Mathematical Modelling of Multi-Agent Systems
- Unit Coordinator: Marco Di Francesco, Antonio Esposito
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
At the end of the course, the student will be familiar with multi-agent systems models of discrete type (both deterministic and stochastic), with their meso-scopic formulation, with their continuum formulation, both of first and second order, and with their formulation on a graph. The students will acquire the mathematical techniques to solve those models suitably and will be able to use those models in various interdisciplinary applications and to adapt them to specific problems in contexts of interest in health-care systems such as diagnostics and imaging, neural networks, genetics, epidemiology, dynamic data management, and biological aggregation phenomena in physiology.
- Content:
- Discrete particle systems for interacting agents. Models with external field. Models with nonlocal aggregation/repulsion forces. Models with alignment, self-propulsion and friction. Swarms models (Vicsek) . Opinion models (Sznajd, Krause). Examples of asymptotic behaviour. The stochastic case.
- Control for discrete models. Mean-field games. Application to optimisation problems. Many species models and models with species transitions. Applications in genetics, imaging, and data science.
- Complementary topics of abstract measure theory. Measure topologies. Transport of measures.
- Mesoscopic models of Vlasov type. Derivation as mean field limits from discrete particle models. Formal derivation of continuum second order models. Derivation of first order models in friction dominated regimes.
- Derivation of linear and nonlinear diffusion models from particle systems. Existence of solutions to the nonlinear diffusion equation. Asymptotic self-similar behavior.
- Aggregation-diffusion equations. Existence of solutions with linear diffusion. Existence in the diffusion-less case and formation of clusters in finite time. Stationary states with quadratic diffusion. The case of many species. Application to epidemiology.
- Introduction to graph modelling of nonlocal type. Applications to neural networks.
- Probabilistic label-switching models and applications.
- Pre-requisites:
Ordinary differential equations, real and functional analysis
- Reading list:
Lecture notes will be provided.
