# Francesco Dell' AccioUniversità della Calabria | Università della Calabria · Dipartimento di Matematica e Informatica

Francesco Dell' Accio

Ph.D. Steklov Mathematical Institute (Moscow) - 1998 к.ф.м.н., МИАН

## About

84

Publications

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Introduction

I was born in Soveria Simeri (near Catanzaro, Calabria, Italy) on the 23rd June 1967.
I obtained my first degree "cum laude" from the Dipartimento di Matematica at the Università degli Studi della Calabria on the 15th April 1991 under the supervision of Prof. Daniele C. Struppa.
On the 8th January 1992 I went to the Department of Complex Analysis at the Steklov Mathematical Institute Russian Academy of Sciences (Moscow) to study for my Ph.D. in Mathematical Analysis under the supervision of Prof. Anatoly G. Vitushkin. During this time I have been an Istituto Nazionale di Alta Matematica two-year Grant recipient (Borsista) and an Università degli Studi della Calabria one-year Grant recipient.
From the 5th November 1995 until the 30th October 1996 I served in the Italian Army at the Scuola del Genio (Roma).
From the 8th Jenuary 1996 until 7th Jenuary 1998 I have been a Consiglio Nazionale delle Ricerche Grant recipient within the Dipartimento di Matematica at the Università degli Studi della Calabria under the supervision of Prof. Paolo A. Oliverio.
On the 11th July 1997 I went back to the Steklov Mathematical Institute for a one year stage. I successfully defended my Ph.D. thesis on the 15th May 1998.
In the Academic Year 1998-99 I have been a lecturer of a course of Mathematical and Statistical Methods in the Facoltà di Farmacia at the Università degli Studi della Calabria.
On the 28th December 1998 I got married to Gemma.
From the 1st November 1999 until the 15th September 2002, I have been a research fellow (Assegnista di Ricerca, S.S.D. A04/A (Analisi Numerica)) within the Dipartimento di Matematica at the Università della Calabria under the supervision of Prof. Francesco A. Costabile.
On the 21st January 2002 my daughter Asia was born.
Since the 16th September 2002 until the15th September 2005 I have been an Assistant Professor of Numerical Analysis (Ricercatore, S.S.D. MAT08, ex A04/A) within the same Department.
Since the 16th September 2005 until the 30th December 2011 I have been a senior Assistant Professor (Ricercatore Confermato).
On the 21st September 2007 my son Moses was born.
Since the 30th December 2011 I am an Associate Professor (Professore Associato).
Since the 25th December 2013 until the 10 November 2019 I have been the Coordinator of the Consiglio di Corso di Studi per la Laurea e la Laurea Magistrale in Matematica (Degree Course in Mathematics), Università della Calabria.
I am a member of the American Mathematical Society, of the Society for Industrial and Applied Mathematics, of the Italian Mathematical Union and of the italian National Group for Scientific Computation.
My primary research interest is in theory of approximation. In particular I am interested in
Mixed polynomial interpolation and regression.
Scattered data approximation and interpolation.
Polynomial approximation and Interpolation
Quadrature formulas.
I collaborated in researches on applications of Mathematics and in particular on
Physics of Fluvial Networks.
Statistical Analysis of Non Linear Processes.
During my Ph.D. studies at the Steklov Mathematical Institute I worked on Complex Analysis and in particular on
Topological approaches to Jacobian Conjecture

Additional affiliations

November 2013 - November 2019

January 2012 - present

September 2002 - December 2011

Education

January 1992 - July 1995

**Steklov Mathematical Institute of the Russian Academy of Sciences**

Field of study

- Mathematical Analysis

November 1986 - September 1990

## Publications

Publications (84)

This paper addresses the challenge of function approximation using Hermite interpolation on equally spaced nodes. In this setting, standard polynomial interpolation suffers from the Runge phenomenon. To mitigate this issue, we propose an extension of the constrained mock-Chebyshev least squares approximation technique to Hermite interpolation. This...

In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix-Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessa...

In computational practice, we often encounter situations where only measurements at equally spaced points are available. Using standard polynomial interpolation in such cases can lead to highly inaccurate results due to numerical ill-conditioning of the problem. Several techniques have been developed to mitigate this issue, such as the mock-Chebysh...

Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitra...

In this paper, we propose two numerical approaches for approximating the solution of the following kind of integral equation
$$f(y)-\mu\int_{-1}^{1} f(x)k(x,y)w(x)\,dx = g(y), \quad y \in [-1,1],$$
where $f$ is the unknown solution, $\mu\in \mathbb{R} \smallsetminus \{0\}$, $k,g$ are given functions not necessarily known in the analytical form, an...

Existence of sufficient conditions for unisolvence of Kansa unsymmet-ric collocation for PDEs is still an open problem. In this paper we make a first step in this direction, proving that unsymmetric collocation matrices with Thin-Plate Splines for the 2D Poisson equation are almost surely nonsingular, when the discretization points are chosen rando...

In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix-Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessa...

ATSF is an international conference series organized to bring together researchers from ALL areas of Approximation Theory and Special Functions to discuss new ideas and new applications. This organization, which has been held seven times so far as mini-symposia, has grown gradually over the years and will be held for the eighth time on September 4-...

INTERNATIONAL CONFERENCE IN APPROXIMATION THEORY AND SPECIAL FUNCTIONS, SEPTEMBER 4-7, 2024, IN ANKARA, TURKEY

Finite element method Enriched finite element method Non-polynomial enrichment Simplicial linear finite element Error estimates a b s t r a c t In this paper, we introduce a new class of finite elements by enriching the standard simpli-cial linear finite element in R d with additional functions which are not necessarily polyno-mials. We provide nec...

We construct cubature methods on scattered data via resampling on the support of known algebraic cubature formulas, by different kinds of adaptive interpolation (polynomial, RBF, PUM). This approach gives a promising alternative to other recent methods, such as direct meshless cubature by RBF or least-squares cubature formulas. 2010 AMS subject cla...

Low-order elements are widely used and preferred for finite element analysis, specifi-
cally the three-node triangular and four-node tetrahedral elements, both based on linear
polynomials in barycentric coordinates. They are known, however, to under-perform when
nearly incompressible materials are involved. The problem may be circumvented by the
us...

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of “pathological” behavior, e.g. “nearly” singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes...

We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and RBF approximation.

The aim of this paper is to unify the ideas and to extend to a more general setting the work done in [1] for a polynomial enrichment of the standard three-node triangular element (triangular
linear element) using line integrals and quadratic polynomials. More precisely, we introduce a new class of nonconforming finite elements by enriching the clas...

In this paper we develop an adaptive algorithm for determining the optimal degree of regression in the constrained mock-Chebyshev least-squares interpolation of an analytic function to obtain quadrature formulas with high degree of exactness and accuracy from equispaced nodes. We numerically prove the effectiveness of the proposed algorithm by seve...

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular Shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of...

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea i...

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea i...

In this paper we use the constrained mock-Chebyshev least squares interpolation to obtain stable quadrature formulas with high degree of exactness and accuracy from equispaced nodes. We numerically prove the effectiveness of the proposed algorithm by several examples.

The constrained mock-Chebyshev least squares interpolation is a univariate polynomial interpolation technique exploited to cut-down the Runge phenomenon. It takes advantage of the optimality of the interpolation on the mock-Chebyshev nodes, i.e. the subset of the uniform grid formed by nodes that mimic the behavior of Chebyshev–Lobatto nodes. The o...

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well...

The multinode Shepard operator is a linear combination of local polynomial interpolants with inverse distance weighting basis functions. This operator can be rewritten as a blend of function values with cardinal basis functions, which are a combination of the inverse distance weighting basis functions with multivariate Lagrange fundamental polynomi...

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well...

The main objective of this paper is to construct an approximant, with cubic precision and quartic approximation order, which interpolates functional values and first order derivatives on a set of scattered data. This approximant is a combination of six-point Shepard basis functions with rational interpolants based on six-tuples of nodes. The numeri...

In this paper we propose a simple procedure for numerically computing the Lagrange interpolation polynomial on a unisolvent set of points in the plane. We suggest the use of the canonical polynomial basis centered at the barycenter of the set of points and the P A = LU decomposition for solving the associated Vandermonde system to compute the coeff...

The triangular Shepard method is a fast and accurate scheme for interpolating scattered data. In this paper, we introduce an improvement of the triangular Shepard method for interpolating functional and first order derivatives values at the scattered points. Theoretical and numerical results show that the proposed method reaches at least cubic appr...

In this paper we present a trivariate algorithm for fast computation of tetrahedral Shepard interpolants. Though the tetrahedral Shepard method achieves an approximation order better than classical Shepard formulas, it requires to detect suitable configurations of tetrahedra whose vertices are given by the set of data points. In doing that, we prop...

The problem of reconstruction of an unknown function from a finite number of given scattered data is well known and well studied in approximation theory. The methods developed with this goal are several and are successfully applied in different contexts. Due to the need of fast and accurate approximation methods, in this paper we numerically compar...

The need of scattered data interpolation methods in the multivariate framework and, in particular, in the trivariate case, motivates the generalization of the fast algorithm for triangular Shepard method. A block-based partitioning structure procedure was already applied to make the method very fast in the bivariate setting. Here the searching algo...

As specified by Little [7], the triangular Shepard method can be generalized to higher dimensions and to set of more than three points. In line with this idea, the hexagonal Shepard method has been recently introduced by combining six-points basis functions with quadratic Lagrange polynomials interpolating on these points and the error of approxima...

The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical res...

The triangular Shepard method, introduced by Little in 1983 [7], is a convex combination of triangular basis functions with linear polynomials, based on the vertices of the triangles, that locally interpolate the given data at the vertices. The method has linear precision and reaches quadratic approximation order [3]. As specified by Little, the tr...

In this paper we present an efficient algorithm for the computation of triangular Shepard interpolation method. More precisely, it is well known that the triangular Shepard method reaches an approximation order better than the Shepard one [1], but it needs to identify useful general triangulation of the node set. Here we propose a searching techniq...

In this paper we discuss an improvement of the triangular Shepard operator proposed by Little to extend the Shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and der...

In this paper we present an efficient scheme for the computation of triangular Shepard method. More precisely, it is well known that the triangular Shepard method reaches an approximation order better than the Shepard one [4], but it needs to identify useful general triangulation of the node set. Here we propose a searching technique used to detect...

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about some function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this paper we split up the initial problem in subproblems...

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this talk we split up the initial problem in subproblems havin...

Shepard’s method is a well-known technique for interpolating large sets of scattered data. The classical Shepard operator reconstructs an unknown function as a normalized blend of the function values at the scattered points, using the inverse distances to the scattered points as weight functions. Based on the general idea of defining interpolants b...

Interpolation problems arise in many areas where there is a need to construct a continuous surface from irregularly spaced data points. This problem has a number of solutions and, among them, the choice of interpolation technique depends on the distribution of points in the data set, the application domain, the approximating function or the method...

The topological interconnection between grid, channel, and Peano networks is investigated by extracting grid and channel networks from high-resolution digital elevation models of real drainage basins, and by using a perturbed form of the equation describing how the average junction degree varies with Horton-Strahler order in Peano networks. The per...

The problem of Hermite-Birkhoff interpolation on scattered data under certain conditions of completeness is considered by using Shepard basis functions in combination with local interpolating polynomials based on the vertices of triangles. This approach allows the construction of interpolation operators with nontrivial polynomial reproduction and a...

Shepard's method is a well-known technique for interpolating large sets of scattered data. The classical Shepard operator
reconstructs an unknown function as a normalized blend of the function values at the scattered points, using the inverse distances
to the scattered points as weight functions. Based on the general idea of defining interpolants b...

The algebraic polynomial interpolation on uniformly distributed nodes can be affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which produces a polynomial that interpolates on a subset of of the gi...

The algebraic polynomial interpolation on uniformly distributed nodes is
affected by the Runge phenomenon, also when the function to be interpolated is
analytic. Among all techniques that have been proposed to defeat this
phenomenon, there is the mock-Chebyshev interpolation which is an interpolation
made on a subset of the given nodes whose elemen...

In this paper we extend the Shepard-Bernoulli operators introduced in [6] to
the bivariate case. These new interpolation operators are realized by using
local support basis functions introduced in [23] instead of classical Shepard
basis functions and the bivariate three point extension [13] of the generalized
Taylor polynomial introduced by F. Cost...

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a...

We show how to combine local Shepard operators with Hermite polynomials on the simplex [C. K. Chui, M.-J. Lai, Multivariate vertex splines and finite elements, J. Approx. Theory 60 (1990) 245–343] so as to raise the algebraic precision of the Shepard–Taylor operators [R. Farwig, Rate of convergence of Shepard’s global interpolation formula, Math. C...

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbiţaş, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These ne...

We introduce the definition of topological turning point of a function ℱ(x, λ): ℝ×ℝ→ℝ, then we propose a numerical method for calculating it. This new definition does not require any regularity for ℱ but its continuity; moreover, topological turning point coincides with turning point when ℱ is sufficiently smooth. The numerical method that we intro...

This work addresses the problem of predicting a binary response associated to a stochastic process. When observed data are of functional type a new method based on the definition of special Random Multiplicative Cascades is introduced to simulate the stochastic process. The adjustment curve is a decreasing function which gives the probability that...

Considering functional data and an associated binary response, a method based on the definition of special Random Multiplicative Cascades to simulate the underlying stochastic process is proposed. It will be considered a class S of stochastic processes whose realizations are real continuous piecewise linear functions with a constrain on the increme...

A network analysis is used to investigate the low connections of natural river channels. At the basin scale, the river networks are analyzed according to the Horton-Strahler hierarchy. We propose a quantitative criterion for the average junction degree as a function of a fixed hierarchical order of the network and independent of the usual scaling l...

We use basic results of the general theory of finite interpolation and linear algebra in order to prove the nonsingularity of a special class of centrosymmetric matrices arising in spectral methods in BVPs.

We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth
real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied
to obtain a new class of embedded boundary-type cubature formulae on the simplex.

The Numerical Analysis Conference Numerical Analysis The State of the Art NAC2005 (dedicated to Professor F.A. Costabile on his 60th birthday) was held in May 19–21, 2005, at the University of Calabria, Italy, with two major aims: investigating the state of the art in Numerical Analysis and encouraging further studies of young researchers on the tr...

We collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only. The problem is considered from the point of view of the existence of explicit formulas, interpolation to boundary data, bounds for the remainder and convergence of the...

In this paper we consider the problem of the approximation of the integral of a smooth enough function f(x,y) on the standard simplex
D2 Ì \mathbbR2\Delta _{2} \subset \mathbb{R}^{2} by cubature formulas of the following kind:
òD2 f( x,y )dxdy = åa = 13 åi,j Aaij \fracai + j axi ayj f( xa ,ya ) + E( f ) {\int\limits_{\Delta _{2} } {f{\left( {x,...

We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the inter...

Six approaches to the theory of Bernoulli polynomials are known;
these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E.
Appell [1], A. H¨urwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal
definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in
particular, we emph...

For a function f∈C2n+1([a,b]) an explicit polynomial interpolant in a and in the even derivatives up to the order 2n-1 at the end-points of the interval is derived. Explicit Cauchy and Peano representations and bounds for the error are given and the analysis of the remainder term allows to find sufficient conditions on f so that the polynomial appr...

In this paper we give a contribution to Lidstone approximation [Proc. Edinburgh Math. Soc. 2 (1929) 16] with a new approximation formula on the simplex. Interpolation conditions satisfied by the proposed formula are studied and sufficient conditions for the uniform convergence are given. In particular, a class of functions for which the sequence of...

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [a,b]R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of C
m...

In this paper we generalize an expansion in Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Starting from this expansion we obtain useful kernels, which are substantially different from Sard's for a wide class of linear functionals that includes the truncation error for cubature formulas.

It is shown that the Klein bottle with two points removed can be embedded in the compactification of ℝ2 by a finite tree.