Abstract: Stellarators are non-axisymmetric magnetic field configurations for confinement of fusion plasmas. In contrast to the more popular axisymmetric tokamak geometries, stellarators rely on symmetry breaking to confine particles. The Simons Collaboration on Hidden Symmetries and Fusion Energy is a team dedicated to advancing the mathematical and computational state of the art in stellarator design, and producing modern stellarator optimization codes and new underlying theory. In this talk, I describe some of the challenges of stellarator optimization, and give examples both of success stories and of challenges that remain.

Abstract: The concepts of polynomial and trigonometric interpolation are well-known. The concepts sparse interpolation (used in computer algebra) and exponential analysis (from digital signal processing) are generalizations thereof, where respectively the powers or the frequencies in the interpolant are not predefined to be $0, 1, 2, \ldots$. Half of the data is then used to determine the appropriate powers or frequencies. \bigskip \noindent We discuss how sparse interpolation and exponential analysis can cross-fertilize and lead to new results. We thereby focus on the overarching inverse problem of identifying, from given values $f_k \in \mathbb{C}$, the nonlinear parameters $\phi_1, \ldots, \phi_n \in\mathbb{C}$, the linear coefficients $\alpha_1, \ldots, \alpha_n \in \mathbb{C}$ and the sparsity $n \in \mathbb{N}$ in the interpolation $$ \sum_{j=1}^n \alpha_j \exp(\phi_j k\Delta) = f_k, $$ for $k=0, 1, \ldots, 2n-1, \ldots,$ and $\Delta \in \mathbb{R}^+. $

Abstract: We solve the nonlinear initial boundary value problem that arises in the study of thin film flows. The generally accepted version is shown not to lead to an energy bound and stability. To deal with that, we firstly consider two modifications of the original equation. Secondly, we develop new splitting techniques for the discretisation of the nonlinear terms. Thirdly, we provide boundary conditions and an implementation procedure that finally leads to provably conservative and stable nonlinear schemes.

Abstract: The 2018-2020 Ebola Virus Disease (EVD) was the first outbreak that occurred in a tumultuous, active conflict and war eastern region of the Democratic Republic of Congo (DRC) characterized by the violence, destruction of Ebola Treatment Centres and escape of patients from hospitals. Likewise, the 2014-2016 West Africa EVD came with an unprecedented challenge in that the outbreak simultaneously arose in three different countries (viz. Guinea, Liberia, and Sierra Leone) to and from which migrations and travels of people by road and air were considerable.

For the 2018-2020 outbreak, we develop a Susceptible-Infective-Recovered (SIR)-type model in which the associated disruptive events and the indirect/slow transmission through the contaminated environment are incorporated. Due to the challenge mentioned above for the 2014-2016 EVD outbreak, we construct for it a metapopulation model in each patch of which, we consider an extended Susceptible-Exposed-Infective-Recovered (SEIR) model modified by adding the disease induced deceased, the Isolated and the Quarantine compartments to account, among others, for travellers who undergo the exit screening intervention at the borders, as recommended by the World Health Organization.

For the two models, our focus is three-fold. We compute the basic reproduction numbers and explicit thresholds, thanks to which the existence and the local/global asymptotic stability (LAS/GAS) of the disease-free, endemic and boundary equilibria (DFE, EE & BE) are investigated. Next, we design nonstandard finite difference (NSFD) schemes that replicate these qualitative properties of the continuous models. Finally, we conduct a global sensitivity analysis and use the real data from the affected regions to provide numerical simulations and undertake a statistical data analytics study, which supports the theory. It is shown that the key parameters, which measure the impact of travels and war, significantly influence the increase in the numbers of infected and deaths individuals of the 2014-2016 and 2018-2020 EVD outbreaks, respectively.

Abstract: We consider the efficient solution of PDEs using hp-Finite Element Methods, which are finite element methods where both the grid size ($h$) and the polynomial order ($p$) may very. A classical quasi-optimal solver for h-FEM (where the polynomial degree is fixed at 1) is The Fast Poisson Solver which achieves $O(N^2\log N)$ operations with $N$ degrees of freedom in each dimension. Very recently a spectral method, equivalent to $p$-FEM with a single element, were introduced by Fortunato and Townsend that achieves quasi-optimal complexity as the polynomial order increases using an Alternating Direction Implicit (ADI) method. In this talk, we extend their results for general hp-FEM, achieving complexity that is optimal independent of the choice of h and p, by using a sparsity respecting solver for piecewise integrated Legendre polynomials as introduced by Babuska combined with ADI. In addition to Poisson and screened Poisson equations on rectangles, we consider s olving PDEs on disks and cylinders which are divided into annular elements using non-classical multivariate orthogonal polynomials on annuli.