# Report on the 23rd Annual Meeting of the SIAM UK and Republic of Ireland Section

The 23rd Annual Meeting of the SIAM UK and Republic of Ireland Section was held on Friday 11th of January 2019 at the Mathematical Institute at the University of Oxford. Around 90 participants attended the meeting.

The invited speakers were Lisa Fauci (Tulane University, SIAM President), Des Higham (Strathclyde University), Carola-Bibiane Schoenlieb (IMA sponsored speaker, University of Cambridge), Kirk Soodhalter (Trinity College Dublin)and Konstantinos Zygalakis (University of Edinburgh), and brief summaries of their lectures are given below.

The meeting also included a poster session for PhD students and postdocs, held over lunchtime, and a preceding "poster blitz" where each poster presenter gave a brief summary of their work.

At the end of the meeting, four prizes for best poster were awarded to Abigail Cocks (University of Nottingham), James Fannon (University of Limerick), Yury Korolev (University of Cambridge) and Carolina Urzua-Torres (University of Oxford).

Des Higham gave a talk with the intriguing title: `Our Friends are Cooler than Us‘. He started by explaining the `Friendship Paradox’ (first identified by Scott Field) which essentially says that "on average our friends have more friends than we do." This is obtained by analysing graphs which model individuals as nodes and friendships as edges. He went on to explain how this can be applied to any mutual pairwise interactions, for example in the "Happiness paradox"—which asserts that our friends are happier than us on average—and how the concept is useful as a method of informing immunization strategies. He then went on to describe the abstraction of this idea into the more general concept of node centrality, and the associated "centrality paradox" and gave examples arising in general principles of linear algebra applied to graphs, such as the eigenvector calculation leading to the Fiedler vector of a graph.

Kirk Soodhalter talked on "Augmented Arnoldi-Tikhonov Methods for Ill-posed Problems." Using a motivating example from image deblurring, the speaker described the ill-posed problem of determining a sharp image from blurred and noisy input data. Given that the problem size of these types of problems is large, the focus was on the use of iterative methods and in particular on projection approaches to update the approximation of the solution vector. Due to the ill-conditioning of the linear systems, some form of regularisation is needed to be able to compute a useful approximation and Tikhonov regularisation was considered. The idea of the augmented approach is to use a low-dimensional user-supplied subspace to augment the projection space. The augmented subspace is constructed from vectors which are able to represent known features of the desired solution. The talk concluded with a numerical example indicating the improved performance of the augmented method for a problem containing a step discontinuity.

The final talk of the day was by Carola-Bibiane Schoenlieb (the IMA sponsored speaker), whose title was "Variational models and partial differential equations for mathematical imaging." The talk started by framing inverse image analysis problems within a variational context. Such problems are inherently ill-posed and require some form of regularisation. An example using total variation regularisation was given for a problem involving the use of undersampled magnetic resonance tomography data. Further examples were presented including the segmentation and classification of biological cells as they change morphology during the cell cycle. The developed algorithm and software is being used by biologists in the development of anti-mitotic treatments for cancer. The talk also highlighted the myriad of mathematical and numerical challenges in the area of imaging and suggested that future hybrid methods would possibly involve a combination with machine learning.