Francesca Mazzia

Cesare Bracco, Carlotta Giannelli, Francesca Mazzia et al.

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.

Cristiano Tamborrino, Antonella Falini, Francesca Mazzia

Density estimation is a fundamental technique employed in various fields to model and to understand the underlying distribution of data. The primary objective of density estimation is to estimate the probability density function of a random variable. This process is particularly valuable when dealing with univariate or multivariate data and is essential for tasks such as clustering, anomaly detection, and generative modeling. In this paper we propose the mono-variate approximation of the density using spline quasi interpolation and we applied it in the context of clustering modeling. The clustering technique used is based on the construction of suitable multivariate distributions which rely on the estimation of the monovariate empirical densities (marginals). Such an approximation is achieved by using the proposed spline quasi-interpolation, while the joint distributions to model the sought clustering partition is constructed with the use of copulas functions. In particular, since copulas can capture the dependence between the features of the data independently from the marginal distributions, a finite mixture copula model is proposed. The presented algorithm is validated on artificial and real datasets.

Luigi Brugnano, Francesca Mazzia, Donato Trigiante

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.