Maximilan Koschade presents his Semester thesis on "Polynomial Chaos Expansion and Active Bayesian Machine Learning for Uncertainty Quantification in the Context of Stochastic Partial Differential Equations"
Abstract:
We first introduce the basics of both intrusive and non-intrusive Polynomial Chaos and spectral expansion methods for uncertainty quantification within the context of Stochastic Partial Differential Equations. We continue by by treating the inference of the Polynomial Chaos coefficients as a supervised learning prob- lem within the Bayesian framework, introduce and review literature to both basic aspects as well as state of the art sparse and active learning. The Sparse Bayesian Learning framework is compared to and benchmarked against deterministic and more established sparse regression methods like Least Angel Regression. Fur- thermore we evaluate the feasibility of Polynomial Chaos based surrogate models for approximation of the likelihood function within the Bayesian inverse problem setting.