Research group in Numerical Analysis

Academic staff
Elisabetta Carlini, Biancamaria Della Vecchia, Maurizio Falcone, Stefano Finzi Vita, Maria Lopez Fernandez, Silvia Noschese

Post-docs and graduate students
Giulio Paolucci, Silvia Tozza (post-doc)

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)
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 evolutionary PDEs with memory terms
Efficient approximation of convolutions with applications to wave scattering problems, fractional differential equations, PDEs with transparent boundary conditions and models of viscoelasticity. Consistency, stability and convergence analysis. Contour integral methods based on Laplace transformation and quadrature. Fast algorithms with reduced memory requirements. Adaptivity in time and space. Sparse approximation of Helmholtz matrices with complex frequencies arising in the time approximation of retarded potentials.
(Maria Lopez Fernandez)
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