Polynomial Chaos Surrogate Construction for Stochastic Models with Parametric Uncertainty

Abstract

Mesoscopic modeling of surface chemistry bridges atomistic and continuum modeling of surface kinetics. Studying these models, typically simulated via kinetic Monte-Carlo (KMC) methods, is crucial for understanding events relevant in surface catalysis, such as species clustering or localized coverage-dependent processes. Further, many of the underlying chemical rate constants are uncertain, with consequential impact on model predictions. The stochastic evolution of the state of the system can be described using Gillespie's stochastic simulation algorithm. Due to the computational expense of obtaining converged estimates of the system state, particularly when including dependence on uncertain rate parameters, uncertainty quantification (UQ) and global sensitivity analysis (GSA) in such systems typically require construction of accurate spatio-temporal surrogates. Development of such a surrogate is further challenged by the large number of input parameters (reaction rates) as well as the dimensionality of the system (number of species and the lattice size of the surface).

Motivated by KMC application in surface kinetics, we develop a general methodology for constructing surrogate representations for high-dimensional stochastic fields. Specifically, we employ the Karhunen-Lo?ve (KL) decomposition for representing spatial variability with respect to both parametric and intrinsic stochasticity of the system. Furthermore, polynomial chaos is employed in conjunction with the Rosenblatt transformation to develop representations for the coefficients corresponding to KL eigen-directions.

The resulting KL-PC surrogate efficiently maps the uncertain rate parameters of a chemical system to random field quantities of interest associated with the surface reactions. Notably, the surrogate representation is in a joint space of both the parametric uncertainty and intrinsic noise allowing efficient re-sampling of the stochastic fields, as well as derivation of expectations of output quantities of interest enabling efficient UQ and GSA. We demonstrate the technique on a few relevant surface reaction models by comparing the surrogate predictions to the underlying KMC results and examining the sensitivity of the output quantities with respect to model parameters.