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Luisa Fermo

Publications and source records attributed to Luisa Fermo.

18 recordsLinked to original sources
arXiv2026

Lagrange interpolation processes based on the zeros of anti-Gauss Jacobi polynomials

This paper introduces and investigates a new Lagrange interpolation process based on the zeros of anti-Gauss Jacobi polynomials. Fundamental properties of anti-Gauss nodes, including their asymptotic distribution, are established, together with estimates for the associated polynomials and their derivatives. These results provide the basis for the construction of an interpolation process whose weighted Lebesgue constants exhibit logarithmic growth, ensuring optimal approximation properties. Compared with previously known interpolation schemes based on Jacobi nodes, the proposed process achieves optimal Lebesgue constants for a shifted range of endpoint weight parameters, allowing the use of smaller endpoint weight exponents. Convergence estimates are established for functions in suitable weighted Sobolev spaces, and numerical experiments support the theoretical findings.

math.NA↗
arXiv2026

Anti-Gauss Lagrange interpolation: Christoffel-Darboux form, barycentric representation, and orthogonal expansion

The paper deals with new formulations of a Lagrange interpolant polynomial based on the nodes of the well-known anti-Gauss rule. A first representation is given in terms of the classical Christoffel-Darboux kernel appropriately modified. The second one closely follows the barycentric form of the classical Lagrange polynomial, while the third formulation represents the interpolant as a combination of an orthonormal family of polynomials with respect to the discrete anti-Gauss inner product. A numerical test shows the performance of the explored forms.

math.NA↗
arXiv2024

Approximation of the Hilbert Transform on the unit circle

The paper deals with the numerical approximation of the Hilbert transform on the unit circle using Szeg\"o and anti-Szeg\"o quadrature formulas. These schemes exhibit maximum precision with oppositely signed errors and allow for improved accuracy through their averaged results. Their computation involves a free parameter associated with the corresponding para-orthogonal polynomials. Here, it is suitably chosen to construct a Szeg\"o and anti-Szeg\"o formula whose nodes are strategically distanced from the singularity of the Hilbert kernel. Numerical experiments demonstrate the accuracy of the proposed method.

math.NA↗
arXiv2024

A global approximation method for second-kind nonlinear integral equations

A global approximation method of Nystr\"om type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered.

math.NA↗
arXiv2023

Anti-Gauss cubature rules with applications to Fredholm integral equations on the square

The purpose of this paper is to develop the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of Fredholm integral equations of the second-kind is also explored. The stability, convergence, and conditioning of the proposed Nystr\"om-type method are studied. The numerical solution of the resulting dense linear system is also investigated and several numerical tests are presented.

math.NA↗
arXiv2023

On the error of best polynomial approximation of composite functions

The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods applied to non-linear problems or for numerical approaches that make use of regularization techniques to cure low smoothness of the solution. This result is obtained through an estimate of the derivatives of composite functions in weighted uniform norm.

math.NA↗
arXiv2023

Averaged Nystr\"om interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nystr\"om methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose, and introduces new stability properties for them.

math.NA↗
arXiv2022

On the numerical solution of Volterra integral equations on equispaced nodes

In the present paper, a Nystrom-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced points. Stability and convergence are studied in the space of continuous functions, and some numerical tests illustrate the performance of the proposed approach.

math.NA↗
arXiv2022

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method to compute the minimal-norm solution in the presence of boundary constraints. The algorithm stems from the Riesz representation theorem and operates in a reproducing kernel Hilbert space. Since the resulting linear system is strongly ill-conditioned, we construct a regularization method depending on a discrete parameter. It is based on the expansion of the minimal-norm solution in terms of the singular functions of the integral operator defining the problem. Two estimation techniques are tested for the automatic determination of the regularization parameter, namely, the discrepancy principle and the L-curve method. Numerical results concerning two artificial test problems demonstrate the excellent performance of the proposed method. Finally, a particular model typical of geophysical applications, which reproduces the readings of a frequency domain electromagnetic induction device, is investigated. The results show that the new method is extremely effective when the sought solution is smooth, but gives significant information on the solution even for non-smooth solutions.

math.NA↗
arXiv2015

Scattering data computation for the Zakharov-Shabat system

A numerical method to solve the direct scattering problem for the Zakharov-Shabat system associated to the initial value problem for the nonlinear Schroedinger equation is proposed. The method involves the numerical solution of Volterra integral systems with structured kernels and the identification of coefficients and parameters appearing in monomial-exponential sums. Numerical experiments confirm the effectiveness of the proposed technique.

math.NA↗
arXiv2014

A fully-discrete-state kinetic theory approach to traffic flow on road networks

This paper presents a new approach to the modeling of vehicular traffic flows on road networks based on kinetic equations. While in the literature the problem has been extensively studied by means of macroscopic hydrodynamic models, to date there are still not, to the authors' knowledge, contributions tackling it from a genuine statistical mechanics point of view. Probably one of the reasons is the higher technical complexity of kinetic traffic models, further increased in case of several interconnected roads. Here such difficulties of the theory are overcome by taking advantage of a discrete structure of the space of microscopic states of the vehicles, which is also significant in view of including the intrinsic microscopic granularity of the system in the mesoscopic representation.

physics.soc-ph↗
arXiv2013

Emerging problems in approximation theory for the numerical solution of nonlinear PDEs of integrable type

In this paper we present some open problems pertaining to the approximation theory involved in the solution of the important class of Nonlinear Partial Differential Equations (NPDEs) of integrable type. For this class of NPDEs, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of this paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis and arouse interest in these problems in order to develop an effective method which works under more general assumptions.

math.NA↗
arXiv2013

Parameter estimation of monomial-exponential sums

We propose a numerical method, based upon matrix-pencils, for the identification of parameters and coefficients of a monomial-exponential sum. We note that this method can be considered an extension of the numerical methods for the parameter estimation of exponential sums. The application of the method is applied to several examples, some already present in the literature and others, to our knowledge, never considered before.

math.NA↗
arXiv2013

A Nystrom method for a boundary integral equation related to the Dirichlet problem on domains with corners

The authors consider the interior Dirichlet problem for Laplace's equation on planar domains with corners. In order to approximate the solution of the corresponding double layer boundary integral equation, they propose a numerical method of Nystrom type, based on a Lobatto quadrature rule.The convergence and stability of the method are proved and some numerical tests are included.

math.NA↗
arXiv2013

Fundamental diagrams for kinetic equations of traffic flow

In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called "spatially homogeneous problem" and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microstates. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.

math-ph↗
arXiv2012

A fully-discrete-state kinetic theory approach to modeling vehicular traffic

This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and velocity) of the vehicles is genuinely discrete. While in the recent literature discrete-velocity kinetic models of car traffic have already been successfully proposed, this is, to our knowledge, the first attempt to account for all aspects of the physical granularity of car flow within the formalism of the aforesaid mathematical theory. Thanks to a rich but handy structure, the resulting model allows one to easily implement and simulate various realistic scenarios giving rise to characteristic traffic phenomena of practical interest (e.g., queue formation due to roadworks or to a traffic light). Moreover, it is analytically tractable under quite general assumptions, whereby fundamental properties of the solutions can be rigorously proved.

math-ph↗