### Heavy-tailed stable priors in Inverse Problems and Imaging

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.

Published on Wednesday 19 July 2017 at 15:30 UTC #publication #inverse-problems #cauchy-distribution

### Inverse Problems Summer School at the Alan Turing Institute

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 (Edinburgh 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

Published on Friday 23 June 2017 at 09:00 UTC #event #inverse-problems

### Ingmar Schuster Joins the UQ Group

It is a pleasure to announce that Ingmar Schuster will join the UQ research group as a postdoctoral researcher with effect from 15 June 2017. He will be working on project A06 “Enabling Bayesian uncertainty quantification for multiscale systems and network models via mutual likelihood-informed dimension reduction” as part of SFB 1114 Scaling Cascades in Complex Systems.

Published on Thursday 15 June 2017 at 08:00 UTC #group #sfb1114

### Preprint: Computing with dense kernel matrices at near-linear cost

Florian Schäfer, Houman Owhadi and I have just uploaded a preprint of our latest paper, “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” to the arXiv. This paper builds upon the probabilistic-numerical ideas of “gamblets” (elementary gables upon the solution of a PDE) introduced by Owhadi (2016) to provide near-linear cost \(\varepsilon\)-approximate compression, inversion and principal component analysis of dense kernel matrices, the entries of which come from Green's functions of suitable differential operators.

**Abstract.** Dense kernel matrices \(\Theta \in \mathbb{R}^{N \times N}\) obtained from point evaluations of a covariance function \(G\) at locations \(\{x_{i}\}_{1 \leq i \leq N}\) arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions elliptic boundary value problems and approximately equally spaced sampling points, we show how to identify a subset \(S \subset \{ 1,\dots, N \} \times \{ 1,\dots,N \}\), with \(\#S = O(N \log(N)\log^{d}(N/\varepsilon))\), such that the zero fill-in block-incomplete Cholesky decomposition of \(\Theta_{i,j} 1_{(i,j) \in S}\) is an \(\varepsilon\)-approximation of \(\Theta\). This block-factorisation can provably be obtained in \(O(N \log^{2}(N)(\log(1/\varepsilon)+\log^{2}(N))^{4d+1})\) complexity in time. Numerical evidence further suggests that element-wise Cholesky decomposition with the same ordering constitutes an \(O(N \log^{2}(N) \log^{2d}(N/\varepsilon))\) solver. The algorithm only needs to know the spatial configuration of the \(x_{i}\) and does not require an analytic representation of \(G\). Furthermore, an approximate PCA with optimal rate of convergence in the operator norm can be easily read off from this decomposition. Hence, by using only subsampling and the incomplete Cholesky decomposition, we obtain at nearly linear complexity the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky decomposition we also obtain a near-linear-time solver for elliptic PDEs.

Published on Tuesday 13 June 2017 at 07:00 UTC #publication #preprint #prob-num

### Probabilistic numerics for PDE-constained inverse problems in MaxEnt

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 36 ^{th} 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 36 ^{th} 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

Published on Friday 9 June 2017 at 11:40 UTC #publication #prob-num #inverse-problems