Tag - Applied mathematics

James Bullock: Rewilding complex ecosystems and landscapes

25th March, 2024

Rewilding aims to create ecosystems and landscapes whose dynamics are driven by natural processes. This idea has been defined verbally, but its theoretical aspects remain at the early stages of development. I will discuss aspects of rewilding that could be explored by theorists, also drawing on the related topic of ecological restoration. 1) Complex systems. Rewilding has been described as an attempt to recreate complex systems, whose key features are complex trophic structures, stochastic disturbances and heightened dispersal. 2) Resilience and critical thresholds. Concepts from complex systems science that are linked to non-linearity, such as regime shifts, ecological resilience and ecological feedbacks, could be employed to help explain variation in rewilding outcomes. 3) Dispersal. Rewilding is focused on ‘natural colonization’, meaning that outcomes depend critically on dispersal abilities and the permeability of landscapes. Ultimately, development of theory on these and related concepts may help us understand the various trajectories that rewilding may take.

Mathieu Lauriere: Deep Backward and Galerkin Methods for Learning Finite State Master Equations

19th March, 2024

We propose two methods to solve the master equation for finite-state mean field games (MFGs). Solving MFGs provides approximate Nash equilibria for stochastic, differential games with finite but large populations of agents. The master equation is a partial differential equation (PDE) whose solution characterizes MFG equilibria for any possible initial distribution. The first method we propose relies on backward induction while the second one directly tackles the PDE without discretizing time. For both approaches, we prove two types of results: there exist neural networks that make the loss functions of the algorithms arbitrarily small and, conversely, if the losses are small, then the neural networks are good approximations of the master equation solution. We conclude with numerical experiments on benchmark problems from the literature in dimension up to 15, and a comparison with solutions computed by a classical method for fixed initial distributions.

Yoichiro Mori: Inextensible interface problem in 2D Stokes fluid

18th March, 2024

We consider the dynamics of a closed intextensible interface immersed in a 2D Stokes fluid, a model that has been used for 2D simulations of vesicle dynamics. In this model, a 1D closed interface exerts a bending force and the interface is subject to an inextensibility constraint. As part of the problem, one must solve for the unknown tension that ensures membrane inextensibility. Given a force exerted on the interface, we first show that the problem of determining the tension is soluble if and only if the interface is not a circle. Using this result, we prove local-in-time well-posedness for this problem. We will finally discuss open questions and future directions.

Katherine Bickerton: Conservation Translocations as a Tool for Rewilding

6th March, 2024

A key component of most rewilding projects is the translocation of plant and animal species to restore ecosystem function, however translocations are a complex process both biologically and socially. Here I will explore the reintroduction process and suggest the areas where mathematical and statistical methods could aid decision making and the understanding of complex biological systems.

Altea Lorenzo-Arribas and Dominic Duckett: Rewilding: stakeholder perceptions and trade-offs

6th March, 2024

There are trade-offs in practice with rewilding particularly around food and nutrition security but also around other societal dimensions as globally represented by the UN Sustainable Development Goals (SDGs). We need solutions that are synergistic.

Building on previous collaborations, we propose transdisciplinary, methodological approaches with a focus on qualitative and quantitative scenario planning to develop both aspirational and plausible solutions based on science and society.

Rafael Bailo: Pedestrian models with congestion effects

28th February, 2024

We study the validity of the dissipative Aw-Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behaviour. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw-Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.

Jakob Möller: Non-linear PDE in semi-classical semi-relativistic quantum physics

22nd February, 2024

Presentation of work on (asymptotic) analysis of non-linear time-dependent PDEs modelling fast, self-interacting charged fermions in relativistic quantum mechanics, from the Dirac-Maxwell to Vlasov/Euler-Poisson equations. I focus on intermediate first and second order in 1/c models, such as the Pauli-Poisswell and Euler-Darwin equations, which are useful e.g. in plasma physics.

Marco Cirant: On the long time behaviour of equilibria in a Kuramoto Mean Field Game

20th February, 2024

In a recent work, R. Carmona, Q. Cormier and M. Soner proposed a mean field game based on the classical Kuramoto model, originally motivated by systems of chemical and biological oscillators. Such MFG model exhibits several stationary equilibria, and the question of their ability to capture long time limits of dynamic equilibria is largely open. I will discuss in the talk how to show that, up to translations, there are two possible stationary equilibria only - the incoherent and the synchronised one - provided that the interaction parameter is large enough. Finally, I will present some local stability properties of the synchronised equilibrium.

Benedetto Piccoli: Conservation Laws, Traffic and Autonomy

12th February, 2024

The problem of control of large multi-agent systems, such as vehicular traffic, poses many challenges both for the development of mathematical models and their analysis and the application to real systems. First, we discuss how conservation laws can be used for macroscopic description of traffic, then present some results for mean-field limit of controlled systems. Finally, we describe on a recent experiment involving 100 autonomous vehicles to dampen stop-and-go waves on an open highway.

Pranava Jayanti: Mass transfer and global solutions in a micro-scale model of superfluidity

5th February, 2024

We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional non-linear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in 𝕋d (d=2,3) for a power-type non-linearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time.

The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates. We present two different approaches (purely energy based, versus a combination of energy estimates and maximal regularity) based on the dimension.