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
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
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
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
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
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)
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
Princeton University
Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM
Numerical Results
Projective Integration
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
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
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
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/
Princeton University
Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM
Thanks!