Multi-scale Mathematical Modeling

Multiscale mathematical models allow us to study emergent behaviors as a consequence of particle-to-particle dynamics in several fields such as biology, gas-dynamics, traffic and pedestrian flow, opinion formation, supply chains, finance and many more.
There are mainly three modeling scales to describe such phenomena. In the microscopic scale, the interaction of individual particles is evolved in time with very large systems of ODEs. The macroscopic scale synthetically describes the evolution of aggregate quantities, such as the density for a gas, or the evolutions of queues for traffic, and typically it is obtained as the solution of PDEs. The kinetic or mesoscopic scale lies in between: here the equations give the evolution of a statistical description of the microscopic states.
We study these scales by linking them through multiscale paradigms, e.g. via mean-field limits and asymptotic or hydrodynamic limits, and design numerical methods for computing solutions to these problems.

Components: Gabriella Puppo and Giuseppe Visconti

Recent works

• M. Piu, G. Visconti, G. Puppo. Second-order multilane traffic flow models: from the microscopic to the macroscopic scale. Submitted. 2025.

• X. Gong, B. Piccoli, G. Visconti. Mean-field limit of a hybrid system for multi-lane multi-class traffic. ESAIM Contr. Op. Ca. Va., 29:71. 2023.

• M. Herty, G. Puppo, G. Visconti. Model of vehicle interactions with autonomous cars and its properties. Discrete Cont. Dyn.-B, 28(2):833-853, 2023.