(University of Oxford)

(University of Durham)

(KU Leuven)

(UNISA)

(University of Bath)

(UWC)

(RWTH Aachen University)

(Università di Bologna)

- Jon Chapman (University of Oxford)
- The making of an Escher: conformal maps and the Droste effect
I will explain the mathematics behind the Escher print "Picture Gallery", and show how it can be used to create new Escher-style pictures.

- The making of an Escher: conformal maps and the Droste effect
- Patrick Dorey (University of Durham)
- Breaking integrability at the boundary
This talk will describe some work on the bouncing of particle-like (“kink”) solutions to a nonlinear wave equation, called the sine-Gordon equation, against a fixed boundary. Away from the boundary, this equation has a property known as integrability, making the scattering of the kinks particularly simple. However, if this integrability is broken at the boundary, then the scattering becomes surprisingly complicated, in ways that will be outlined in the talk with the help of some movies.

- Breaking integrability at the boundary
- Daan Huybrechs (KU Leuven)
- The benefits and pitfalls of redundancy in the approximation of functions
Continuous functions, for example solutions to partial differential equations, are usually represented in a basis. This is convenient once a basis is available, but unfortunately bases are inflexible and often hard to construct. What if the domain geometry is complicated? What if the function has singularities? What if you know properties about the function, that do not match well with the basis you're using? Such questions do have accepted answers, for example the use of meshes, and adaptive refinement. In this talk we present a simple alternative that enables a lot of flexibility: redundancy. With some redundancy, it becomes simple -in fact, as we will show, nearly trivial- to represent functions on domains of arbitrary shape, or to add features to an approximation space. On the other hand, redundancy rapidly leads to ill-conditioning in algorithms. Surprisingly, with the right approach and with suitable restrictions, high accuracy can be achieved in a numerically stable manner. Moreover, efficient algorithms can be devised for many cases. In this talk we explore the introduction of redundancy in approximations, and we give practical guidelines about what is the 'right approach' and what are the 'suitable restrictions'.

- The benefits and pitfalls of redundancy in the approximation of functions
- Kerstin Jordaan (UNISA)
- Properties of orthogonal polynomials characterized by structural relations
In this talk we consider orthogonal polynomials that are characterized either by a structural relation of type \begin{equation}\pi(x)SP_n(x)=\sum_{k=-r}^{s}a_{n,n+k}P_{n+k}(x), \label{1}\end{equation}where $\pi(x)$ is a polynomial and $S$ is a linear operator that maps a polynomial of precise degree $n$ to a polynomial of degree $n-1$ or by \begin{equation}\label{2}\Pi(x)TP_n(x)=\sum_{k=-p}^{q}a_{n,n+k}P_{n+k}(x),\end{equation}where $\Pi(x)$ is a polynomial and $T$ is a linear operator that maps a polynomial of precise degree $n$ to a polynomial of degree $n-2$. It is understood that $S$ annihilates constants while $T$ annihilates polynomials of degree $1$. When $S=\displaystyle{\tfrac{d}{dx}}$, the structure relation (1) characterizes semiclassical orthogonal polynomials. We discuss properties of certain classes of semiclassical orthogonal polynomials. We characterize Askey-Wilson polynomials and their special or limiting cases as the only monic orthogonal polynomial solutions of (2) when $T=\mathcal{D}_q^2$ where $\mathcal{D}_q$ is the Askey-Wilson divided difference operator and $\Pi(x)$ is a polynomial of degree at most $4$. We use the structure relation (2) to derive bounds for the extreme zeros of Askey-Wilson polynomials.

- Properties of orthogonal polynomials characterized by structural relations
- Paul Milewski (University of Bath)
- Understanding the Complex Dynamics of Faraday Pilot Waves
Faraday pilot waves are a newly discovered hydrodynamic object that consists a bouncing droplet which creates, and is propelled by, a Faraday wave. These pilot waves can behave in extremely complex ways and result in dynamics mimicking quantum mechanics. I will show some of this fascinating behaviour and will present a surface wave-droplet fluid model that captures many of the features observed in experiments, particularly focussing on the emergence, under chaotic dynamics, of statistical of states with surprising structure.

- Understanding the Complex Dynamics of Faraday Pilot Waves
- Kailash Patidar (UWC)
- Modelling and robust simulations of slow-fast dynamical systems
Deterministic models are often indispensable tools for studying biological and ecological systems. Many of these models are nonlinear and highly complex systems of differential equations for which analytical studies are limited to describing the underlying dynamics semi-qualitatively. In recent years, we have studied a class of slow-fast dynamical systems through various types of differential equation models. These include the mathematical models of co-infections of HIV-TB and HIV-Malaria. In this talk, we will consider a class of problems arising in mathematical ecology. Construction of numerical schemes for such problems exhibiting multiple time scales within a system has always been a challenging task. To this end, using geometric singular perturbation theory, we will discuss appropriate decoupling of full system into slow and fast sub-systems. This will be followed by construction of sub-algorithms for these sub-systems. The algorithm for the full problem will then be obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step.

- Modelling and robust simulations of slow-fast dynamical systems
- Holger Rauhut (RWTH Aachen University)
- Compressive sensing and its use for the numerical solution of parametric PDEs
Compressive sensing enables accurate recovery of approximately sparse vectors from incomplete information. After giving a short introduction to this field, we move to an application to the numerical solution of parametric operator equations where the parameter domain is high-dimensional. In fact, one can show that the solution of certain parametric operator equations (parametric PDEs) is analytic in the parameters which can be exploited to show convergence rates for nonlinear (sparse) approximation. Building on this fact, we show that methods from compressive sensing can be used to compute approximations from samples (snapshots) of the parametric operator equation for randomly chosen parameters, which in turn can be computed by standard techniques including Petrov-Galerkin methods. We provide theoretical approximation rates for this scheme. Moreover, we then pass to a multilevel version of this scheme, similarly to multilevel Monte Carlo methods, and show that the overall complexity of computing parametric solutions can be further reduced.

- Compressive sensing and its use for the numerical solution of parametric PDEs
- Valeria Simoncini (Università di Bologna)
- On the numerical solution of large-scale linear matrix equations
Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation. More recently, matrix equations have emerged as a natural linear algebra framework for the discretized version of (systems of) deterministic and stochastic partial differential equations ((S)PDEs), and new challenges have arisen.

In this talk we will review some of the key methodologies for solving large scale linear matrix equations. Emphasis will be put on rank-structured and sparsity-structured problems, as they occur in applications. If time allows, we will also discuss recent strategies for the numerical solution of advanced linear matrix equations, such as multiterm equations and bilinear systems of equations, which are currently attracting great interest due to their occurrence in new application models associated with (S)PDEs.

- On the numerical solution of large-scale linear matrix equations