Modul:   MAT870  Zurich Colloquium in Applied and Computational Mathematics

Bayesian inverse problems, Gaussian processes, and partial differential equations

Vortrag von Prof. Dr. Richard Nickl

Datum: 30.09.20  Zeit: 16.15 - 17.45  Raum: Online ZHACM

The Bayesian approach to inverse problems has become very popular in the last decade after seminal work by Andrew Stuart (2010) and collaborators. Particularly in non-​linear applications with PDEs and when using Gaussian process priors, this can leverage powerful MCMC methodology to tackle difficult high-​dimensional and non-​convex inference problems. Little is known in terms of rigorous performance guarantees for such algorithms. After laying out the main ideas behind Bayesian inversion, we will discuss recent progress providing both statistical and computational guarantees for these methods. We will touch upon issues such as how to prove posterior consistency and how to objectively validate posterior uncertainty quantification methods. A main focus will be on very recent results about mixing times of high-​dimensional Langevin dynamics that establish the polynomial time computability of posterior measures in some non-​linear model examples arising with PDEs.