Bayesian uncertainty quantification in process modelling

Abstract

This thesis promotes the Bayesian approach to uncertainty quantification. It allows the consideration of a priori information about the estimated parameters and the estimation result is an a posteriori distribution that captures all the effects of non-linear dependency between observables and parameters. In contrast, the frequentist paradigm limits itself to the exclusive use of data and its very popular method of confidence ellipsoids assumes a linear dependency between observables and parameters. Despite these important drawbacks, the frequentist method has the practical advantage of being much less computationally demanding than the Bayesian counterpart. Even so, the improved modelling tools and Monte Carlo sampling techniques combined with the speed, memory, and parallel architecture of the present processors are sufficient ingredients for solving real-life Bayesian parameter estimation problems. But the combination of these ingredients is missing and it represents one of the main goals of this work. Another goal is to apply the nested sampling technique in the context of set-membership estimation and feasibility analysis in order to obtain an inner-approximation of a feasible set. The resulting methods are applicable to any type of mathematical model because only the value of the constraints is needed. Moreover, the methods consider both robustly and probabilistically -satisfied constraints hence they are applicable to a variety of feasible sets. The first two chapters offer an introduction to the role of statistics in the mathematical modelling exercise and recall the Bayesian paradigm advantages. Also, various Monte Carlo techniques for posterior distribution sampling are presented briefly and with a focus on the algorithms that are implemented in gPROMS. Chapter 3 describes the work performed to integrate seamlessly within gPROMS a Markov Chain Monte Carlo solver (BPEMC) and a Nested Sampling solver (BPENS) that sample the parameter posterior distribution and provide summary information about the marginal posterior distribution. Chapter 4 demonstrates through case studies the capability of the solvers to tackle practically relevant problems. The first case study – based on two datasets consisting of tens of steady state experiments – estimates five to eleven parameters for four alternative kinetic models of methanol and DME synthesis from syngas. The second case study focuses on the issue of parameter non-identifiability in case of a Goodwin biological oscillator, which is a challenging example of parameter estimation despite the small model and dataset size. Chapter 5 extends the nested sampling method applicability to characterization of feasible sets defined by robustly or probabilistically -satisfied constraints. Set-membership estimation is an example of problem with robustly-satisfied constraints. Stochastic flexibility analysis is an example of problem with probabilistically-satisfied constraints. Illustrative and more real-world cases are solved using our algorithms’ implementation in the Python package DEUS. The final chapter contains overall conclusions and future research directions.