# Abstracts

Title: *Serrated trailing edges and the reduction of aerofoil noise*

Abstract: Aerodynamic noise is a fundamental concern facing the aviation industry, whether it's the noise generated by a passenger aeroplane, or by a delivery drone. The key feature linking all aerodynamic designs is aerofoils/blades which generate both leading- and trailing-edge noise through interaction with unsteady fluid flows. This talk will first discuss the basics of noise generation by aerofoils in unsteady subsonic flows, followed by discussing new bio-inspired adapted blade designs for reducing trailing-edge noise. In particular this talk will present mathematical models that are capable of quickly predicting the generated noise and can be used to infer the noise-reduction mechanisms of blade adaptations.

Zofia Trstanova (Edinburgh)

Title: *Langevin dynamics with general kinetic energies*

Sampling Boltzmann distributions is crucial in many scientific domains including molecular dynamics and Bayesian interence. The multi-modality and the high-dimensionality of the underlying distribution makes this task very challenging. In this talk, I will discuss Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling. In particular, this kinetic energy can be non-globally Lipschitz, which raises issues for the stability of discretizsations of the associated Langevin dynamics. I will present numerical schemes which are stable and of weak order two, by considering splitting strategies where the discretisations of the fluctuation/dissipation are corrected by a Metropolis procedure. I will explain how the metastability can be reduced on some toy models using non-globally Lipschitz kinetic energies. At the end of my talk I will also discuss some alternative sampling strategies based on geometric information about the underlying manifold.

Michael Tretyakov (Nottingham)

Title: *Uncertainty Quantification for moving boundary problems*

Abstract: The considered UQ problems are motivated by modelling of one of the main manufacturing processes for producing advanced composites - resin transfer moulding (RTM). We consider one-dimensional and two-dimensional models of the stochastic resin transfer moulding process, which are formulated as random moving boundary problems. We study their properties, analytically in the one-dimensional case and numerically in the two-dimensional case. We show how variability of time to fill depends on correlation lengths and smoothness of a random permeability field. We will also briefly discuss Bayesian inversion for random moving boundary problems. The talk is based on joint works with Minho Park and Marco Iglesias.

Title: *Variational analysis for dipoles of topological singularities in two dimensions*.

Abstract: We present two continuous models for the study of topological singularities in 2D: the core-radius approach and the Ginzburg-Landau theory.

It is well known that - at zero temperature and under suitable regimes - the energies associated to these models tend to concentrate, as the length scale parameter epsilon goes to zero, around a finite number of points, the so-called vortices.

We focus on low energy regimes that prevent the formation of vortices in the limit as epsilon tends to zero, but that are compatible (for positive epsilon) with configurations of short (in terms of epsilon) dipoles, and more in general with short clusters of vortices having zero average.

By using a Gamma-convergence approach, we provide a quantitative analysis of the energy induced by such configurations on a continuous range of length scales.

Title: *Solving PDEs on triangles with multivariate orthogonal polynomials*

Abstract: Univariate orthogonal polynomials have a long history in applied and computational mathematics, playing a fundamental role in quadrature, spectral theory and solving differential equations with spectral methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials exist, they have an unfair reputation of being unwieldy on non-tensor product domains, and their use in applications has been limited. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators, fast evaluation of expansions using Clenshaw’s algorithm and the ability to construct sparse partial differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast solution of general partial differential equations.

Title: Analysis of p-Laplacian Regularization in Semi-Supervised Learning

Abstract: This talk concerns a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of n sample points, provided a small training subset of N labelled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabelled data when assigning labels. In this talk the geometry is represented by the random geometric graph model with connection radius r(n). The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian regularization.

The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of r(n) for the asymptotic consistency to hold. For standard approaches used thus far there is a restrictive upper bound on how quickly r(n) must converge to zero as n goes to infinity. I will present a new model which overcomes this restriction. It is as simple as the standard models, but converges as soon as r(n) -> 0. This is joint work with Dejan Slepcev (CMU).

Title: Non-equilibrium self-organization of motile bacteria

Abstract: Active materials can self-organize in many more ways than their equilibrium counterparts. For example, self-propelled particles with density dependend motility can display motility-induced phase separation (MIPS), resulting in novel routes to pattern formation. In this talk it is shown how internal fluctuations in the population size and swimming speed of motile bacteria have a significant impact on the way they self-organize. Two nontrivial regimes are identified, depending on the population carrying capacity. Below a certain threshold, the fluctuations make bacteria clusters appear and disappear periodically in time at random locations in space, with a period that is roughly independent of the noise amplitude. Above the threshold, bacteria organize in metastable clusters, and fluctuations lead to transitions between those at random times that are exponentially long in the noise amplitude, following specific out-of-equilibrium pathways. Both in the quasi-periodic and the metastable regimes, these findings can be explained by combining tools from large deviation theory with a bifurcation analysis in which the mean bacteria density, assumed to vary slowly via birth and death, plays the role of control parameter.

Per-Gunnar Martinsson (Oxford)

Title: Fast Direct Solvers for Linear Elliptic PDEs

Abstract: That the linear systems arising upon the discretization of elliptic PDEs can be solved very efficiently is well-known, and many successful iterative solvers with linear complexity have been constructed (multigrid, Krylov methods, etc). Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will survey some recent work in the field and will argue that direct solvers have several advantages, including improved stability and robustness, and dramatic improvements in speed in certain environments. Moreover, the direct solvers being proposed have low communication costs, and are very well suited to parallel implementations.