## Research group in Numerical Analysis

**Academic staff**

Elisabetta Carlini, Biancamaria Della Vecchia, Maurizio Falcone, Stefano Finzi Vita, Silvia Noschese, Gabriella Puppo, Giuseppe Visconti

**Post-docs and graduate students**

Luca Saluzzi

#### Research areas

**Approximation Theory and Positive Operators**

Analysis of the approximation properties of rational operators of Shepard type, applications to scattered data interpolation. Construction of a new class of Bernstein type polynomials.

(Biancamaria Della Vecchia)**Approximation of Viscosity Solutions for Nonlinear PDEs**

Construction and analysis of approximation schemes for HamiltonJacobi type equations of first and second order. Finite difference, Finite volumes and semi-Lagrangian schemes: accuracy, stability and convergence. A priori and a posteriori error estimates. Adaptive methods. Highorder accurate approximation schemes. Acceleration methods: fast marching, fast sweeping, domain decomposition. Homogenization and approximation of the effective hamiltonian.

(Elisabetta Carlini, Maurizio Falcone, Giulio Paolucci, Smita Sahu, Silvia Tozza)**Approximation of Differential Models for Granular Materials**

Analysis and approximation of two layer models. Consistency, stability and convergence for the approximation schemes. Evolution and equilibria for deposition models: open table, partially open table and silos. Dune evolution via two layer models.

(Maurizio Falcone, Stefano Finzi Vita)**High Order Finite Volume Schemes for Conservation laws**

Many phenomena in science are described by means of systems of nonlinear hyperbolic balance laws. They include gas dynamics, shallow water flows, plasmas. These equations are usually not solvable analytically. It is thus necessary to design robust numerical schemes to compute their approximate solutions. We work on the development and study of efficient and high-order accurate finite volume methods for the numerical evolution of hyperbolic PDEs, such as CWENO schemes and their implicit formulation (Quinpi).

(Gabriella Puppo, Giuseppe Visconti)**Multiscale Mathematical Modeling**

Multiscale mathematical models allow 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.

In these applications, interacting particles can be birds or fishes, gas particles, vehicles and pedestrians, supply chain goods, voters, etc. From their interactions, global phenomena arise, and the purpose is to investigate the emergent behaviour in such complex systems.

There are mainly three modeling scales to describe such phenomena. In the microscopic scale, the interaction of individual particles are 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 partial differential equations.

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.

(Gabriella Puppo, Giuseppe Visconti)**Numerical Schemes for Optimal Control Problems and Differential Games**

Efficient approximation schemes for highdimensional optimal control problems and games via Dynamic Programming schemes. Approximation of optimal feedbacks and optimal trajectories for free and constrained problems. Patchy domain decomposition. Pursuit-evasion games, surveillance games. Approximation of Nash equilibria. Approximation of PDE systems for Mean Field Games. Approximation of the game p-Laplacian.

(Elisabetta Carlini, Maurizio Falcone)**Numerical Methods for Nonlinear PDEs in Image Processing**

Single image ShapefromShading for Lambertian and non Lambertian surfaces. 3D reconstruction using several images (Photometric ShapefromShading). Segmentation via active contours, leve-lset methods. Nonlinear filters. Mean Curvature Motion (MCM) and related models in image processing.

(Elisabetta Carlini, Maurizio Falcone, Silvia Tozza)**Numerical Linear Algebra**

Structured matrix nearness problems. Eigenvalue sensitivity: Structured eigenvalue conditioning, Structured pseudospectra and defectivity measures. Iterative methods for the solution of large linear systems and Preconditioners for Toeplitz systems. Numerical solution of linear discrete ill-posed problems: Tikhonov-type regularization, Fractional regularization matrices, Lavrentiev-type regularization methods, TSVD and TGSVD-type regularization methods.

(Silvia Noschese)

**Weekly seminar**

The seminar **Modelllistica Differenziale Numerica** typically meets on Tuesday, 3.00-4.00 pm, in Aula di Consiglio, (1st floor of the Math Department), the seminar is open to everyone and Master students are particularly welcome. For more informations on the seminar (program, abstracts, slides) click here