Han Cheng Lie and I have just uploaded a preprint of our latest paper, “Equivalence of weak and strong modes of measures on topological vector spaces” to the arXiv. This addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in infinite-dimensional spaces, which are defined either strongly (a la Dashti et al. (2013), via a global maximisation) or weakly (a la Helin & Burger (2015), via a dense subspace): when are strong and weak modes equivalent?
Abstract. Modes of a probability measure on an infinite-dimensional Banach space \(X\) are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of \(X\), and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces.
The final version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has now been published online in Inverse Problems and Imaging; the print version will be available in October.
From 29 August–1 September 2017, the Alan Turing Institute will host a summer school on Mathematical Aspects of Inverse Problems organised by Claudia Schillings (Mannheim) and Aretha Teckentrup (Edingburgh and Alan Turing Institute), two of my former colleagues at the University of Warwick. The invited lecturers are:
- Masoumeh Dashti (Sussex): Bayesian Approach to Inverse Problems
- Michela Ottobre (Maxwell Institute, Edinburgh): Hamiltonian Monte Carlo
- Carola-Bibiane Schönlieb and Clarice Poon (Cambridge): A Comparison of Variational Methods and Deep Neural Networks for Inverse Problems
- Alison Fowler (Reading): The Value of Observations for Numerical Weather Prediction
Jon Cockayne, Chris Oates, Mark Girolami and I have just had our paper “Probabilistic numerical methods for PDE-constrained Bayesian inverse problems” published in the Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering. This paper complements our more extensive work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” and gives a more concise presentation of the main ideas, aimed at a general audience.
J. Cockayne, C. Oates, T. J. Sullivan & M. Girolami. “Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems” in Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. G. Verdoolaege. AIP Conference Proceedings 1853:060001-1–060001-8, 2017. doi:10.1063/1.4985359
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.