Uncertainty analysis and robust optimization of multiscale process systems with application to epitaxial thin film growth

Multiscale modeling of materials growth involves inherently coupled processes that span over a wide range of time and length scales. As a common practice, a multiscale model is adopted to simulate the thin film deposition process that augments Partial Differential Equations (PDEs), describing the macro-scale phenomena, with a high-order lattice-based kinetic Monte Carlo (KMC) simulator, which aims to capture thin film microstructure. Although such a model is a fair representation of the system, the evolution of thin film encompasses processes that are subject to model parameter uncertainty that can significantly affect the control and optimization objectives of this process, e.g. film׳s roughness and thickness. Thus, to provide a robust and more realistic strategy, it is crucial to perform an uncertainty analysis for this process. This work explores a systematic framework to analyze model parameter uncertainty for robust control and optimization of multiscale models. Such an analysis is extremely challenging due to (i) the lack of a closed formulation between the process optimization objective (i.e. film thickness) and the model parameters and (ii) the computational costs incurred to model the fine-scale events, typically performed using KMC simulation. To tackle these challenges, Power Series Expansion (PSE) is employed to analyze model uncertainty propagation. The PSE-based method for uncertainty analysis is more computationally efficient in comparison to the traditional Monte Carlo approach. The distributional uncertainty in rates of microscopic events is characterized by series expansions of the uncertain parameters. The fitted distributions to the rate of events at a given confidence level are employed to estimate upper and lower bounds on the desired outputs (e.g. film roughness). The computational efficiency of the approach is achieved by employing multiple reduced-order lattices in the KMC simulator. The potential application of the proposed method is illustrated through an optimization problem that aims to specify the robust optimal substrate temperature profile that maximizes the endpoint thin film thickness in the presence of uncertainty.