Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Bayesian probabilistic numerical methods” to the arXiv. Following on from our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems”, our aim is to provide some rigorous theoretical underpinnings for the emerging field of probabilistic numerics, and in particular to define what it means for such a method to be “Bayesian”, by connecting with the established theories of Bayesian inversion and disintegration of measures.
Abstract. The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented.
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.
Next week's colloquium at the Einstein Center for Mathematics Berlin will be on the topic of Optimisation. The speakers will be:
- Sebastian Sager (Magdeburg): Mathematical Optimization for Clinical Diagnosis and Decision Support
- Werner Römisch (HU Berlin): Stochastic Optimization: Complexity and Numerical Methods
- Karl Kunisch (Graz): Sparsity in PDE-constrained Open and Closed Loop Control
Time and Place. Friday 20 January 2017, 14:00–17:00, Humboldt-Universität zu Berlin, Main Building Room 2.097, Unter den Linden 6, 10099 Berlin
Published on Tuesday 10 January 2017 at 12:00 UTC #event
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 \( C \geq 1 \).
This semester, Winter Semester 2016–2017, I will be teaching the third-semester course Stochastik I for mathematics bachelors' degree students at the Free University of Berlin. Exercise sheets, announcements, etc. for this course will all be posted on this page, as well as on the official FU Berlin webpages such as KVV.