Bayesian Approach to Inverse Problems
Masoumeh Dashti (University of Sussex, UK)

We consider the inverse problem of recovering an unknown parameter from a noisy and indirect vector of observations. We adopt a Bayesian approach and study how well-posed the solution in this framework is, in the case that the unknown is a functional parameter. We will then see how this approach is useful in quantifying the propagation of uncertainty under solution operators of partial differential equations.

Masoumeh Dashti is a lecturer in the Mathematics Department of the University of Sussex. Prior to that she was a PhD student and then a postdoctoral research fellow at the University of Warwick. Her research is in Bayesian approach to inverse problems and partial differential equations.



Hamiltonian Monte Carlo
Michela Ottobre (Heriot-Watt University, UK, Edinburgh)
The project will be split in two steps: 1) proof of some of the properties which lie at the root of HMC-type algorithms 2) some useful generalizations of the basic HMC algorithm. Appropriate material will be provided for both parts.

Michela Ottobre is a lecturer at the Maxwell Institute (Edinburgh). She has been Chapman Fellow at Imperial College and RA at Warwick University. Her interests concern several problems involving the interplay of probability, statistics, stochastic analysis and (functional) analysis.


A Comparison of Variational Methods and Deep Neural Networks for Inverse Problems
Carola-Bibiane Schoenlieb and Clarice Poon (University of Cambridge,  UK)

In the past decade, there has been a vast quantity of work in the use of variational methods for inverse problems. This has been fuelled by the observation that many natural images/signals are sparse. For example, they can be efficiently represented using sparse representation systems such as wavelets [1] or modelled using the space of bounded variation [2]. On the other hand, more recently, deep neural networks (DNN) have been successfully applied for a wide range of applications, including reconstruction problems in medical imaging [3,4,5]. The goal of this project is to investigate the use of deep neural networks for solving inverse problems, and to make some comparisons with the more classical approaches of variational methods.

[1] Mallat, S. (2008). A wavelet tour of signal processing: the sparse way. Academic press.
[2] Chambolle, A., Caselles, V., Cremers, D., Novaga, M., & Pock, T. (2010). An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery, 9(263-340), 227.
[3] Jin, K. H., McCann, M. T., Froustey, E., & Unser, M. (2016). Deep Convolutional Neural Network for Inverse Problems in Imaging. arXiv preprint arXiv:1611.03679.
[4] Adler, J., & Öktem, O. (2017). Solving ill-posed inverse problems using iterative deep neural networks. arXiv preprint arXiv:1704.04058.
[5] Antholzer, S., Haltmeier, M., & Schwab, J. (2017). Deep Learning for Photoacoustic Tomography from Sparse Data. arXiv preprint arXiv:1704.04587.



The Value of Observations for Numerical Weather Prediction
Alison Fowler (University of Reading, UK)

The assimilation of observations into atmospheric models has proven crucial to the accuracy of modern day weather forecasting. The atmosphere is observed by a vast array of observations providing information about different variables and different scales.
Developing, manufacturing, deploying, collecting and assimilating the data for each of the associated observations is hugely expensive, both in terms of money and time. In order to optimise the observation network, the question is how can we quantify which of these observations have the greatest value? To address this question, it is necessary to understand how the model utilizes the information provided by the observations and how this depends upon the inherent scales of the model, the model uncertainty and the uncertainty of the observations.

Doctor Alison Fowler’s research into the theory of data assimilation for understanding the Earth system has spanned the last ten years. In close collaboration with the UK Met Office she has worked on the development of algorithms for the treatment of positional errors in coherent features, and theory to advance the development of coupled atmosphere-ocean data assimilation. Her current role is senior research scientist at the National Centre for Earth Observation. Her remit is to perform fundamental research into data assimilation theory in order to maximize the value of Earth observation data. This work focuses on understanding and quantifying the uncertainty associated with the observations and developing metrics to quantify their value. Most recently she has been involved in a European Space Agency bid to develop a novel satellite instrument to measure cloud top winds.