Past lectures

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Abstract

Common examples for non-linear inverse problems range from parameter identification in PDEs to tomography and data assimilation problems. They naturally involve high- or infinite dimensional parameter spaces and appropriate statistical noise models lead to a class of non-convex inference problems that present substantial challenges in contemporary data science. In influential work, Andrew Stuart (2010) has proposed a unified Bayesian approach to solve such problems. It is computationally feasible via Gaussian process priors and high-dimensional MCMC algorithms and provides important uncertainty quantification methodology ('error bars' or confidence regions) based on posterior distributions. Despite evident empirical success, the theoretical understanding of the performance of such methods has been limited until recently. Specifically in non-linear settings Bayesian methods are distinct from optimisation based algorithms and their analysis requires a very different set of mathematical ideas.

In these lectures we will summarise recent developments that allow to give rigorous statistical and computational guarantees for the use of these algorithms in high-dimensional and non-convex settings. The general theory will be illustrated in two non-linear model examples arising with elliptic partial differential equations.

A standard background in probability and measure, statistics and real analysis will be sufficient to follow this course.