arXiv subjects

Antonella Falini

Publications and source records attributed to Antonella Falini.

6 recordsLinked to original sources

arXiv2024

Empirical Density Estimation based on Spline Quasi-Interpolation with applications to Copulas clustering modeling

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.

stat.ML↗

arXiv2022

IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth conformal multi-patch representation of their finite boundary surface. The discretization space is formed by $C^0$ inter-patch continuous basis functions whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch parameterization. For both regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function--by--function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes.

math.NA↗

arXiv2022

The Object Oriented c++ library QIBSH++ for Hermite spline Quasi Interpolation

The library QIBSH++ is a C++ object oriented library for the solution of Quasi Interpolation problems. The library is based on a Hermite Quasi Interpolating operator, which was derived as continuous extensions of linear multistep methods applied for the numerical solution of Boundary Value Problems for Ordinary Differential Equations. The library includes the possibility to use Hermite data or to apply a finite difference scheme for derivative approximations, when derivative values are not directly available. The generalization of the quasi interpolation procedure to surfaces and volumes approximation by means of a tensor product technique is also implemented. The method has been also generalized for one dimensional vectorial data, periodic data, and for two dimensional data in cylindrical coordinates, periodic with respect to the angular argument. Numerical tests show that the library could be used efficiently in many practical problems.

math.NA↗

arXiv2022

Construction of $G^2$ planar Hermite interpolants with prescribed arc lengths

In this paper we address the problem of constructing $G^2$ planar Pythagorean--hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree $7$, which results in having a closed form solution of the $G^2$ interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is $5$.

math.NA↗

arXiv2018

A study on spline quasi-interpolation based quadrature rules for the isogeometric Galerkin BEM

Two recently introduced quadrature schemes for weakly singular integrals [Calabr\`o et al. J. Comput. Appl. Math. 2018] are investigated in the context of boundary integral equations arising in the isogeometric formulation of Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand is approximated by a suitable quasi--interpolation spline. In the second scheme the regular part is approximated by a product of two spline functions. The two schemes are tested and compared against other standard and novel methods available in literature to evaluate different types of integrals arising in the Galerkin formulation. Numerical tests reveal that under reasonable assumptions the second scheme convergences with the optimal order in the Galerkin method, when performing $h$-refinement, even with a small amount of quadrature nodes. The quadrature schemes are validated also in numerical examples to solve 2D Laplace problems with Dirichlet boundary conditions.

math.NA↗

arXiv2018

An adaptive IGA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes

The isogeometric formulation of Boundary Element Method (BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The new quadrature schemes are based on a spline quasi-interpolant (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual based error estimator. Numerical examples show that the optimal convergence rate of the BEM solution is recovered by the proposed adaptive method.

math.NA↗