Tim Sullivan

ECMath Colloquium

ECMath Colloquium

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

Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors

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 #cauchy-distribution

Free University of Berlin

Stochastik I at FU Berlin

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.

Published on Monday 17 October 2016 at 08:00 UTC #stochastik-1 #fu-berlin

UQ Talks: Jon Cockayne

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 [1].

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.

[1] 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

Published on Monday 3 October 2016 at 10:00 UTC #event #uq-talk #prob-num

Cameron-Martin theorems for Cauchy-distributed random sequences

Preprint: Cameron-Martin theorems for Cauchy-distributed random sequences

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv. Inspired by questions of prior robustness left unanswered in this earlier paper on α-stable Banach space priors, this paper addresses the basic probabilistic question: when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation? In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite. We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\). More precisely, if the Cauchy distribution on the nth term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| < \infty \)

and, with the usual convention that \(0 \log 0 = 0\),

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \log \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| \right| < \infty . \)

We also discuss similar results for dilation of the scale parameters, i.e. \(\gamma_{n} \mapsto \sigma_{n} \gamma_{n}\) for some real sequence \(\sigma = (\sigma_n)_{n = 1}^{\infty}\).

Published on Monday 15 August 2016 at 10:00 UTC #publication #preprint

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