Giacomo Elefante, PhD

The main goal of this work is to provide an explicit algorithm for the estimation of the segmental Lebesgue constant, an extension of the nodal Lebesgue constant that arise, for instance, in histopolation problems. With the help of two simple but efficacious lemmas, we reverse the already known technology and sensibly speed up the numerical estimation of such quantities. Results are comparable with the known literature, although cpu time of the presented method is sensibly smaller. It is worth pointing out that the numerical approach is the only known for analyzing the majority of families of supports.

This paper investigates the behavior of microstructured viscoelastic metamaterials with complex topologies, focusing on their wave propagation characteristics, specifically the behavior of damped Bloch waves. Fourier-based methods are used to solve the governing dynamic equations, taking into account both spatial and temporal damping effects. The study addresses eigenproblems related to Bloch wave dispersion, with a particular emphasis on rational eigenproblems, which are solved using an enhanced derationalization technique previously proposed by the authors. To efficiently approximate the Fourier coefficients, the technique utilizes the quasi-Monte Carlo integration method, which is particularly effective for complex geometries. An illustrative example based on triply periodic minimal surface structures is provided to demonstrate the effectiveness of the proposed approach. The results highlight the potential of these metamaterials for applications in noise reduction, impact resistance, and other advanced engineering fields.

The paper presents an augmented curvilinear virtual element method to determine homogenized in-plane shear material moduli of long-fibre reinforced composites in the framework of asymptotic homogenization method. The new virtual element combine an exact representation of the curvilinear computational geometry for complex fibre cross section shapes through an innovative two-dimensional cubature suite for NURBS-like polygonal domains. A selection of representative numerical tests supports the accuracy and efficiency of the proposed approach for both doubly periodic and random fibre arrangement with matrix domain.

We discuss a “bottom-up” algorithm for Tchakaloff-like compression of QMC (QuasiMonteCarlo) integration on surfaces that admit an analytic parametrization. The key tools are Davis-Wilhelmsen theorem on the so-called “Tchakaloff sets” for positive linear functionals on polynomial spaces, and Lawson-Hanson algorithm for NNLS. This algorithm shows remarkable speed-ups with respect to Caratheodory-like subsampling, since it is able to work with much smaller matrices. We provide the corresponding Matlab code Qsurf, together with integration tests on regions of different surfaces such as sphere, torus, and a smooth Cartesian graph.

We present an algorithm for Tchakaloff-like compression of quasi-Monte Carlo volume and surface integration on an arbitrary union of balls, via non-negative least squares. We also provide the corresponding Matlab codes together with several numerical tests.

In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and Hormann (2018) for the Floater–Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentiation matrix. Finally, we conclude with various examples and a numerical study of the convergence at equidistant nodes and conformally mapped nodes.

Recently, \((\beta,\gamma)\)-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of \([-1,1]\), which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of \((\beta,\gamma)\)-Chebyshev functions in the corresponding orthogonality interval.

An electrically-tunable metamaterial is herein designed for the active control of damped elastic waves. The periodic device is conceived including both elastic phases and a piezoelectric phase, shunted by a dissipative electric circuit whose impedance/admittance can be adjusted on demand. As a consequence, the frequency band structure of the metamaterial can be modified to meet design requirements, possibly changing over time. A significant issue is that in the presence of a dissipative circuit, the frequency spectra are obtained by solving eigen-problems with rational terms. This circumstance makes the problem particularly difficult to treat, either resorting to analytical or numerical techniques. In this context, a new derationalization strategy is proposed to overcome some limitations of standard approaches. The starting point is an infinite-dimensional rational eigen-problem, obtained by expanding in their Fourier series the periodic terms involved in the governing dynamic equations. A special derationalization is then applied to the truncated eigen-problem. The key idea is exploiting a LU factorization of the matrix collecting the rational terms. The method allows to considerably reduce the size of the problem to solve with respect to available techniques in literature. This strategy is successfully applied to the case of a three-phase metamaterial shunted by a series RLC circuit with rational admittance.

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the context of approximation theory. We will first prove necessary and sufficient conditions on point sets which guarantee the existence and uniqueness of an interpolant. We will then study the Reproducing Kernel Hilbert Spaces (or native spaces) of these kernels and their norms, and provide inclusion relations between spaces corresponding to different kernel parameters. With these spaces at hand, it will be further possible to derive generic error estimates which apply to sufficiently smooth functions, thus escaping the native space. Finally, we will show how to employ an efficient stable algorithm to these kernels to obtain accurate interpolants, and we will test them in some numerical experiment. After this analysis several computational and theoretical aspects remain open, and we will outline possible further research directions in a concluding section. This work builds some bridges between kernel and polynomial interpolation, two topics to which the authors, to different extents, have been introduced under the supervision or through the work of Stefano De Marchi. For this reason, they wish to dedicate this work to him in the occasion of his 60th birthday.

An improved version of Compressed Quasi-MonteCarlo cubature (CQMC) on large low-discrepancy samples is implemented, on 2D and 3D regions with complex shape. The algorithms rests on the concept of Tchakaloff set and on NNLS solution of a sequence of “small” moment-matching systems. Examples of area and volume integration are provided.

When an approximant is accurate on an interval, it is only natural to try to extend it to multi-dimensional domains. In the present article we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several-times differentiable functions to interpolate them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions–up to a logarithmic factor–and with an order limited only by the order of differentiability for real functions (provided that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit one to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects.

The mapped bases or Fake Nodes Approach (FNA), introduced in De Marchi et al. (J Comput Appl Math 364:112347, 2020c), allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied for mitigating the Runge's phenomenon, using the S-Runge map, or the Gibbs phenomenon, with the S-Gibbs map. However, the original S-Gibbs suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge’sphenomenon. Here, we propose a novel approach, termed Gibbs–Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge’s and Gibbs phenomena are mitigated simultaneously. After providing a theoretical analysis of the Lebesgue constant associated with the mapped nodes, we test the new approach by performing various numerical experiments which confirm the theoretical findings.

In this paper, we introduce the class of \((\beta,\gamma)\)-Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev polynomials and points. For the \((\beta,\gamma)\)-Chebyshev functions, we prove that they are orthogonal in certain subintervals of \([-1,1]\) with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of \((\beta,\gamma)\)-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters \(\beta\) and \(\gamma\).

A well-known result in linear approximation theory states that the norm of the operator, known as the Lebesgue constant, of polynomial interpolation on an interval grows only logarithmically with the number of nodes, when these are Chebyshev points. Results like this are important for studying the conditioning of the approximation. A cosine change of variable shows that polynomial interpolation at Chebyshev points is just the special case for even functions of trigonometric interpolation (on the circle) at equidistant points. The Lebesgue constant of the latter grows logarithmically, also for functions with no particular symmetry. In the present work, we show that a linear rational generalization of the trigonometric interpolant enjoys a logarithmically growing Lebesgue constant for more general sets of nodes, namely periodic well-spaced ones, patterned after those introduced for an interval by Bos et al. (2013) few years ago. An important special case are conformally shifted equispaced points, for which the rational trigonometric interpolant is known to converge exponentially.

We investigate the use of the so-called mapped bases or fake nodes approach in the framework of numerical integration. We show that such approach is able to mitigate the Gibbs phenomenon when integrating functions with steep gradients. Moreover, focusing on the optimal properties of the Chebyshev-Lobatto nodes, we are able to analytically compute the quadrature weights of the fake Chebyshev-Lobatto nodes. Such weights, quite surprisingly, coincide with the composite trapezoidal rule. Numerical experiments show the effectiveness of the proposed method especially for mitigating the Gibbs phenomenon without the need of resampling the given function.

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater–Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an accurate approximation of discontinuous functions.

Consider the set of equidistant nodes in \([0,2\pi)\), $$\theta_k := k \cdot \frac{2\pi}{n},\qquad k=0,\dots,n-1.$$ For an arbitrary \(2\pi\)-periodic function \(f(\theta)\), the barycentric formula for the corresponding trigonometric interpolant between the \(\theta_k\)'s is $$ T[f](\theta)=\frac{\sum_{k=0}^{n-1} (-1)^k \,\mathrm{cst}(\frac{\theta-\theta_k}{2}) \,f(\theta_k)}{\sum_{k=0}^{n-1} (-1)^k \,\mathrm{cst}(\frac{\theta-\theta_k}{2})},$$ where \(\mathrm{cst}(\cdot):=\mathrm{ctg}(\cdot)\) if the number of nodes $n$ is even, and \(\mathrm{cst}(\cdot):=\csc(\cdot)\) if \(n\) is odd. Baltensperger [3] has shown that the corresponding barycentric rational trigonometric interpolant given by the right-hand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward \(f\) when the nodes are the images of the \(\theta_k\)'s under a periodic conformal map. In the present work, we present a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts. Despite its simplicity, this map allows for a very accurate approximation of smooth periodic functions with steep gradients.

In this paper we consider two sets of points for Quasi-Monte Carlo integration on two-dimensional manifolds. The first is the set of mapped low-discrepancy sequence by a measure preserving map, from a rectangle to the manifold. The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold. Thanks to the Poppy-seed Bagel Theorem we know that the classes of points with minimal Riesz s-energy, under suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure. They can then be considered as quadrature points on manifolds via the Quasi-Monte Carlo (QMC) method. On the other hand, we do not know if the greedy minimal Riesz s-energy points are a good choice to integrate functions with the QMC method on manifolds. Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions.

The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.