting, Boston July 12, 2006 Yu Zou Ioannis G. Kevrekidis Department of Chemical Engineering and PACM Princeton University

Upload: megan

Post on 02-Feb-2016

58 views

Category:

Documents


0 download

DESCRIPTION

Equation-Free Uncertainty Quantification on Stochastic Chemical Reactions. Yu Zou Ioannis G. Kevrekidis Department of Chemical Engineering and PACM Princeton University. SIAM Annual Meeting, Boston July 12, 2006. Outline. Stochastic Catalytic Reactions Uncertainty Quantification - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Equation-Free Uncertainty Quantification on Stochastic Chemical Reactions

SIAM Annual Meeting, Boston

July 12, 2006

Yu Zou Ioannis G. Kevrekidis Department of Chemical Engineering and PACM

Princeton University

Page 2: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Outline

• Stochastic Catalytic Reactions

• Uncertainty Quantification

• Equation-Free Uncertainty Quantification

• Numerical Results

• Conclusions and Remarks

Page 3: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Input(random parameter)

Response

System

Stochastic Catalytic Reactions

A (CO) +1/2 B2 (O2) → AB (CO2)

CO O2 CO2

vacancy

*

*

*

( , , , )

( , , , )

1

AA A B

BB A B

A B

df

dtd

fdt

: random parameter set

parameter response ),( t

Page 4: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Uncertainty Quantification

Monte Carlo Simulation

( , )ξ t

ξ

Stochastic Galerkin (Polynomial Chaos) Method (Ghanem and Spanos, 1991)

0

( , ) ( ) ( )

P

i i

i

ξ t a t ξ

ijji )(),(

0( , ), (0) d

fdt

( ), ( ) ( ) ( ) ( )

g h g h p d

+ 0( ), (0)

d

Fdt

0 1( , ,..., ) TPa a a

• exponential convergence rate• model reduction• correlation between parameter and solution• F(Θ) ?

parameter response ),( t

• convergence rate ~ O(1/M1/2)• time-consuming

Page 5: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Equation-Free Uncertainty Quantification

Θ(t)

θ(ξ,t)

0

( , ) ( ) ( )

P

i i

i

ξ t a t ξ

lifting

( , )d

fdt

θ(ξ,t+Δt)

Θ(t+Δt)

micro-simulation

restriction ( , ), ( )( )

( ), ( )

i

ii i

ta t

Equation Free: Quantities estimated on demand (Kevrekidis et al., 2003, 2004)

( ) ( )

d t t t

dt t

θ(ξ,t): mean coverages of reactants

in catalytic reactions A (CO) +1/2 B2 (O2) → AB (CO2)

*

2*

/

/ 2

A A r A B

B r A B

d dt k

d dt k

x

NA(t), NB(t), N*(t) NA(t+Δt), NB(t+Δt), N*(t+Δt)

lifting NA=int(NtotθA)+1 with pA1

int(NtotθA) with pA0

The same to NB

θA=<NA>/Ntot

θB=<NB>/Ntot

restriction

Time-stepper

Gillespie

1

4

1

i

i

r

r

p1

2

4

1

i

i

r

r

3

4

1

i

i

r

r

4

4

1

i

i

r

r

reaction time Gillespie Algorithm

jik

ji

ii

jiji

ii

gABBA

gAA

BBgB

AgA

r

)(

)(

)(

)(

,,

,

,,2

,

1 *

22 *

3

4

tot

tot

A tot

r A B tot

r N

r N

r N

r k N

4

12 /ln

iirp

Page 6: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Equation-Free Uncertainty Quantification

Projective Integration (Kevrekidis et al., 2003, 2004)

gPC coefficients

Mean coverages

Number of sites

gPC coefficients

Mean coverages

Number of sites

gPC coefficients

Mean coverages

Number of sites

gPC coefficients

Mean coverages

Number of sites

gPC coefficients

Mean coverages

Number of sites

lifting

lifting restriction

restriction

restriction

restriction

restriction

restriction lifting

lifting

integrate

Δtf

Δtcc(adaptive stepsize control)

Random Steady-state Computation(Kevrekidis et al., 2003, 2004)

gPC coefficients

Mean coverages

Number of sites

gPC coefficients

Mean coverages

Number of sites

lifting

lifting

restriction

restriction

T

ΦT

Θ=ΦT(Θ)

• Newton’s Method• Newton-Krylov GMRES (Kelly, 1995)

Δts(≥trelaxation+hopt)

Page 7: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Numerical Results

α=1.6, γ= 0.04, kr=4, β=6.0+0.25ξ, ξ~U[-1,1]gPC coefficients computed by ensemble average

Number of gPC coefficients: 12

Ne of θA, θB and θ* : 40,000

Ne of NA, NB and N*: 1,000Ntot of surface sites: 2002

Projective Integration

Page 8: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Numerical Results

Projective Integration

Page 9: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Numerical Results

Projective Integration

α=1.6, γ = 0.04, kr=4, β=6.0+0.25ξ, ξ~U[-1,1]gPC coefficients computed by Gauss-LegendrequadratureNumber of gPC coefficients: 12

Ne of θA, θB and θ* : 200Ne of NA, NB and N*: 1,000Ntot of surface sites: 2002

Page 10: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Numerical Results

Random Steady-State Computation

α=1.6, γ= 0.04, kr=4β=<β>+0.25ξ,, ξ~U[-1,1]gPC coefficients computed by ensemble average

Number of gPC coefficients: 12

Ne of θA, θB and θ* : 40,000

Ne of NA, NB and N*: 1,000Ntot of surface sites: 2002

Page 11: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Numerical Results

Random Steady-State Computation

α=1.6, γ = 0.04, kr=4β=<β>+0.25ξ,, ξ~U[-1,1]gPC coefficients computed byGauss-Legendre quadrature

Number of gPC coefficients: 12

Ne of θA, θB and θ* : 200Ne of NA, NB and N*: 1,000Ntot of surface sites: 2002

Page 12: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Conclusions and remarks

• An EF UQ approach involving three levels is proposed to quantify propagation of uncertainty for mean coverages in stochastic catalytic reactions.

• Computation of random steady states near turning zones should be treated carefully. In the discrete simulations, relationship functions of the parameter and response may not be continuous. More work needs to be done along this line.

• Possible extension to situations with multiple random parameters – Quasi Monte Carlo or other efficient sampling techniques.

Reference

Yu Zou and Ioannis G. Kevrekidis, Equation-Free Uncertainty Quantification on Heterogeneous Catalytic Reactions, in preparation, available at http://arnold.princeton.edu/~yzou/

Page 13: Equation-Free Uncertainty Quantification on  Stochastic Chemical Reactions

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Thanks!