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, Incoming 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 preceeding `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).

Attendees of the meeting enjoyed the excellent facilities of the Andrew Wiles Building

at the Mathematical Institute, University of Oxford, and excellent lunch and refreshments. The local organisation was handled most ably by Dr Alberto Paganini and Dr Abdul-Lateef Haji-Ali.

**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 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.

The meeting welcomed **Lisa Fauci** on **one of the first official engagements of her SIAM Presidency**. Lisa’s title was`**Complex dynamics of fibers in flow at the microscale**‘. She started by showing experiments detailing the behaviour of micro-organisms (e.g. sperm cells, E. Coli or diatom chains) immersed in fluids. She described theoretical work on the role of flexibility in diatom chains and explained the difficulty of computation of elastic properties of such complex media in the microscopic context. She then described mathematical work on the modeling of fibers in flow governed by the Navier-Stokes and Stokes equations, in the latter case using boundary potentials and fast summation techniques for speeding up the computation of the composite potential. The talk then went on to present beautiful computational simulations and to describe their relation to lab experiments for several examples including helical swimmers, passive fibers in shear flow and in straining flow.

**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.

**Konstantinos Zygalakis** gave a talk on `**Explicit stabilised Runge-Kutta methods and their application to Bayesian inverse problems**‘. This talk explored how ordinary and stochastic differential equations can be used as a basis for designing algorithms to solve optimisation and sampling tasks in Bayesian inverse problems. These kinds of problems frequently appear in applied mathematics and machine learning, for example

in the context of imaging. Utilising the connection to (stochastic) differential equations allows for the design of novel optimisation and sampling algorithms based on the numerical analysis of differential equations. The talk showed how a gradient flow naturally leads to optimisation algorithms, and how Langevin dynamics can be used for sampling. The focus of the talk was on incorporating sophisticated numerical time stepping methods with good stability properties in order to obtain computationally efficient algorithms. This is of great importance in problems of practical interest, where the objective function is often very high-dimensional and expensive to evaluate.

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.