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2020-21

Organiser: Susana Gomes

The Applied Maths Seminars are held on Fridays 12:00-13.00 on MS Teams. You can send a membership request via MS Teams or email susana dot gomes at warwick dot ac dot uk if you want to be added to the team. Once you have joined the team, you will receive an Outlook invitation to the seminar itself every week, and all you will have to do is join the meeting at the right time, either via the browser or app interface. 

Please contact Susana Gomes if you have any speaker suggestions.

Seminar Etiquette: 

  • Please join the calls muted and without video, especially if late, to avoid interrupting and increase video quality. 
  • You will be added to the meeting as an attendee. This means you can't share screens but you should be able to share video and sound. If you want to ask a question during the presentation, you can either raise your hand or unmute yourself and ask it - just make sure you mute your microphone after. 
  • At the end of the talk, we will have oral questions as usual, and people can unmute themselves (and turn on their video).
  • After the seminar, we will also stay around for virtual lunch and informal discussions.

Term 1


Abstracts

Week 1. Mikhail Poluetkov (WMG) - Non-parametric models in a form of function trees and methods of their identification using experimental data

This talk introduces a conceptually novel approach to data modelling. The data is considered to be a set of approximately known records of a physically deterministic system, where the output is assumed to be a result of the multiple inputs. The model is sought as a composition of multiple functions of one variable (a function tree), which is an equivalent replacement of a continuous multivariate function and is known as the Kolmogorov-Arnold representation. Furthermore, the model is non-parametric, which implies that the shapes of the involved functions are directly determined in the process of identification.

The foundation of the suggested algorithm is the identification of the kernel of the Urysohn operator (a non-linear integral operator), using known inputs and outputs of the operator. Such identification can be classified as solving an inverse problem for the integral equation of the Urysohn type. This basic method is then generalised to a composition of the Uryshon operators in a discrete form, with “hidden” intermediate variables, which constitutes the Kolmogorov-Arnold representation.

Week 2. Matthew Hennessy (Oxford) - Mathematical modelling of phase separation in hydrogels

A hydrogel is a soft, two-phase system consisting of a deformable solid matrix that is swollen with fluid. Slightly altering the temperature or pH of the surrounding environment can trigger the spontaneous formation of structures within and on the surface of the gel. This process, called phase separation, is of fundamental importance to materials science and cell biology due to its relevance in smart materials and neurodegenerative diseases. In this talk, I will discuss a flexible framework for deriving phase-field models of hydrogels using non-equilibrium thermodynamics. Numerical simulations and phase-plane analyses will be used to explore the wealth of scenarios that occur when a hydrogel undergoes phase separation. These include the propagation of fronts that destabilise and then rupture, and the formation of highly localised, highly swollen phases that are ejected from the free boundary of the gel one by one. Asymptotic methods will then be used to examine the electric double layer that forms at the interface between a polyelectrolyte gel and solvent bath. Phase transitions can lead to double layers with finite thickness or periodic structures while producing large compressive stresses that may result in instability. The talk will conclude with a discussion of viscoelastic and granular materials and current challenges in this area.

Week 3. Josephine Evans (Warwick) - Existence of a non-equilibrium steady state for the non-linear BGK model on an interval.

The BGK (Bhatnagar, Gross and Crook) equation models the behaviour of a dilute gas. It is similar to but simpler than the Boltzmann equation. We study the BGK equation where the space variable is in an interval with boundary walls which are heated to different temperatures. We show the existence of a non-equilibrium steady state for this equation when the boundary temperatures are large. I will discuss our results and also the interest of non-equilibrium steady states for non-linear gas equations and the challenges related to their analysis.

Week 4. Stefan Engblom (Uppsala) - Computational Bayesian modeling for disease control

Data-driven disease modeling and prediction are currently under intense pressure to deliver answers on a wide array of questions. In this talk I will therefore present work on Bayesian modeling of disease spread at national scales.

I first consider an endemic situation, namely the spread of shiga toxin-producing E. coli in Swedish cattle, and for which we have data over extended periods of time: this includes sample disease measurements (prevalence estimates), as well as a detailed transport network over about 10 years. The measurements are low-informative on the epidemiological state and distributed sparsely in both space and time. Nevertheless, we develop a Bayesian simulation-driven approach which performs convincingly, thus producing an in silico replica of the disease with predictive value. Some of the efforts that went into judging the identifiability and robustness of the overall procedure will be detailed, as well as tentative applications towards disease control.

I will conclude by highlighting an ongoing modeling effort concerning the Covid-19 situation in Sweden. The aim is to make use of several different data sources and produce predictions for the demands on hospitals, for early detection of hot-spots, and provide insight on the effectiveness of mitigation measures.

Co-authors: Samuel Bronstein (ENS Paris), Robin Eriksson, Alexander Medvedev, and Håkan Runvik (Department
of Information Technology, Uppsala University, Sweden), Stefan Widgren (Department of Disease Control and epidemiology, National Veterinary Institute, Sweden)

Week 5. Rafael Bailo (Lille) - Energy-Dissipating Schemes for Aggregation-Diffusion Gradient Flows

We propose fully-discrete, implicit-in-time finite-volume schemes for general non-linear non-local Fokker-Planck equations with a gradient flow structure. The schemes verify the positivity-preserving and energy-dissipating properties, done conditionally by the second order scheme and unconditionally by the first order counterpart. Dimensional splitting allows for the construction of these schemes with the same properties and a reduced computational cost in any dimension. We will showcase the handling of complicated phenomena: free boundaries, meta-stability, merging, and phase transitions.

Week 6. Elisa Davoli (TU Vienna) - Nonlocal-to-local convergence of Cahn-Hilliard equations

In this talk, we will consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential will be allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel will not fall in any available existence theory under Neumann boundary conditions. We will prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we will show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behavior will be analyzed by means of monotone analysis and Gamma-convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type. This is based on a series of joint works with H. Ranetbauer, L. Scarpa, and L. Trussardi.

Week 7. Franziska Weber (Carnegie Melon) - Numerical approximation of statistical solutions of hyperbolic systems of conservation laws

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions for multi-dimensional hyperbolic systems of conservation laws. We present a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of 'weak statistical scaling', which is inspired by Kolmogorov's 1941 turbulence theory, the approximations converge in an appropriate topology to statistical solutions. Numerical experiments indicate that the assumption might hold true in practice.

Week 8. Svetlana Dubinkina (CWI Amsterdam) - Shadowing approach to data assimilation

Data assimilation is broadly used in atmosphere and ocean science to correct error in the state estimation by incorporating information from measurements (e.g. satellites) into the mathematical model. The widely-used variational data assimilation method has a drawback of a drastic increase of the number of local minima of the corresponding cost function as the number of measurements increases. The shadowing approach to data assimilation, which was pioneered by K. Judd and L. Smith in Physica D (2001), aims at estimating the whole trajectory at once. It has no drawback of several local minima. However, it is computationally expensive, requires measurements of the whole trajectory, and has an infinite subspace of solutions.

We propose to decrease the computational cost by projecting the shadowing approach to the unstable subspace that typically has much lower dimension than the phase space. Furthermore, we propose a novel shadowing-based data assimilation method that lifts up the requirement of a fully-observed state. We prove convergence of the method and demonstrate in numerical experiments with Lorenz models that the developed data assimilation method substantially outperforms the variational data assimilation method.

Week 9. Bernhard Schmitzer (Goettingen) - Barycenters for the Hellinger-Kantorovich distance

The Hellinger--Kantorovich metric is an unbalanced generalization of the Wasserstein distance, allowing the comparison of non-negative measures of arbitrary mass. The Wasserstein barycenter is a geometrically
intuitive way to form an average between probability measures. In this talk we study the barycenter of the Hellinger--Kantorovich metric. We find that it differs substantially from the Wasserstein barycenter by exhibiting a local clustering behaviour that depends on the length scale of the input measures.
Joint work with Gero Friesecke and Daniel Matthes.
Preprint: https://arxiv.org/abs/1910.14572

Week 10. Siri Chongchitnan (Warwick) - Cosmological inflation

How do cosmologists model the early Universe? In this non-specialist talk, I will give an introduction to the leading theory of the early Universe known as cosmological inflation, which postulates a brief period of exponential cosmic expansion 13.8 billion years ago. I will explain why inflation is needed, and give some insights into its mathematical details. I will describe some unsolved problems in inflation, and describe how future experiments (such as space-based gravitational-wave satellites) could constrain and rule out competing inflation models, hence giving new insights into the origin of the Universe.