1. Equation-Free (EF) Uncertainty Quantification (UQ): Techniques and Applications Ioannis Kevrekidis and Yu Zou Princeton University September 2005

2. Background for Uncertainty Quantification Uncertain Phenomena in science and engineering ◊Inherent Uncertainty: Uncertainty Principle of quantum mechanics, Kinetic theory of gas, … ◊Uncertainty due to lack of knowledge: randomness of BC, IC and parameters 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 random 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 Monte Carlo, Latin Hypercube Sampling, Quadrature/Cubature rules,… ◊Non-sampling methods (intrusive) : Second-order analysis, higher-order moment analysis, stochastic Galerkin method, …

3. Dynamical Systems and Their Representation Model strong correlation poor correlation  P deterministic system stochastic system

4. 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 a j ’s are PC coefficients, Ψ j ’s are orthogonal polynomial functions with =0. if i≠j. The inner product is defined as is a probability measure of. Notes ◊ The coefficients a j fully determines. ◊ Selection of Ψ j is dependent on the probability measure or distribution of, e.g., if is a Gaussian measure, then Ψ j are Hermite polynomials and this leads to the Wiener-Hermite polynomial chaos (Homogeneous Chaos) expansion if is a Lesbeque measure, then Ψ j are Legendre polynomials

5. 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 f j (t), j=1,2,…,M Model Input: IC, BC, Parameters Response: Solution

6. Stochastic Galerkin Method Solution technique: Galerkin projection resulting in coupled ODE’s for f j (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.

7. 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 LiftingRestriction Micro Simulation Evaluation of temporal derivative

8. Equation-Free Analysis on ODE’s EF techniques ◊ Coarse Projective Integration ◊ Coarse Steady-State Analysis ◊ Coarse Bifurcation, Coarse Dynamic Renormalization, etc. LiftingRestriction

9. Equation-Free Galerkin-Free Uncertainty Quantification Principle Assuming smoothness of Polynomial Chaos coefficients in a large temporal 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 sampling,…): ◊ Microsimulation: ◊ Restriction: For standard Monte Carlo sampling, For quadrature/cubature points sampling, is the weight associated with each sampling point.

10. Equation-Free Galerkin-Free Uncertainty Quantification Domain of applications corresponding to EF analysis on ODE’s EF analysis on ODE’sEF 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

11. Example 1: Continuous Stirred-tank Chemical Reactor x 1 : the conversion x 2 : the dimensionless temperature D a : Damkoehler number B: heat of reaction β: heat transfer coefficient γ: activation energy x 2c : coolant temperature Sampling technique in lifting: standard Monte Carlo Evolution of x 1, B=22, D a =0.07, β=3 ( 1+0.1 ξ) Evolution of x 1, B=22,β=3 D a varies

12. Example 1: Continuous Stirred-tank Chemical Reactor Random steady states of x 1, B=22,β=3(1+0.1 ξ), D a varies Random steady states of x 1, B=22,β=3,D a = (1+0.1 ξ), varies

13. Example 1: Continuous Stirred-tank Chemical Reactor Evolution of x 1, B=22,β=3 D a varies B=22,β=3, D a ~(0.083,0.084)

14. Example 2: CO oxidation in a P t surface Chemical reaction: A(CO)+1/2B 2 (O 2 ) -> AB (CO 2 ) Model: θ A : mean coverage of A θ B : mean coverage of B θ * : mean coverage of vacant sites α=1.6; γ=0.04; k r = 4; β=6+0.25 ξ Sampling method 1: standard Monte Carlo (Ne=40,000)

15. Example 2: CO oxidation in a P t surface Sampling method 2: Gauss-Legendre quadrature points (Ne=200)

16. Example 2: CO oxidation in a P t surface Standard Monte Carlo (Ne=40,000) β= (1+0.05 ξ) G-L quadrature (Ne=200) β= (1+0.05ξ)