Applications of Optimal Transport Theory to Process Optimization and Monitoring
Creators: Kammammettu, Sanjula
Publisher: name: University of Alberta Library
Year: 2024

<p>Process optimization and process monitoring form two major cornerstones of the field of process systems engineering. This thesis tackles selected problems from these foci through the lens of optimal transport theory, a powerful mathematical methodology that is receiving renewed attention in recent years. The optimal transport problem seeks to transport probability mass from one probability distribution to another at the least total cost. This thesis uses this underlying concept in three main ways. </p>

<p>Firstly, the optimal transport distance is used as a measure of similarity between probability distributions. This thesis explores the use of entropy-regularized optimal transport to accomplish optimal reduction of large datasets that may be further used for scenario-based stochastic optimization. Entropy regularization for optimal transport proves to be advantageous in this case due to the availability of a numerical iterative solution scheme, alleviating the curse of dimensionality encountered in large-dimensional optimization problems. This work is further extended to generate optimal scenario trees for multistage stochastic programming problems. Results from case studies demonstrated that the proposed algorithms provide an efficient, iterative method to reduce the computational burden in scenario-based stochastic optimization, while also preserving the solution quality.</p>

<p>Secondly, the optimal transport distance is used to construct ambiguity sets used in distributionally robust optimization. This thesis explores the use of optimal transport between Gaussian mixtures for distributionally robust optimization, which seeks to retain desirable features of both stochastic and robust optimization frameworks. In this work, the optimization problem is considered fraught with distributional ambiguity on multimodal uncertainty that is modeled as a Gaussian mixture. An optimal transport variant for Gaussian mixtures is further used to construct an ambiguity set of distributions around this reference model, and a tractable formulation is presented. The superior performance of this proposed formulation is contrasted with the established Wasserstein method on an illustrative study, as well as on a portfolio optimization problem. The thesis then uses the proposed formulation to tackle chance-constrained optimization in a distributionally robust setting, wherein the worst-case expected constraint violation is restricted to a user-defined limit. In a similar vein, this formulation is shown to outperform the conventional Wasserstein method.</p>

<p>Finally, optimal transport distance is used as a measure of similarity in a process monitoring framework. This thesis presents the applicability of the optimal transport distance as a metric for change-point and fault detection in multivariate processes and compares its performance with that of conventional fault detection metrics. The final component of this thesis tackles the fault detection problem through the lens of distributional ambiguity. In this work, distributional ambiguity is considered in the context of a multimodal process, and the worst-case performance of a fault detection system is evaluated on the basis of two performance metrics - false alarm rate, and fault detection rate. The evolution of worst-case performance metrics is tracked for varying levels of ambiguity, using the distributionally robust optimization formulation proposed in earlier chapters. The thesis concludes with a summary of the work conducted, the knowledge gaps addressed, and some future directions.</p>