1. Princeton University 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

2. Department of Chemical Engineering and PACM Outline Stochastic Catalytic Reactions Uncertainty Quantification Equation-Free Uncertainty Quantification Numerical Results Conclusions and Remarks

3. Princeton University Department of Chemical Engineering and PACM Input (random parameter) Response System Stochastic Catalytic Reactions A (CO) +1/2 B 2 (O 2 ) → AB (CO 2 ) COO2O2 CO 2 vacancy : random parameter set parameterresponse

4. Princeton University Department of Chemical Engineering and PACM Uncertainty Quantification Monte Carlo Simulation Stochastic Galerkin (Polynomial Chaos) Method (Ghanem and Spanos, 1991) + exponential convergence rate model reduction correlation between parameter and solution F(Θ) ? parameterresponse convergence rate ~ O(1/M 1/2 ) time-consuming

5. Princeton University Department of Chemical Engineering and PACM Equation-Free Uncertainty Quantification Θ(t) θ(ξ,t) lifting θ(ξ,t+Δt) Θ(t+Δt) micro-simulation restriction 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 B 2 (O 2 ) → AB (CO 2 ) x N A (t), N B (t), N * (t)N A (t+Δt), N B (t+Δt), N * (t+Δt) lifting N A =int(N tot θ A )+1 with p A1 int(N tot θ A ) with p A0 The same to N B θ A = /N tot θ B = /N tot restriction Time-stepper Gillespie p1p1 reaction time Gillespie Algorithm

6. Princeton University Department 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 restriction lifting integrate ΔtfΔtf Δt cc (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 restriction T ΦTΦT Θ=ΦT(Θ)Θ=ΦT(Θ) Newton’s Method Newton-Krylov GMRES (Kelly, 1995) Δt s (≥t relaxation +h opt )

7. Princeton University Department of Chemical Engineering and PACM Numerical Results α=1.6, γ= 0.04, k r =4, β=6.0+0.25ξ, ξ~U[-1,1] gPC coefficients computed by ensemble average Number of gPC coefficients: 12 N e of θ A, θ B and θ * : 40,000 N e of N A, N B and N * : 1,000 N tot of surface sites: 200 2 Projective Integration

8. Princeton University Department of Chemical Engineering and PACM Numerical Results Projective Integration

9. Princeton University Department of Chemical Engineering and PACM Numerical Results Projective Integration α=1.6, γ = 0.04, k r =4, β=6.0+0.25ξ, ξ~U[-1,1] gPC coefficients computed by Gauss-Legendre quadrature Number of gPC coefficients: 12 N e of θ A, θ B and θ * : 200 N e of N A, N B and N * : 1,000 N tot of surface sites: 200 2

10. Princeton University Department of Chemical Engineering and PACM Numerical Results Random Steady-State Computation α=1.6, γ= 0.04, k r =4 β= +0.25ξ,, ξ~U[-1,1] gPC coefficients computed by ensemble average Number of gPC coefficients: 12 N e of θ A, θ B and θ * : 40,000 N e of N A, N B and N * : 1,000 N tot of surface sites: 200 2

11. Princeton University Department of Chemical Engineering and PACM Numerical Results Random Steady-State Computation α=1.6, γ = 0.04, k r =4 β= +0.25ξ,, ξ~U[-1,1] gPC coefficients computed by Gauss-Legendre quadrature Number of gPC coefficients: 12 Ne of θ A, θ B and θ * : 200 N e of N A, N B and N * : 1,000 N tot of surface sites: 200 2

12. Princeton University Department 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/

13. Princeton University Department of Chemical Engineering and PACM Thanks!