(315e) Multilevel Monte Carlo Applied for Efficient Estimation of Observables in Multiscale Stochastic Systems

The characteristic trait of stochastic systems is the noise in their observables. Insufficient knowledge about the variability in the observables limits the ability to design processes robust to the uncertainty. The simplest way to accurately quantify the observables is to collect many samples from the system of interest and conduct a statistical analysis of the observed variables. However, this brute-force approach is resource- and/or time-intensive and thus decreases the utility of the results obtained from it, especially in time-sensitive situations. Many methods of efficient quantification of noise in the observables have been developed, such as Latin Hypercube sampling, Power Series and Polynomial Chaos Expansions [1]. A more recent computational technique is Multilevel Monte Carlo (MLMC) sampling [2], where many coarse inexpensive samples are used to obtain a crude estimate and assess its variability, and progressively fewer samples of increasing accuracy (higher discretization “levels”) are used to refine the approximation. MLMC has been developed in computational finance, and has since then been applied in many other fields.

In our previous work, we applied MLMC to uncertainty quantification in classic chemical engineering systems [3]. In this work, we extend and adapt MLMC to perform accurate estimation of observables in stochastic multiscale systems. We used as a case study a model of thin film formation by Chemical Vapour Deposition (CVD) [4], which was also previously used to quantify uncertainty and estimation of observables in multiscale systems [5],[6]. In this model, a kinetic Monte Carlo (kMC) solid-on-solid simulation represents the microscale formation of a thin film, while continuum mass, energy and momentum transfer equations are used to represent the macroscale supply of the chemical vapor to the substrate where the film forms. The non-closed-form microscale representation and the deterministic macroscale equations are coupled through a boundary condition and thus form a stochastic multiscale system.

Most of the studies that use MLMC, including our previous work [3], discretize the time domain of continuous equations and use finer discretization for higher levels of approximation accuracy. In the CVD system, the accuracy of approximation increases with larger lattice sizes of the kMC simulation. We established an empirical relationship that relates the kMC time step to the lattice size and used it to control the definition of higher accuracy levels in the MLMC framework. We demonstrate that MLMC can be used to accurately estimate the observables of this multiscale stochastic system in a fraction of the time required when conducting the conventional Monte Carlo sampling of the full multiscale simulation.

References

[1] Eldred, M.S., “Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design”, In Proceedings of the 50^th AIAA Structures, Structural Dynamics and Materials Conference, Palm Springs, California, 2009.

[2] Giles, M. “Multilevel Monte Carlo Path Simulation”, Operations Research, 56, 3, 2008.

[3] Kimaev, G. and Ricardez-Sandoval, L.A., “Multilevel Monte Carlo Applied to Chemical Engineering Systems Subject to Uncertainty”, AIChE Journal, 64, 5, 2018.

[4] Vlachos, D., “Multiscale Integration Hybrid Algorithms for Homogeneous–Heterogeneous Reactors”, AIChE Journal, 43, 11, 1997.

[5] Rasoulian, S. and Ricardez-Sandoval, L.A. “Uncertainty Analysis and Robust Optimization of Multiscale Process Systems With Application to Epitaxial Thin Film Growth”, Chemical Engineering Science, 116, 590, 2014.

[6] Rasoulian, S. and Ricardez-Sandoval, L.A. “Robust Multivariable Estimation and Control in an Epitaxial Thin Film Growth Process Under Uncertainty”, Journal of Process Control, 34, 70, 2015.