# Plenary Talks

**Nira Chamberlain,***What's the Point: Why Maths Really Matters*(public lecture)All around the world, children and even some adults ask the question: “What is the point of mathematics?”

The field of mathematical modelling not only helps answer this question, it can even help quench the human thirst for knowledge.

Join Dr. Nira Chamberlain, Chartered Mathematician, Chartered Scientist, and President of the Institute of Mathematics and its Applications (United Kingdom) as he shares applications of mathematical modelling, from serious to fun. From the edge of our solar system to battling an artificial intelligence takeover, we will see the point of mathematics after all.**Ginestra Bianconi,**Multilayer networks are emerging as a novel and powerful way to describe complex systems. Uncovering the interplay between multilayer network structure and function is a big theoretical challenge with a vast realm of applications. On the other side the urgency of understanding real-world multilayer network problems requires novel theoretical approaches. In this talk we will show how the statistical mechanics theory beyond multi-layer networks reveals the information encoded in these structures and its effect on multilayer network robustness.**Multilayer Networks: Structure and Function**

**Carola-Bibiane Schönlieb,***Deep Learning for Solving Inverse Imaging Problems*

Inverse problems are about the reconstruction of an unknown physical quantity from indirect measurements. In imaging, they appear in a variety of places, from medical imaging, for instance MRI or CT, to remote sensing, for instance Radar, to material sciences and molecular biology, for instance electron microscopy. Here, imaging is a tool for looking inside specimen, resolving structures beyond the scale visible to the naked eye, and to quantify them. It is a mean for diagnosis, prediction and discovery. Most inverse problems of interest are ill-posed and require appropriate mathematical treatment for recovering meaningful solutions.

Classically, such approaches are derived almost conclusively in a knowledge driven manner, constituting handcrafted mathematical models. Examples include variational regularization methods with Tikhonov regularization, the total variation and several sparsity-promoting regularizers such as the L1 norm of Wavelet coefficients of the solution. While such handcrafted approaches deliver mathematically rigorous and computationally robust solutions to inverse problems, they are also limited by our ability to model solution properties accurately and to realise these approaches in a computationally efficient manner.

Recently, a new paradigm has been introduced to the regularization of inverse problems, which derives solution to inverse problems in a data driven way. Here, the inversion approach is not mathematically modelled in the classical sense, but modelled by highly over-parametrised models, typically deep neural networks, that are adapted to the inverse problems at hand by appropriately selected (and usually plenty of) training data. Current approaches that follow this new paradigm distinguish themselves through solution accuracies paired with computational efficieny that were previously unconceivable. In this talk I will provide a glimpse into such deep learning approaches and some of their mathematical properties. I will finish with open problems and future research perspectives.**Elisabetta Rocca,****A Phase-Field-Based Graded Material Topology Optimization with Stress Constraint**

In the talk we introduce a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraints and potentially multiple materials or multiscales. First-order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible work flow to obtain a 3D-printed object from the numerical solutions is described and the nal structure is printed using a fused deposition modeling (FDM) 3D printer.

This is a joint work with F. Auricchio, E. Bonetti, M. Carraturo, D. Hömberg, A. Reali.**Mark McGuinness,**Moisture measurement in real time as bauxite is offloaded to be processed to produce alumina might be possible using low-level microwave radiation. This talk is motivated by a Study Group problem brought to MACSI in Limerick in 2017. It will be about modelling the propagation of electromagnetic waves, analysing the data, and using phase shift and attenuation to read the moisture content of a conducting electrolyte. There will be luxuriously linear differential equations, squircles, XOR operations, transmission, and some reflection, both physical and contemplative.**Detecting Moisture Content in Bauxite Using Microwaves****Andrea Bertozzi,****Minimal Surface Configurations for Microparticles**

Drop-Carrier Particles (DCPs) are solid microparticles designed to capture uniform microscale drops of a target solution without using costly microfluidic equipment and techniques. DCPs are useful for automated and high-throughput biological assays and reactions, as well as single cell analyses. The ability of the DCPs to enable templated uniform-sized droplets can be understood theoretically using surface energy minimization for multiple droplet interactions. We compare the theoretical prediction for the volume distribution to macroscale experiments of pairwise droplet splitting, with good agreement. This leads to a theory for the number of pairwise interactions of DCPs needed to reach a uniform volume distribution. We develop a probabilistic pairwise interaction model for a system of such DCPs exchanging fluid volume to minimize surface energy. Heterogeneous mixtures of DCPs with different sized particles require fewer interactions to reach a minimum energy distribution for the system. We present ideas for the optimization of the DCP geometry for minimal required target solution and uniformity in droplet volume.

**Anna-Karin Tornberg,**Starting with the electrostatic potential, we introduce a unified treatment for fast and spectrally accurate evaluation subject to periodic boundary conditions in any or none of the three spatial dimensions. Ewald decomposition is used to split the problem into a real-space and Fourier-space part, and the FFT-based Spectral Ewald (SE) method is used to accelerate computation of the latter, yielding the total runtime O(N log(N)) for N sources and targets.**Fast Summation: FFT Based Spectral Ewald Methods for Arbitrary Periodicity**

An FFT based method is most natural for fully periodic problems, but is here extended to treat also problems with periodity in only some or none of the spatial dimensions. Key components are an FFT-based solution technique for the free-space Poisson problem (in different dimensions) and an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The Gaussian window function previously used in the SE method is compared with a new piecewise polynomial approximation of the Kaiser-Bessel window, which further reduces the runtime. We present error estimates and a parameter selection scheme for all parameters of the method, and compare the runtime of the SE method with that of the Fast Multipole Method. We also discuss the extension of the method to the evaluation of Stokes potentials, for which the kernel is not a radial function.**Eddie Wilson,**It is an exciting time in the world of transport modelling. On the one hand, we have complex and inter-dependent challenges: reducing carbon and becoming more sustainable; improving air quality and health; yet maintaining personal mobility and economic activity - in a time of rapid demographic change. On the other hand, we have technical opportunities: such as more-electric operations, intelligent information systems, and autonomy. The challenges and opportunities now align and rapid changes that were due over the next 10-15 years will be accelerated by the COVID crisis - very likely, ECMI will continue to be a virtual conference, at least in part!**Mathematical Problems in the Modelling of Transport and Traffic**

In this talk, I will try to pick out a handful of topics where modelling and simulation can usefully contribute to the conversation, and where there is a prospect for some interesting applied mathematics. So far as it relates to my own work, I will talk (for example) about highway traffic flow and autonomy; control of unmanned air traffic; smart city operations and design; mobility apps; and how to plan walking and cycling infrastructure.**William Lee,**Snacks made from potato dough are cooked by being submerged in hot oil. A critical parameter for practitioners is the time it takes for the snacks to become buoyant. This occurs by evaporation of the water content of the dough which both reduces the mass of the snack and increases its effective volume by forming a vapour layer at the base of the snack. This can be modelled as a Stefan problem, in which fronts separating wet and dry dough propagate into the snack, coupled with a thin film equation describing the vapour layer. The model allows the time taken to lift off the be estimated in terms of the parameters of the frier.**Mathematical Modelling of Frying Potato Snacks****Luis Nunes Vicente,**In the application of machine learning to real life decision-making systems, e.g., credit scoring and criminal justice, the prediction outcomes might discriminate against people with sensitive attributes, leading to unfairness.**Accuracy and Fairness Trade-offs in Machine Learning: A Stochastic Multi-Objective Approach**

The commonly used strategy in fair machine learning is to include fairness as a constraint or a penalization term in the minimization of the prediction loss, which ultimately limits the information given to decision-makers.

In this talk, we introduce a new approach to handle fairness by formulating a stochastic multi-objective optimization problem for which the corresponding Pareto fronts uniquely and comprehensively define the accuracy-fairness trade-offs.

We have then applied a stochastic approximation-type method to efficiently obtain well-spread and accurate Pareto fronts, and by doing so we can handle training data arriving in a streaming way.**Paul Dellar,****Vector Lattice Boltzmann Equations in Science and Industry**

Solutions to partial differential equations of practical interest, notably the Navier-Stokes equations, can be usefully computed by embedding the desired PDEs within a larger linear constant coefficient hyperbolic system. All nonlinearity can be confined to algebraic source terms, and treated locally at grid points when discretised. The larger system for Navier-Stokes resembles a discrete version of Boltzmann's equation from the kinetic theory of gases. The resulting lattice Boltzmann algorithms have been widely adopted in commercial fluid simulation software by the oil, automotive, and aerospace industries.

The extension to magnetic fields and Maxwell's equations requires a different representation of vector fields, inspired by an industrial study of magnetic resonance imaging for flow in porous media. The resulting lattice Boltzmann magnetohydrodynamics algorithm has been used as the basis for commercial software development, and was adopted by DARPA as a benchmark for optimising software for high performance computers. Fuller exploitation of kinetic degrees of freedom in the larger system leads to more realistic descriptions of plasmas, and shows that a system designed only to capture magnetohydrodynamics and the pre-Maxwell equations fully supports electromagnetic waves.

Maxwell's equations include the no-monopoles condition div B = 0. Lattice Boltzmann simulations maintain a kinetic variable, the consistent approximation to div B at zero to near machine precision. Recent work inspired by machine learning ideas has identified the optimal finite difference analogue of div B = 0 satisfied by the macroscopic magnetic field on the grid. Finally, the close connection between the magnetohydrodynamics equations and Jeffery's equation for evolving the orientation of suspended ellipsoidal particles in viscous fluids opens new application areas in suspension mechanics and active matter.

Additionally, the two award winners will be given the opportunity to give a plenary lecture on Wednesday: **Anile-ECMI Prize for Mathematics in Industry**

**Bernadette Stolz,***Global and local persistent homology for the shape and classification of biological data*

**Hansjörg Wacker Memorial Prize**

**Halvor Snersrud Gustad,***Using Artificial Neural Networks for Predicting Bending Moments of Riser Structures***Jan Brekelmans,***The volume-of-fluid method applied to vertical slug flow using an axial-symmetric and a fully three-dimensional approach*