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.
Next week Jon Cockayne (University of Warwick) will give a talk on “Probabilistic Numerics for Partial Differential Equations”.
Time and Place. Friday 14 October 2016, 12:00–13:00, ZIB Seminar Room 2006, Zuse Institute Berlin, Takustraße 7, 14195 Berlin
Abstract. Probabilistic numerics is an emerging field which constructs probability measures to capture uncertainty arising from the discretisation which is often necessary to solve complex problems numerically. We explore probabilistic numerical methods for Partial differential equations (PDEs). We phrase solution of PDEs as a statistical inference problem, and construct probability measures which quantify the epistemic uncertainty in the solution resulting from the discretisation .
We analyse these probability measures in the context of Bayesian inverse problems, parameter inference problems whose dynamics are often constrained by a system of PDEs. Sampling from parameter posteriors in such problems often involves replacing an exact likelihood with an approximate one, in which a numerical approximation is substituted for the true solution of the PDE. Such approximations have been shown to produce biased and overconfident posteriors when error in the forward solver is not tightly controlled. We show how the uncertainty from a probabilistic forward solver can be propagated into the parameter posteriors, thus permitting the use of coarser discretisations while still producing valid statistical inferences.
 Jon Cockayne, Chris Oates, Tim Sullivan, and Mark Girolami. “Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems.” arXiv preprint, 2016. arXiv:1605.07811