Bayesian inverse problems, Gaussian processes, and partial differential equations

Abstract: 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.

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Bio: Richard Nickl is Professor of Mathematical Statistics at the University of Cambridge. He is author of the book `Mathematical foundations of infinite-dimensional statistical models’ published in 2016 at Cambridge University Press, and among other things recipient of the 2017 Ethel Newbold Prize of the Bernoulli Society and the 2017 PROSE Award of the American Association of Publishers.”