Learning theory & Gaussian Process Regression for surrogate modeling, and a novel framework for Design Optimization under uncertainty. Application to an early-stage aircraft wing design

The aim of this thesis is to study the problem of regression for surrogate modeling, to develop a novel framework for design optimization that makes no assumptions on the model, and to give guidelines to apply the aforementioned theory to an early-stage aircraft wing design. The first part provides a summary of the mathematical foundations of learning theory for regression. The theory of Reproducing Kernel Hilbert Spaces is broadly covered. In addition, Gaussian Process regression is explained in detail. The second part introduces a novel framework for design optimization. Sampling from a probability distribution is at the core of this framework. Therefore,algorithms for simulating different distributions are described in detail. Furthermore, rejection sampling, the theory of Markov chains,and Metropolis-Hasting are explained as methods for simulating arbitrary distributions. The framework aims to optimize the parameters of a Probability Density Function in the input space of a surrogate model in order to satisfy prescribed performance in the output. Stochastic Optimization is suggested as the optimization process and a description of Simulated Annealing is included. This MPhil is part of a project in collaboration with Airbus. They provided a dataset with the goal of optimizing the jig twist of an aircraft wing. The last part analyzes this data and provides future researchers in the project with guidelines to train a Gaussian Process and apply the novel framework mentioned above to tackle the optimization proble