### UQ Talks: Hanne Kekkonen

In two weeks Hanne Kekkonen (University of Warwick) will give a talk on “Large noise in variational regularisation”.

**Time and Place.** Monday 12 June 2017, 11:00–12:00, ZIB Seminar Room 2006, Zuse Institute Berlin, Takustraße 7, 14195 Berlin

**Abstract.** We consider variational regularisation methods for inverse problems with large noise, which is in general unbounded in the image space of the forward operator.
We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operator.
As an example we study the particularly important cases of one- and *p*-homogeneous regularisation functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance.
As an example we study total variation prior.
This is joint work with Martin Burger and Tapio Helin.

Published on Wednesday 31 May 2017 at 16:00 UTC #event #uq-talk #inverse-problems

### Preprint: Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic numerical methods for PDE-constrained Bayesian inverse problems” to the arXiv. This paper is intended to complement our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” and to give a more concise presentation of the main ideas, aimed at a general audience.

Published on Wednesday 18 January 2017 at 12:00 UTC #publication #preprint #prob-num #inverse-problems

### Preprint: Bayesian inversion with heavy-tailed stable priors

A revised version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has been released on arXiv today. Among other improvements, the revised version incorporates additional remarks on the connection to the existing literature on stable distributions in Banach spaces, and generalises the results of the previous version of the paper to quasi-Banach spaces, which are like complete normed vector spaces in every respect except that the triangle inequality only holds in the weakened form

\( \| x + y \| \leq C ( \| x \| + \| y \| ) \)

for some constant \( C \geq 1 \).

Published on Monday 21 November 2016 at 11:30 UTC #publication #preprint #inverse-problems

### Preprint: Bayesian inversion with heavy-tailed stable priors

Just uploaded to the arXiv: “Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors”. This article builds on the function-space formulation of Bayesian inverse problems advocated by Stuart et al. to allow the prior to be heavy-tailed: not only may it not be exponentially integrable, as is the case for a Gaussian or Besov measure, it might not even have a well-defined mean, as in the case of the famous Cauchy distribution on \(\mathbb{R}\).

**Abstract.** This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (*Acta Numer.* 19:451–559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined.
It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures.
Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

Published on Friday 20 May 2016 at 09:00 UTC #publication #preprint #inverse-problems