# Research

Numerical methods, modeling and control of  non-linear dynamics
I am interested in the study of self-organized system, from a numerical and analytical point of view.
Usually these models take in account the evolution of a large number of interacting individuals interacting each other, and they can be described by an ODEs system with the following structure $$\left\{\begin{array}{l}\dot{x}_i=v_i\\\dot{v}_i=\sum_{j=1}^Nm_jF(x_i,x_j,v_i,v_j)\end{array}\right.\qquad i=1,\ldots, N$$ where $(x_i,v_i)\in\mathbb{R}^{2d}$, $d\geq 1$ and $F(\cdot)$ describes the possibly non-linear interaction among agents.
Solving such systems implies a huge computational effort when the number of agents $N$ is large, and this cost increase further for large values of the  dimensional parameter $d$.
A first approach to reduce such computational effort, consists in the derivation of a mesoscopic description of the system. Thus  the density function $f=f(x,v,t)$ of agents at time $t$ in position $(x,v)\in\mathbb{R}^{2d}$ is introduced, whose evolution is described by the following kinetic equation $$\partial_t f +v\cdot\nabla_x f=-\nabla_v\cdot\left(\mathcal{F}[f]f\right).$$  One of the main goal of my research consists in the development of efficient numerical methods for the solution of such models.
A further direction we are investigating is the action of few individuals among a large set of individuals, examples run from  the study of predator-prey models in biological system, to the influence of opinion leaders in a social network.
For more insights look at the following links: