Dettagli sull'Insegnamento per l'A.A. 2019/2020
Nome:
Mathematical Models for Collective Behaviour / Mathematical Models for Collective Behaviour
Informazioni
Crediti:
: Master Degree in Mathematics 6 CFU (c)
: Master Degree in Mathematical Engineering 6 CFU (b)
Erogazione:
Master Degree in Mathematical Engineering 2nd anno curriculum Comune Elective
Lingua:
Inglese
Prerequisiti
Nozioni base di analisi funzionale.
Soluzioni classiche per equazioni alle derivate parziali del primo ordine, metodo delle caratteristiche.
Obiettivi
L'obiettivo del corso e' di presentare alcuni modelli matematici utilizzati in letteratura per l'analisi di fenomeni collettivi quali: traffico stradale e pedonale, fenomeni di "flocking"
(formazione di gruppi con caratteristiche simili). Verranno discussi problemi specifici originati da problemi di tipo applicativo.
Sillabo
- Macroscopic traffic models. LWR model, its derivation. Fundamental diagrams.
The Riemann problem, examples.
Second order models for traffic flow: Payne-Whitham model, description, drawbacks; Aw-Rascle model, shocks description, domains of invariance, instabilities near vacuum.
- Theory: systems of conservation laws, strict hyperbolicity, Rankine-Hugoniot conditions; Lax admissibility condition.
The Riemann problem for systems: the linear case; GNL and LD fields; rarefactions and contact discontinuities.
BV functions, examples and properties. A compactness theorem.
- Wave front tracking algorithm: approximate rarefactions,
possible types of interactions.
Bounds on number of waves and on total variation. Compactness of approximate solutions.
The initial-boundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunov-type functional. The Toll gate problem.
- Networks, basic definitions, conservation of the flux. Examples. Distributions along the roads, maximization of the flux. Riemann problem on a junction composed by 2 incoming roads and 2 outgoing roads.
The case of 2 incoming roads and 1 outgoing road: the "right of way" rule.
Junction between one incoming and one outgoing road, different fluxes.
- Pedestrian flow: normal and panic situation. Macroscopic models for evacuation, conservation of "mass", eikonal equation.
The Hughes model for pedestrian flow. The eikonal equation: non uniqueness, viscosity solutions, relation with vanishing viscosity approximation.
The Hughes model in one space dimension. Curve of turning points, Rankine-Hugoniot conditions.
The case of constant initial density and of symmetric initial data; conservation of the left and right mass; an example with mass exchange across the turning point.
Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain.
Linearized stability around a smooth solution.
- Introduction to the theory of flocking. Examples:
Krause model for opinion dynamics, Cucker-Smale model, model for attraction-repulsion phenomena.
The Cucker-Smale flocking model: basic properties, estimates on the kinetic energy. A "flocking theorem": proof by bootstrapping method (Ha and Tadmor).
Some drawbacks of the model.
Introduction to the kinetic limit for flocking: the N-particle distribution function, Liouville equation, marginal distribution, continuity equation.
The formal mean-field limit: a Vlasov-type equation.
Descrittori di Dublino
Alla fine del corso, lo studente dovrebbe
- Acquaint with basic mathematical models that describe collective phenomena, and with standard techniques in solving specific features
- Demostrate skill in analyzing and interpret properly a mathematical model for collective behaviour
- Understand and explain the meaning of other models using proper mathematical notation
- Demonstrate capacity for reading and understanding other texts on related topics.
Testi di riferimento
- M.D. Rosini, Macroscopic models for vehicular flows and crowd dynamics: theory and applications , Springer. 2013. http://link.springer.com/book/10.1007/978-3-319-00155-5/page
- / M. Garavello, B. Piccoli, Traffic flow on networks. Conservation laws models , AIMS Series on Applied Mathematics. 2006. http://www.aimsciences.org/books/am/AMVol1.html
Modalità d'esame
Prova scritta, prova orale o tesina
Aggiornamenti alla pagina del corso
Le informazioni sulle editioni passate di questo corso sono disponibili per i seguenti anni accademici:
Per leggere le informazioni correnti sul corso, se ancora erogato, consulta il catalogo corsi di ateneo.
Ultimo aggiornamento delle informazioni sul corso: 28 ottobre 2016, 11:21