The Numerical Analysis Group has as its main objective the construction of explicit formulas for functions approximation, also implicitly defined, which make use of polynomial, rational or differentiable basis functions with compact support in one or more varibles, making use of special functions.
Moreover, the group works on the applications of these formulas to:
the problem of interpolation of large sets of scattered data;
the global approximation of the solution of boundary value problems or initial value problems;
the numerical quadrature and cubature;
bisection-exclusion algorithms to find zeros of a function.
The research focus, essentially, on the Umbral Calculus. Departing from the modern theory of Rota and his collaborators, Garsia has observed that: “unfortunately the notation and the proofs in that very original papers in some instance have something to be desired, and tend to obscure the remarkable simplicity and beauty of the results”. Then, the attention moved to looking for approaches different from the Rota’s one, from which it is possible to deduce both known results in simpler way and new characterization and algorithms. There have been identified new recursive formulas and equivalent determinantal formulas for Sheffer polynomials of A-zero type, including the Appell and bynomial polynomial sequences. Moreover, there have been considered new applications, especially in interpolating real functions with not classical interpolation conditions, even in the bivariate case.
The problem of interpolation of large sets of scattered data.
In many applications are known the value of a function (including some derivative values) at the boundary of the function domain and there is the need to give an approximation of the function in the interior points. This is a classical problem, very studied in literature due to its variuos applications. In this topic, the contribution of our research group concerns the definition of new univariate expansions and the extension of classical and new expansions to rectangular and triangular domain by means of classical or defined ad hoc techniques. In the framework of approximation and interpolation of scattered data, the univariate polynomial expansions and the corresponding expansion over the simplex, have been used to solve specific Hermite-Birkhoff interpolation problems through sufficiently differentiable rational or compactly supported functions. The use of Shepard or multipoint basis functions, in combination with interpolation polynomials based on the triangle, has allowed to develop accurate algorithm, with rate of convergence at least quadratic, for the interpolation of bivariate functions. Others applications of the univariate and bivariate polynomials have been realized for the computation of the zeros of a function by algorithm with ensured convergence and high index of efficiency and for the numerical quadrature/cubature by embedded formulas.
The global approximation of the solution of boundary value problems or initial value problems.
The reseach on this topic focus on the application of particular polynomial basis, also coming from the other research area of the group, with the aim of defining spectral methods, and more precisely collocation methods. In particular, pseudospectral methods are discretization methods for differential or integral operators, to add to the classical methods (such as finite difference, control volumes, finite elements,..), and are especially suitable for eigenvalue problems due to their high accuracy. The need of these methods is then justified be the high accuracy, often requested in the applications. Boyd stated that “When many decimal places of accuracy are needed, the contest between pseudospectral algorithms and finite difference and finite element methods is not an even battle but a rout: pseudospectral methods win hands-down. This is part of the reason that physicists and quantum chemists, who must judge their calculations against experiments accurate to as many as fourteen decimal places (atomic hydrogen maser), have always preferred spectral methods.”
Computation of the zeros of a function by algorithm with ensured convergence and high index of efficiency.
Many root refiners algorithms for the computation of roots of polynomials are proposed in literature. These kind of algorithms have in general an high index of efficiency. The researches on this topic have approached the problem by combining the special function approximation, in particular with Bernoulli polynomials, with bracketing techniques, obtaining methods of ensured convergence with high index of efficiency.