(110b) Multilevel Monte Carlo Sampling for Thin Film Deposition Under Uncertainty | AIChE

(110b) Multilevel Monte Carlo Sampling for Thin Film Deposition Under Uncertainty


Kimaev, G. - Presenter, University of Waterloo
Ricardez-Sandoval, L. A., University of Waterloo
Deterministic systems can exhibit variability in their observables due to uncertainties in system parameters and/or model structure. However, stochastic systems exhibit inherent noise in their observables even without these sources of uncertainty. This noise contributes to the variability in the observables of stochastic systems subject to uncertainty. Efficient methods of parametric uncertainty quantification, such as Power Series and Polynomial Chaos Expansions (PSE and PCE, respectively) [1], rely on the knowledge of the observable values at specific realizations of the uncertain parameters. When these techniques are applied to stochastic systems for the design of robust processes, uncertainty quantification is complicated by the inherent noise in the observables because the expected values of the observables have to be obtained prior to constructing the PSE and PCE. It has been common practice to use heuristic rules regarding the necessary number of samples to arrive at the expected values. However, a recently developed computational technique called Multilevel Monte Carlo (MLMC) sampling [2] can obtain such estimates by assessing the variability of the estimates of varying accuracy instead of relying on pure heuristics. MLMC uses many coarse inexpensive samples to find an initial estimate of the observable and then refines that estimate with progressively fewer samples of higher accuracy and computational cost. MLMC originated from computational finance and has been applied to various other fields.

In our previous works, we applied MLMC to estimate the observables of classic chemical engineering systems subject to parametric uncertainty [3] and of a stochastic multiscale system of Chemical Vapour Deposition (CVD) [4] where the variability in the observables was due to noise rather than uncertainty [5]. In both [3] and [5], we demonstrated that MLMC can be used to accurately estimate the expected values of the observables in a fraction of the time required for conventional Monte Carlo sampling. However, for stochastic multiscale systems, parametric uncertainty, rather than stochastic noise, is the greater contributor to the variability in the observables, and such uncertainty was not considered in [5]. To the best of our knowledge, an application of MLMC for uncertainty quantification in a stochastic multiscale system is absent from the literature.

In our previous work [6], we relied on heuristics to obtain the expected values of the observables from the stochastic multiscale CVD model [4] and subsequently employed that data to construct PSE and PCE for efficient uncertainty quantification [6]. It was found that the stochastic noise negatively affected the ability of PCE to approximate the expected values of the observables and that more data was necessary to improve the accuracy. In this work, using a methodology similar to [5], we apply MLMC to a stochastic multiscale model of a heterogeneous catalytic flow reactor system [7] to obtain its expected observables and use them to develop PSE and PCE expressions. The expressions are then employed to conduct parametric uncertainty quantification. The PSE and PCE expressions are also derived using traditional heuristics. The study compares the results of PSE and PCE derived using both techniques to the results of Monte Carlo sampling and establishes whether the use of MLMC instead of heuristics can decrease the variability in PSE sensitivities and PCE coefficients and consequently provide more accurate estimates of the probability distributions of the observables of the catalytic flow reactor system under uncertainty.


[1] Eldred, M.S. Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design, In Proceedings of the 50th 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] Kimaev, G. and Ricardez-Sandoval, L. A. Multilevel Monte Carlo for noise estimation in stochastic multiscale systems, Chem. Eng. Res. Des., vol. 140, pp. 33–43, 2018.

[6] Kimaev, G. and Ricardez-Sandoval, L. A. A comparison of efficient uncertainty quantification techniques for stochastic multiscale systems, AIChE Journal, vol. 63, no. 8, pp. 3361–3373, 2017.

[7] Chaffart, D., Rasoulian, S. and Ricardez-Sandoval, L. A. Distributional Uncertainty Analysis and Robust Optimization in Spatially Heterogeneous Multiscale Process Systems, AIChE J., vol. 62, no. 7, pp. 2374–2390, 2016.


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