Multilevel Monte Carlo for noise estimation in stochastic multiscale systems

The final publication is available at Elsevier via https://doi.org/10.1016/j.cherd.2018.10.006� 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The purpose of this study is to adapt Multilevel Monte Carlo (MLMC) sampling technique for random noise estimation in stochastic multiscale systems and evaluate the performance of this method. The system under consideration was a simulation of thin film formation by chemical vapour deposition, where a kinetic Monte Carlo solid-on-solid model was coupled with partial differential equations that represented mass, energy and momentum transport. The noise in the expected value of the system�s observable (film roughness) was estimated using MLMC and standard Monte Carlo (MC) sampling. The MLMC technique achieved conservative estimates of noise in the observable at an order of magnitude lower computational cost than standard MC sampling. This study highlights the nuances of adapting the MLMC technique to the stochastic multiscale system and provides insight on the benefits and challenges of using MLMC for noise estimation in stochastic multiscale systems.Natural Sciences and Engineering Research Council of Canad

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with MATLAB implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community

On the Techniques for Efficient Sampling, Uncertainty Quantification and Robust Control of Stochastic Multiscale Systems

In order to better understand and leverage natural phenomena to design materials and devices (e.g. biomedical coatings, catalytic reactors, thin conductive films for microprocessors, etc.), stochastic multiscale models have been developed that explicitly model the interactions and feedbacks between the electronic, atomistic/molecular, mesoscopic and macroscopic scales. These models attempt to use the accurate results from the fine scales to inform industrially relevant domain sizes and thereby improve product quality through optimal control actions during industrial manufacturing. However, the presence of stochastic calculations increases the computational cost of such modeling approaches and makes their direct application in uncertainty quantification, optimization and online control challenging. Uncertainty cannot be ignored from simulations, otherwise there will be model-plant mismatch and loss in performance. The added computational intensity necessitates the development of more efficient computational methods that can leverage the accurate predictions of stochastic multiscale models in the industrial setting where accuracy, efficiency and speed are of utmost importance. A lot of research has been done in the area of stochastic multiscale models over the past few decades, but some gaps in knowledge remain. For instance, the performance of traditional uncertainty quantification techniques such as power series (PSE) and polynomial chaos expansions (PCE) has not been compared in the context of stochastic multiscale systems. Furthermore, a novel sampling technique called Multilevel Monte Carlo (MLMC) sampling emerged from the field of computational finance with the aim of preserving accuracy of estimation of model observables while decreasing the required computational cost. However, its applications in the field of chemical engineering and in particular for stochastic multiscale systems remain limited. Also, the advancements in computing power caused the usefulness of machine learning methods such as Artificial Neural Networks (ANNs) to increase. Because of their flexibility, accuracy and computational efficiency, ANNs are experiencing a resurgence of research interest, but their application for stochastic multiscale chemical engineering systems are still limited at the moment. This thesis aims to fill the identified gaps in knowledge. The results of the conducted research indicate that PCE can be more computationally efficient and accurate than PSE for stochastic multiscale systems, but it may be vulnerable to the effects of stochastic noise. MLMC sampling provides an attractive advantage over the heuristic methods for uncertainty propagation in stochastic multiscale systems because it allows to estimate the level of noise in the observables. However, the stochastic noise imposes a limit on the maximum achievable MLMC accuracy, which was not observed for continuous systems that were originally used in MLMC development. ANNs appear to be a very promising method for online model predictive control of stochastic multiscale systems because of their computational efficiency, accuracy and robustness to large disturbances not seen in the training data