Research Field Applied Mathematics, Numerical Analysis.
Research Interests Numerical methods for nonlinear Partial Differential Equations, hyperbolic and parabolic problems.
Keywords hyperbolic problems; asymptotic high-order schemes; finite difference methods; partial integro-differential equations; option pricing; traffic flow models; shallow-water simulations; fluid-dynamic models on networks.
Fundings: I am currently funded by PRIN 2022 PNRR 'Hero-MAC' (PI: Roberta Bianchini - IAC CNR), PRIN 2022 (PI: Gabriella Puppo - Sapienza Roma).
Scientific organizations: I am a member of GNAMPA.
My research in brief
During my scientific career, my research has mainly focused on the study of numerical methods for Partial Differential Equations (PDEs) and of their applications to several research fields such as computational fluid dynamics, vehicle traffic models, computational finance, biomathematics and combustion.
Below is a brief description of my main research interests and achievements.
Approximation of the solution of dissipative hyperbolic systems
My main contribution was to introduce a new class of finite difference schemes that take into account the long-term behavior of the solutions. Using precise analytical estimates of the time decay of the local truncation error, it is possible to design increasingly accurate schemes for long times. This technique allows to obtain accurate solutions, in particular when calculating small perturbations of stable asymptotic states around stationary solutions and in the diffusion limit.
Approximation of solutions of partial integro-differential equations
The main difficulty is to deal with the non-local integral term. The use of implicit-explicit approximation techniques (IMEX) for the time integration of PIDEs, has allowed me to treat the integral term adequately (with an explicit approximation in time) and to obtain a class of higher order accuracy schemes under weak time step stability restrictions. I have also applied these types of finite-difference techniques to the numerical calculation of European and American option prices in jump-diffusion stochastic volatility models. From a computational point of view, the most delicate point is the treatment of the dynamics of the volatility process. My contribution has been to define a ”hybrid” backward algorithm combining finite difference schemes to handle the jump-diffusion price process and Markov chains (in particular, multiple jump trees) to approximate the other random sources, such as stochastic volatility and stochastic interest rate. These numerical methodologies have been implemented in the platform for pricing financial derivatives PREMIA, developed by the MathRisk team from INRIA, Ecole des Ponts ParisTech, University of Marne la Valle (France).
Vehicular traffic on road networks
One of my most recent research interests concerns the simulation of vehicular traffic on road networks. As part of the project with the Autovie Venete company, I developed a fluid dynamic model describing the dynamics of heavy and light vehicles on a motorway network. The model was calibrated by means of real data measured by fixed sensors, and numerical results have shown that it is able to capture some characteristic phenomena of the interaction between the two classes of vehicles. Thus, it has been integrated (I was in charge of its implementation) into the traffic forecasting software developed as part of the project and it is currently used by Autovie Venete operators as a decision support tool. While studying vehicle traffic models, I became interested in its impact on air pollution. The claim would be to design a tool that links traffic models to the diffusion of gases in the atmosphere, combining consecutive modules: a traffic simulation module, an emission module, a module for the main chemical reactions leading to particular gases production, and a module for their diffusion in the atmosphere. This project is still in its infancy, the first results concern the traffic module based on second-order traffic flow models.
Shallow water equations in channel networks
More recently, I have started to study different aspects of the numerical approximation of shallow
water equations in channel networks. The first result was to propose a discontinuous Galerkin Runge-Kutta scheme as an efficient and compact numerical approach for the calculation of water flow in a channel and to apply it to the channel network. Then, I came across the problem of junction conditions. In this context, the main challenge was to study the solution in the case where there is a phase transition between the fluvial and torrential regime at the intersection of the channels. The proposed solution was established by assigning a Riemann solver and giving a new definition of the admissible junction values. Recently, new junction conditions have been studied which take into account the angles formed by the channels.
