  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.
