Equation-Free Uncertainty Quantification on Stochastic Chemical Reactions

1. 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

2. Outline • Stochastic Catalytic Reactions • Uncertainty Quantification • Equation-Free Uncertainty Quantification • Numerical Results • Conclusions and Remarks

3. A (CO) +1/2 B2 (O2) → AB(CO2) vacancy CO O2 CO2 : random parameter set System Response Input (random parameter) parameter response Stochastic Catalytic Reactions

4. parameter response Stochastic Galerkin (Polynomial Chaos) Method (Ghanem and Spanos, 1991) + Uncertainty Quantification Monte Carlo Simulation • convergence rate ~ O(1/M1/2) • time-consuming • exponential convergence rate • model reduction • correlation between parameter and solution • F(Θ) ?

5. Equation Free: Quantities estimated on demand (Kevrekidis et al., 2003, 2004) θ(ξ,t):mean coverages of reactants in catalytic reactions A (CO) +1/2 B2 (O2) → AB(CO2) lifting restriction NA=int(NtotθA)+1 with pA1 int(NtotθA) with pA0 The same to NB θA=<NA>/Ntot θB=<NB>/Ntot NA(t), NB(t), N*(t) NA(t+Δt), NB(t+Δt), N*(t+Δt) Equation-Free Uncertainty Quantification Time-stepper Θ(t+Δt) Θ(t) lifting restriction micro-simulation θ(ξ,t) x θ(ξ,t+Δt) Gillespie Gillespie Algorithm p1 reaction time

6. Random Steady-state Computation (Kevrekidis et al., 2003, 2004) T Number of sites Number of sites lifting restriction ΦT Mean coverages Mean coverages lifting restriction gPC coefficients gPC coefficients Equation-Free Uncertainty Quantification Projective Integration (Kevrekidis et al., 2003, 2004) Δtf Number of sites Number of sites Number of sites Number of sites Number of sites lifting restriction restriction restriction lifting Mean coverages Mean coverages Mean coverages Mean coverages Mean coverages lifting lifting restriction restriction restriction integrate gPC coefficients gPC coefficients gPC coefficients gPC coefficients gPC coefficients Δts(≥trelaxation+hopt) Δtcc(adaptive stepsize control) Θ=ΦT(Θ) • Newton’s Method • Newton-Krylov GMRES • (Kelly, 1995)

7. 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 ensemble average Number of gPC coefficients: 12 NeofθA, θBandθ*:40,000 Neof NA, NB and N*: 1,000 Ntotof surface sites: 2002

8. Numerical Results Projective Integration

9. 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-Legendre quadrature Number of gPC coefficients: 12 NeofθA, θBandθ*:200 Neof NA, NB and N*: 1,000 Ntotof surface sites: 2002

10. Numerical Results Random Steady-State Computation α=1.6, γ= 0.04, kr=4 β=<β>+0.25ξ,, ξ~U[-1,1] gPC coefficients computed by ensemble average Number of gPC coefficients: 12 NeofθA, θBandθ*:40,000 Neof NA, NB and N*: 1,000 Ntotof surface sites: 2002

11. Numerical Results Random Steady-State Computation α=1.6, γ= 0.04, kr=4 β=<β>+0.25ξ,, ξ~U[-1,1] gPC coefficients computed by Gauss-Legendre quadrature Number of gPC coefficients: 12 Ne ofθA, θBandθ*:200 Neof NA, NB and N*: 1,000 Ntotof surface sites: 2002

12. 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/

13. Thanks!