Stefano De Marchi

University of Padova | UNIPD · Department of Mathematics "Tullio Levi-Civita"

The paper deals with polynomial interpolation, least-square approximation and cubature of functions defined on the rectangular cylinder, K=D×[-1,1]K=D×[-1,1], with D the unit disk. The nodes used for these processes are the Approximate Fekete Points (AFP) and the Discrete Leja Points (DLP) extracted from suitable Weakly Admissible Meshes (WAMs) of...

We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature. KeywordsPadua points-Fast algorithms-Lagrange interpolation-Fast Fourier transform -Algebraic cubature

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation...

We consider the interpolation problem with the inverse multiquadric radial basis function. The problem usually produces a large dense linear system that has to be solved by iterative methods. The efficiency of such methods is strictly related to the computational cost of the multiplication between the coefficient matrix and the vectors computed by...

It is a common practice in multimodal medical imaging to undersample the anatomically-derived segmentation images to measure the mean activity of a co-acquired functional image. This practice avoids the resampling-related Gibbs effect that would occur in oversampling the functional image. As sides effect, waste of time and efforts are produced sinc...

In recent years, various kernels have been proposed in the context of persistent homology to deal with persistence diagrams in supervised learning approaches. In this paper, we consider the idea of variably scaled kernels, for approximating functions and data, and we interpret it in the framework of persistent homology. We call them Variably Scaled...

The approximation properties of the Aldaz-Kounchev-Render (AKR) operators are discussed and classes of functions for which these operators approximate better than the classical Bernstein operators are described. The new results are then extended to the bivariate case on the square $[0,1]^2$ and compared with other existing results known in literatu...

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 functio...

It is a common practice in multimodal medical imaging to undersample the anatomically-derived segmentation images to measure the mean activity of a co-acquired functional image. This practice avoids the resampling-related Gibbs effect that would occur in oversampling the functional image. As sides effect, waste of time and efforts are produced sinc...

In recent years, various kernels have been proposed in the context of persistent homology to deal with persistence diagrams in supervised learning approaches. In this paper, we consider the idea of variably scaled kernels, for approximating functions and data, and we interpret it in the framework of persistent homology. We call them Variably Scaled...

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 functio...

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. Ho...

We discuss a generalization of Berrut’s first and second rational interpolants to the case of equally spaced points on a triangle in R2.

Kernel-based schemes are state-of-the-art techniques for learning by data. In this work we extend some ideas about kernel-based greedy algorithms to exponential-polynomial splines, whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we introduce two algorithms which...

In this paper, we introduce the class of (β,γ)-Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev polynomials and points. For the (β,γ)-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...

To analyze multimodal three-dimensional medical images, interpolation is required for resampling which—unavoidably—introduces an interpolation error. In this work we describe the interpolation method used for imaging and neuroimaging and we characterize the Gibbs effect occurring when using such methods. In the experimental section we consider thre...

It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as a smooth approximant to f(r)=r with higher accuracy and better convergence properties than the multiq...

The mapped bases or Fake Nodes Approach (FNA), introduced in [10], allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied in preventing the appearance of the Gibbs phenomenon when interpolating discontinuous functions. However, the originally proposed S-Gibbs map suffers of a subtle...

Abstract: To analyse multimodal 3-dimensional medical images, interpolation is required for resampling which-unavoidably-introduces an interpolation error. In this work we consider three segmented 3-dimensional images resampled with three different neuroimaging software tools for comparing undersampling and oversampling strategies and to identify w...

In this paper, we introduce the class of (β, γ)-Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev poly-nomials and points. For the (β, γ)-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 investi...

The main goal of the present paper is to extend the interpolation via the so-called mapped bases without resampling to any basis and dimension. So far indeed, we investigated the univariate case, its extension to rational polynomial interpolation and its natural application to numerical integration. The concept of mapped bases has been widely studi...

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerica...

We propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix. Classical bounds in terms of Lebesgue constants a...

We investigate the use of the so-called mapped bases or fake nodes approach in the framework of numerical integration. The main contribution consists in showing that by using the fake Chebyshev-Lobatto nodes, we are able to analytically compute the quadrature weights, that turn out to be very similar to the ones obtained via the true Chebyshev-Loba...

The main goal of the present paper is to extend the interpolation via the so-called mapped bases without resampling to any basis and dimension. So far indeed, we investigated the univariate case, its extension to rational polynomial interpolation and its natural application to numerical integration. The concept of mapped bases has been widely studi...

The article Jumping with variably scaled discontinuous kernels (VSDKs) written by S. De Marchi, F. Marchetti and E. Perracchione was originally published electronically on the publisher’s Internet portal on [date of OnlineFirst publication] without open access. With the author(s)’ decision to opt for Open Choice, the copyright of the article change...

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon i...

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 appr...

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 ap-proximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in [12]. Numerical tests show that it yields an accurate approximation of disco...

The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due...

An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and...

In this work we propose a new method for univariate polynomial interpolation based on what we call mapped bases. As theoretically shown, constructing the interpolating function via the mapped bases, i.e. in the mapped space, turns out to be equivalent to map the nodes and then construct the approximant in the classical form without the need of resa...

For over a half millennium the life and work of the Polish astronomer and mathematician Nicolaus Copernicus has been described and discussed in countless publications. And yet there are some aspects of his life, as well as the whereabouts of his remains that are relatively unknown to the general public. One of the less explored episodes is the per...

The rescaled localized RBF method was introduced in [3] for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. As the basis functions are scaled appropriately, the method is easy and stably computable. Numerical evidence provided in [3] shows good approximation...

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocat...

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of...

In this work we propose a new method for univariate polynomial interpolation based on what we call mapped bases. As theoretically shown, constructing the interpolating function via the mapped bases, i.e. in the mapped space, turns out to be equivalent to map the nodes and then construct the approximant in the classical form without the need of resa...

It is well known that the classical polynomial interpolation gives bad approximation if the nodes are equispaced. A valid alternative is the family of barycentric rational interpolants introduced by Berrut in [4], analyzed in terms of stability by Berrut and Mittelmann in [5] and their extension done by Floater and Hormann in [8]. In this paper fir...

The aim of this chapter is to discuss some applications of mathematics: in oenology and in food and wine pairing. We introduce and study some partial differential equations for the correct definition of a wine cellar and to the chemical processes involved in wine aging. Secondly, we present a mathematical method and some algorithmic issues for anal...

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by discontinuous kernels lead to a very flexible tool which sensibly reduces the well-known Gibbs phenomenon in reconstructing fun...

We perform a local computation via the Partition of Unity (PU) method of rational Radial Basis Function (RBF) interpolants. We investigate the well-posedness of the problem and we provide error bounds. The resulting scheme, efficiently implemented by means of the Deflation Accelerated Conjugate Gradient (DACG), enables us to deal with huge data set...

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...

We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this pap...

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in (Formula presented.). These inequalities provide a basic tool for the discretization of the Lp norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Rec...

Some years ago the authors, who are both mathematicians and wine sommeliers, were invited by a local wine producer to organize a wine tasting course on the relationship between famous mathematicians who taught at the ancient and prestigious University of Padova (founded in 1222) in the past and the wines they may have during their lifetimes. After...

In the recent paper [1], a new method to compute stable kernel-based interpolants has been presented. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investig...

Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an effective way for a fast image reconstruction in Magnetic Particle Imaging. Due to the nature of spectral methods, a Gibbs phenomenon occurs in the reconstructed image if the underlying function has discontinuities. A possible s...

For a∈Z>0d we let ℓa(t):=(cos (a1t),cos (a2t),⋯,cos (adt)) denote an associated Lissajous curve. We study such Lissajous curves which have the quadrature property for the cube [-1,1]d that. ∫[-1,1]dp(x)dμd(x)=1π∫0πp(ℓa(t))dt for all polynomials p(x)∈V where V is either the space of d-variate polynomials of degree at most m or else the d-fold tensor...

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical...

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, f...

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...

Computerized tomography requires suitable numerical methods for the approximation of a bivariate function f from a finite set of discrete Radon data, each of whose data samples represents one line integral of f. In standard reconstruction methods, specific assumptions concerning the geometry of the Radon lines are usually made. In relevant applicat...

In applied sciences, such as physics and biology, it is often required to model the evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models in case of multi-stability. We propose to reconstruct the domains of attraction via an implicit interpolant using stable r...

In the recent paper ``On certain Vandermonde determinants whose variables separate'' [{\it Linear Algebra and its Applications} 449 (2014) pp. 17--27], there was established a factorized formula for some bivariate Vandermonde determinants (associated to almost square grids) whose basis functions are formed by Hadamard products of some univariate po...

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs.

We present an algorithm to approximate large dataset by Radial Basis Function (RBF) techniques. The method couples a fast domain decomposition procedure with a localized stabilization method. The resulting algorithm can efficiently deal with large problems and it is robust with respect to the typical instability of kernel methods.

We study Lissajous curves in the three-dimensional cube that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single one-dimensional Fast Chebyshev Transform (by the Chebfun package) and to compute discrete extremal sets of Fek...

Introduction. It is well known that resolution on a gamma camera varies as a function of distance, scatter and the camera’s characteristics (collimator type, crystal thickness, intrinsic resolution etc). Manufacturers frequently provide only a few pre-calculated resolution values (using a line source in air, 10–15 cm from the collimator surface and...