arXiv subjects

Stefano De Marchi

Publications and source records attributed to Stefano De Marchi.

At least 19 recordsLinked to original sources

arXiv2026

Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods

Classical approximation methods are usually improved by changing the sampling set, increasing the number of data points, or selecting a different basis. We advocate a complementary viewpoint: keep the sampled values fixed and modify the representation through a suitable mapping. This viewpoint unifies several constructions that were originally introduced for different purposes. Rescaled radial basis function interpolation maps the interpolation operator through a normalization that enforces exact reproduction of constants. Variably scaled kernels map the geometry by lifting the data to a higherdimensional manifold determined by a scale function. Mapped bases and fake nodes map the approximation space without resampling the data, while rational kernel expansions can be interpreted as nonlinear mappings of the trial space. We develop a common notation for these mechanisms, summarize their approximation and stability properties, and explain how discontinuous and rational variants fit the same framework. Reproducible numerical experiments illustrate constant reproduction, error reduction through data-mimicking scale functions, and the suppression of the Runge phenomenon by mapped polynomial bases. The results support a general principle: mappings provide a flexible way to adapt approximation spaces to data geometry and regularity without modifying the original observations

math.NA↗

arXiv2026

Results, challenges and new steps on RBF approximation and computation

We present an up-to-date overview of approximation methods based on Radial Basis Function (RBF) techniques, which have recently attracted attention across various computational tools and application fields. This review study presents relevant results on RBF techniques, highlighting the associated computational challenges and stability issues that must be addressed when high-performance or parallel computation is required.

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arXiv2026

Geometric properties of the Lebesgue function

We present a collection of observations concerning the peculiar behavior of the Lebesgue function in the setting of the interval $[-1,1]\subset \mathbb{R}$ and the square $[-1,1]^2\subset \mathbb{R}^2$. We provide numerical results and formulate several open problems related to the geometry of the Lebesgue function.

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arXiv2025

Polynomial approximation from diffused data: unisolvence and stability

In this work, we address the problem of polynomial interpolation of non-pointwise data. More specifically, we assume that our input information comes from measurements obtained on diffuse compact domains. Although the nodal and the diffused problems are related by the mean value theorem, such an approach does not provide any concrete insights in terms of well-posedness and stability. We hence develop a different framework in which {\it unisolvence} can be again recovered from nodal results, for which a wide literature is available. To analyze the stability of the so-obtained diffused interpolation procedure, we characterize the norm of the interpolation operator in terms of a Lebesgue constant-like quantity. After analyzing some of its features, such as invariance properties and sensitivity to support overlapping, we numerically verify the theoretical findings.

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arXiv2025

A Note on the Direct Approximation of Derivatives in Rational Radial Basis Functions Partition of Unity Method

This paper proposes a Direct Rational Radial Basis Functions Partition of Unity (D-RRBF-PU) approach to compute derivatives of functions with steep gradients or discontinuities. The novelty of the method concerns how derivatives are approximated. More precisely, all derivatives of the partition of unity weight functions are eliminated while we compute the derivatives of the local rational approximants in each patch. As a result, approximate derivatives are obtained more easily and quickly than those obtained in the standard formulation. The corresponding error bounds are briefly discussed. Some numerical results are presented to show the technique's potential.

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arXiv2024

On the Lebesgue constant of the Morrow-Patterson points

The study of interpolation nodes and their associated Lebesgue constants are central to numerical analysis, impacting the stability and accuracy of polynomial approximations. In this paper, we will explore the Morrow-Patterson points, a set of interpolation nodes introduced to construct cubature formulas of a minimum number of points in the square for a fixed degree $n$. We prove that their Lebesgue constant growth is ${\cal O}(n^2)$ as was conjectured based on numerical evidence about twenty years ago in the paper by Caliari, M., De Marchi, S., Vianello, M., {\it Bivariate polynomial interpolation on the square at new nodal sets}, Appl. Math. Comput. 165(2) (2005), 261--274.

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arXiv2024

Fast-Decaying Polynomial Reproduction

Polynomial reproduction plays a crucial role in deriving error estimates for various approximation schemes. In particular, local polynomial reproduction is a key ingredient in both error estimation and stability analysis. However, for certain computationally relevant methods, such as Rescaled Localized Radial Basis Functions (RL-RBF), this requirement constitutes a limitation. To enable the analysis of a broader class of approximation methods in a unified and efficient manner, the present work introduces a framework based on fast-decaying polynomial reproduction. In this approach, we do not restrict ourselves to compactly supported basis functions. Instead, we allow the basis functions to decay to zero at infinity, with the decay rate controlled as a function of the separation distance. The adoption of fast-decaying polynomial reproduction yields stable and convergent approximation schemes. These methods can achieve smoothness when used in conjunction with moving least squares. All theoretical results presented in this paper regarding the rate of convergence, the Lebesgue constant and the smoothness of the approximant have been numerically validated, including in the multivariate setting.

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arXiv2024

Persistence kernels for classification: A comparative study

The aim of the present work is a comparative study of different persistence kernels applied to various classification problems. After some necessary preliminaries on homology and persistence diagrams, we introduce five different kernels that are then used to compare their performances of classification on various datasets. We also provide the Python codes for the reproducibility of results.

cs.LG↗

arXiv2024

$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators

In this work, we study the Kantorovich variant of max-min neural network operators, in which the operator kernel is defined in terms of sigmoidal functions. Our main aim is to demonstrate the $L^{p}$-convergence of these nonlinear operators for $1\leq p<\infty$, which makes it possible to obtain approximation results for functions that are not necessarily continuous. In addition, we will derive quantitative estimates for the rate of approximation in the $L^{p}$-norm. We will provide some explicit examples, studying the approximation of discontinuous functions with the max-min operator, and varying additionally the underlying sigmoidal function of the kernel. Further, we numerically compare the $L^{p}$-approximation error with the respective error of the Kantorovich variants of other popular neural network operators. As a final application, we show that the Kantorovich variant has advantages compared to the sampling variant of the max-min operator and Kantorovich variant of the max-product operator when it comes to approximate noisy functions as for instance biomedical ECG signals.

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arXiv2023

More properties of $(β,γ)$-Chebyshev functions and points

Recently, $(β,γ)$-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 $(β,γ)$-Chebyshev functions in the corresponding orthogonality interval.

math.CA↗

arXiv2023

Mapped Variably Scaled Kernels: Applications to Solar Imaging

Variably scaled kernels and mapped bases constructed via the so-called fake nodes approach are two different strategies to provide adaptive bases for function interpolation. In this paper, we focus on kernel-based interpolation and we present what we call mapped variably scaled kernels, which take advantage of both strategies. We present some theoretical analysis and then we show their efficacy via numerical experiments. Moreover, we test such a new basis for image reconstruction tasks in the framework of hard X-ray astronomical imaging.

math.NA↗

arXiv2023

Moving Least Squares Approximation using Variably Scaled Discontinuous Weight Function

Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable {\it piecewise} Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.

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arXiv2022

Computational issues by interpolating with inverse multiquadrics: a solution

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 the solver at each iteration. We propose an efficient technique for the calculation of the product of the coefficient matrix and a generic vector. This computation is mainly based on the well-known spectral decomposition in spherical coordinates of the Green's function of the Laplacian operator. We also show the efficiency of the proposed method through numerical simulations.

math.NA↗

arXiv2022

Aldaz-Kounchev-Render operators and their approximation properties

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 literature. Several numerical examples, illustrating the relevance and supporting the theoretical findings, are presented

math.NA↗

arXiv2022

Polynomial mapped bases: theory and applications

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.

math.NA↗

arXiv2022

Reducing the Gibbs effect in multimodal medical imaging by the Fake Nodes Approach

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 since the anatomical segmentation at full resolution is performed in many hours of computations or manual work. In this work we explain the commonly-used resampling methods and give errors bound in the cases of continuous and discontinuous signals. Then we propose a Fake Nodes scheme for image resampling designed to reduce the Gibbs effect when oversampling the functional image. This new approach is compared to the traditional counterpart in two significant experiments, both showing that Fake Nodes resampling gives smaller errors.

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arXiv2022

Variably Scaled Persistence Kernels (VSPKs) for persistent homology applications

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 Persistence Kernels (VSPKs). These new kernels are then tested in different classification experiments. The obtained results show that they can improve the performance and the efficiency of existing standard kernels.

math.NA↗

arXiv2021

Greedy algorithms for learning via exponential-polynomial splines

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 perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals ($f$-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($λ$-greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e. the spline nodes. However, while the {$f$-greedy} selection is tailored to one specific target function, the $λ$-greedy algorithm is independent of the function values and enables us to define a priori optimal interpolation nodes.

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