Uncertainty Quantification for Atomistic Reaction Models: An Equation-Free Stochastic Simulation Algorithm Example

Abstract

We describe a computational framework linking Uncertainty Quantification (UQ) methods for continuum problems depending on random parameters with Equation-Free (EF) methods for performing continuum deterministic numerics by acting directly on atomistic/stochastic simulators. Our illustrative example is a heterogeneous catalytic reaction mechanism with an uncertain atomistic kinetic parameter; the "inner" dynamic simulator of choice is a Gillespie Stochastic Simulation Algorithm (SSA). We demonstrate UQ computations at the coarse-grained level in a nonintrusive way, through the design of brief, appropriately initialized computational experiments with the SSA code. The system is thus observed at three levels: (a) a fine scale for each stochastic simulation at each value of the uncertain parameter; (b) an intermediate coarse-grained state for the expected behavior of the SSA at each value of the uncertain parameter; and (c) the desired fully coarse-grained level: distributions of the coarse-grained behavior over the range of uncertain parameter values. The latter are computed in the form of generalized Polynomial Chaos (gPC) coefficients in terms of the random parameter. Coarse projective integration and coarse fixed point computation are employed to accelerate the computational evolution of these desired observables, to converge on random stable/unstable steady states, and to perform parametric studies with respect to other (nonrandom) system parameters.