Bayesian optimal experimental design in inverse problems: applications and stability

Abstract

The focus of this thesis is in the area of Bayesian D-Optimal Experimental Design (DOED) for Bayesian inverse problems. Here, we introduce some theoretical results of stability and applied some standard numerical algorithms in a study case of a chromatography model. Our first main contribution is an analysis of stability with respect to the expected utility function. This analysis includes results about perturbations with respect to the likelihood function for non-Gaussian noise problems, with emphasis on the converge ratio when the likelihood function is approximated using a surrogate model. Additionally, we obtained results about the convergence of maximizers in the expected utility function using Gamma-convergence theory. Some numerical simulations corroborated the stability results.

Our second main contribution is an analysis of Bayesian D-OED in the area of chromatography modeled with nonlinear partial differential equations, specifically for the equilibrium dispersive model with the Langmuir isotherm model. We defined the injection time and initial concentration as design variables and we found ideal values to estimate some parameters with small uncertainty in the structure of the partial differential equation. Additionally, we accelerated the calculus by introducing a technique of a surrogate model based on a piecewise sparse linear interpolation method.