Interpolazione multivariata con polinomi, RBF e altre basi e applicazioni (PRIN 2009)

Data inizio
15 luglio 2011
Durata (mesi) 
Responsabili (o referenti locali)
Bos Leonard Peter
Parole chiave
Interpolazione, polinomi, basi id funzioni radiali

Ours is a project on Multivariate Interpolation, in all its important forms, ranging
from purely theoretical aspects of polynomial interpolation, through methods of
applied interpolation, to the very specific application of interpolation procedures in
differential equation solvers.

Here are some of the highlights. First of all we are a team of four units with complementary
skills and experience. Our individual projects are each on different aspects of multivariate
interpolation, with significant constructive overlap. The Cosenza unit will study interpolation
methods for data given only on the boundary of the domain, with applications to quadrature
and cubature formulas for similar data, to exclusion algoritms for searching for
solutions of systems of nonlinear equations as well as to innovative methods for solving
certain boundary value problems in (Partial) Differential Equations. The Padova unit will
continue their recent work on constructing sufficiently good discrete models of a
compact set (Weakly Admissible Meshes) from which to extract extremal sets (Fekete or Leja points)
for polynomnial interpolation. This technology is not restricted to standard domains
like cubes or simplexes -- it works on virtually any domain! Hence
they can construct, for example, effective quadrature formulas on "designer"
domains, as well as provide good points for non-standard spectral methods for PDEs.
The Torino unit will study Radial Basis Functions (RBF) for scattered data interpolation,
and their generalizations, with a view to important applications. One such problem is the
efficient recording of medical images. Another is to detect important features
such as faults, in for example, seismic images. A third, with many applications to the
Earth Sciences, is the reconstruction of images from large, (irregularly) scattered data
on a sphere. The Verona unit will push forward the theoretical aspects of multivariate
polynomial and RBF interpolation, and use these developments to study interpolatory (in space)
exponential integrators (in time) to solve time dependent PDEs.

How do these projects bear one on the other? There are specific areas of intersection and cooperation.
The Verona and Padova units havea long standing and fruitful cooperation
(as evidenced from the CVs of the researchers involved).
The Torino, Padova and Verona projects all involve RBF interpolation, both theory and applications.
The Cosenza, Padova and Torino units will all be working on numerical cubature problems.
The Cosenza, Padova and Verona units will be studying the application of interpolation for the
numerical solution of PDEs. The common theme of multivariate interpolation is
abundantly clear, and we are hopeful of good cooperation and good results!

Enti finanziatori:

Finanziamento: richiesto
Programma: COFIN - Progetti di Ricerca di Interesse Nazionale

Partecipanti al progetto

Leonard Peter Bos
Incaricato alla ricerca
Marco Caliari
Professore ordinario

Collaboratori esterni

Stefano De Marchi
Padova Matematica Pura ed Applicata
Marco Vianello
Padova Matematica Pura ed Applicata
Giamppietro Allasia
Francesco Aldo Costabile