arXiv subjects

Federico Nudo

Publications and source records attributed to Federico Nudo.

At least 19 recordsLinked to original sources

arXiv2026

Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation

In the current work, we study univariate polynomial weighted histopolation on $[-1,1]$, where the data are weighted integrals over a family of intervals. After choosing a polynomial basis, the weighted moment conditions lead to a histopolation matrix whose structure depends on the weight and on the geometry of the cells. We investigate its nonsingularity, which guarantees unisolvence, together with exact diagonality of its Gram matrix, which allows its singular values and spectral condition number to be determined explicitly. For families of intervals whose endpoints belong to a fixed grid, we characterize unisolvence in terms of the connectedness of the associated endpoint graph. In the unisolvent case, this graph is a tree, and the unique paths joining consecutive grid points provide an explicit expression for the inverse matrix. This identity gives explicit formulas for the singular values of both matrices, and shows that their condition numbers in the two-norm coincide and grow linearly with the matrix size. Moreover, it yields the limiting singular value distributions of the two matrix sequences. We also establish a general diagonalization criterion based on discrete weighted orthogonality. The criterion recovers the first kind Chebyshev construction and leads to a diagonal configuration for the constant weight based on discrete sine orthogonality. For interval families with a connected endpoint graph, the corresponding moment vectors define an inner product on the polynomial space and lead to a monic basis with a diagonal weighted Gram matrix. Finally, we derive reduction formulas for cell moments associated with generalized Jacobi weights and introduce an alternative basis for shifted Jacobi weights. Applied to the Chebyshev weight of the fourth kind, this basis, together with a correction of one nonconstant element, yields an exactly diagonal Gram matrix.

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arXiv2026

Weighted Derivative Histopolation on Arbitrary Grids: Admissibility and Exact Factorizations

In this paper, we introduce a weighted derivative histopolation framework on families of intervals. The degrees of freedom consist of one scalar normalization and weighted integral moments of the derivative over a prescribed family of subintervals. We prove that the resulting scheme is unisolvent on $\Pi_N$ when the interval family separates polynomials of degree at most $N-1$ through weighted moments and the normalization is nonzero on constants. Thus, the derivative moments determine the polynomial up to an additive constant, and the scalar normalization fixes this remaining degree of freedom. This gives a sharp criterion for the well-posedness of the interpolation problem and a complete characterization of the admissible scalar normalizations. We then show how admissible families of intervals can be constructed from a fixed grid. When the endpoints of the intervals belong to the grid, admissibility is reduced to the nonsingularity of an interval matrix associated with the family, which depends only on the representation of the intervals in terms of consecutive cells. For Jacobi weights, the associated data matrices have a natural block structure in Jacobi polynomial bases, and the reduced derivative matrix can be expressed in terms of shifted Jacobi moment matrices. We next study Chebyshev configurations in which this structure becomes explicit. For the four classical Chebyshev families, suitable polynomial bases lead to diagonal Gram matrices for the reduced derivative matrices. We show that this diagonal structure depends on the simultaneous choice of the weight, the basis, and the grid. Numerical experiments on equispaced and Chebyshev--Lobatto nodes show the behaviour of the method for different interval families and for different Jacobi parameters.

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arXiv2026

Spectral distribution of Jacobi weighted histopolation matrices via GLT theory

In this paper we study a weighted histopolation problem on $[-1,1]$ associated with Jacobi weights. In the first part of the present work we prove results in approximation theory, while in the second we analyze the resulting matrices from an asymptotic linear algebra perspective. More in detail, in the first part, given weighted cell averages, we construct a reconstruction operator based on weighted primitives of Jacobi polynomials and investigate the resulting discretization matrices. At any fixed discretization level, we derive an exact factorization of the histopolation matrix through a backward-difference operator and a sampling operator of Jacobi weighted primitives. Combining a sharp integration by parts identity with the three-term recurrence of Jacobi polynomials, we further show that the primitive sampling operator admits an explicit decomposition involving a tridiagonal coupling matrix in the Jacobi spectral index. This yields a tridiagonal factor representation of the histopolation matrix. In the second part, under standard mesh-regularity assumptions, we show that all the various induced matrix sequences belong to the Generalized Locally Toeplitz (GLT) class, by describing in detail the related GLT symbols. As a consequence, we provide the corresponding spectral distributions and discuss their implications for numerical stability when solving the associated linear systems.

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arXiv2026

Polynomial interpolation--regression on the sphere

We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed subset of nodes and uses the remaining sampling nodes to minimize the residual in a least squares sense. Under natural rank assumptions on the associated Vandermonde matrices, the approximant is unique and is characterized by an orthogonality condition with respect to the discrete inner product on the sampling set. We then focus on the case in which the sampling and interpolation nodes are antipodally symmetric. In this setting, when the polynomial is expressed in real spherical harmonics, the constrained problem can be decomposed into independent even and odd components. In the same framework, we prove equivariance under the antipodal map and, more generally, under orthogonal transformations preserving the node sets. We also consider spherical designs. In this case, the normal matrix is a scalar matrix. Consequently, the spectral condition number of the associated KKT matrix can be written explicitly. Numerical experiments in both antipodal and non-antipodal settings illustrate the effectiveness of the proposed method.

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arXiv2026

Spectral Schur analysis of structured moment matrices for quadratic histopolation

In this paper we study parameter-dependent structured moment matrices with a canonical block form arising from weighted quadratic histopolation on simplicial meshes. For a strictly positive density on a simplex, we construct compatible face densities and an orthogonal decomposition of the quadratic polynomial space into face and interior components, which induces a natural face-interior block structure. A reduced Schur complement is identified that fully characterizes enrichment and well-posedness and provides a sharp spectral stability result. We show that this quantity coincides with the square root of the smallest eigenvalue of a low-dimensional symmetric positive definite operator. This matrix-based viewpoint yields simple spectral criteria for the invertibility of local moment systems and motivates spectrally preferable choices of face and interior bases with improved conditioning. Using the resulting degrees of freedom together with density and scaling parameters as design variables, we formulate a small eigenvalue optimization problem aimed at improving stability and reducing the condition number of the global reconstruction system. Three-dimensional experiments on uniform and quasi-uniform simplicial meshes illustrate the predicted stability, conditioning, and convergence behaviour of the enriched quadratic reconstruction.

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arXiv2025

Generalized Probability Density Approach to Histopolation Schemes of Arbitrary Order

In this paper, we investigate the reconstruction of a bivariate function from weighted edge integrals on a triangular mesh, a problem of central importance in tomography, computer vision, and numerical approximation. Our approach is based on local histopolation methods defined through unisolvent triples, where the edge weights are induced by probability densities. We present a general strategy that applies to arbitrary polynomial order~$k$, in which edge moments are taken against orthogonal polynomials associated with the chosen densities. This yields a systematic framework for weighted reconstructions of any degree, with theoretical guarantees of unisolvency and fully explicit basis functions. As a concrete and flexible instance, we introduce a two-parameter family of Jacobi-type distributions on $[-1,1]$, together with its symmetric Gegenbauer subclass, and show how these densities generate new quadratic reconstruction operators that generalize the standard linear histopolation scheme while preserving its simplicity and locality. We employ an adaptive parameter selection algorithm for Jacobi densities, which automatically tunes the distribution parameters to minimize the global reconstruction error. This strategy enhances robustness and adaptivity across different function classes and mesh resolutions. The effectiveness of the proposed operators is demonstrated through extensive numerical experiments, which confirm their superior accuracy in approximating both smooth and highly oscillatory functions. Finally, the framework is sufficiently general to accommodate any admissible edge density, thus providing a flexible and broadly applicable tool for weighted function reconstruction.

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arXiv2025

A General Probability Density Framework for Local Histopolation and Weighted Function Reconstruction from Mesh Line Integrals

In this paper, we study the reconstruction of a bivariate function from weighted integrals along the edges of a triangular mesh, a problem of central importance in tomography, computer vision, and numerical approximation. Our approach relies on local histopolation methods defined through unisolvent triples, where the edge weights are induced by suitable probability densities. In particular, we introduce two new two-parameter families of generalized truncated normal distributions, which extend classical exponential-type laws and provide additional flexibility in capturing local features of the target function. These distributions give rise to new quadratic reconstruction operators that generalize the standard linear histopolation scheme, while retaining its simplicity and locality. We establish their theoretical foundations, proving unisolvency and deriving explicit basis functions, and we demonstrate their improved accuracy through extensive numerical tests. Moreover, we design an algorithm for the optimal selection of the distribution parameters, ensuring robustness and adaptivity of the reconstruction. Finally, we show that the proposed framework naturally extends to any bivariate function whose restriction to the edges defines a valid probability density, thus highlighting its generality and broad applicability.

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arXiv2025

Quadratic Weighted Histopolation on Tetrahedral Meshes with Probabilistic Degrees of Freedom

In this paper we introduce three complementary three-dimensional weighted quadratic enrichment strategies to improve the accuracy of local histopolation on tetrahedral meshes. The first combines face and interior weighted moments (face-volume strategy), the second uses only volumetric quadratic moments (purely volumetric strategy), and the third enriches the quadratic space through edge-supported probabilistic moments (edge-face strategy). All constructions are based on integral functionals defined by suitable probability densities and orthogonal polynomials within quadratic trial spaces. We provide a comprehensive analysis that establishes unisolvence and derives necessary and sufficient conditions on the densities to guarantee well-posedness. Representative density families, including two-parameter symmetric Dirichlet laws and convexly blended volumetric families, are examined in detail, and a general procedure for constructing the associated quadratic basis functions is outlined. For all admissible densities, an adaptive algorithm automatically selects optimal parameters. Extensive numerical experiments confirm that the proposed strategies yield substantial accuracy improvements over the classical linear histopolation scheme.

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arXiv2025

A $C^{\infty}$ rational quasi-interpolation operator for functions with jumps without the Gibbs phenomenon

The study of quasi-interpolation has gained significant importance in numerical analysis and approximation theory due to its versatile applications in scientific and engineering fields. This technique provides a flexible and efficient alternative to traditional interpolation methods by approximating data points without requiring the approximated function to pass exactly through them. This approach is particularly valuable for handling jump discontinuities, where classical interpolation methods often fail due to the Gibbs phenomenon. These discontinuities are common in practical scenarios such as signal processing and computational physics. In this paper, we present a $C^{\infty}$ rational quasi-interpolation operator designed to effectively approximate functions with jump discontinuities while minimizing the issues typically associated with traditional interpolation methods.

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arXiv2025

Multinode Shepard Functions and Tensor Product Polynomial Interpolation: Applications to Digital Elevation Models

The paper presents an in-depth exploration of the multinode Shepard interpolant on a regular rectangular grid, demonstrating its efficacy in reconstructing surfaces from DEM data. Additionally, we study the approximation order associated to this interpolant and present a detailed algorithm for reconstructing surfaces. Numerical tests showcase the effectiveness of the proposed algorithm.

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arXiv2025

Weighted and unweighted enrichment strategies for solving the Poisson problem with Dirichlet boundary conditions

In this paper, we propose weighted and unweighted enrichment strategies to enhance the accuracy of the linear lagrangian finite element for solving the Poisson problem with Dirichlet boundary conditions. We first recall key examples of admissible enrichment functions, specifically designed to overcome the limitations of the linear lagrangian finite element in capturing solution features such as sharp gradients and boundary-layer phenomena. We then introduce two novel three-parameter families of weighted enrichment functions and derive an explicit error bound in $L^2$-norm. Numerical experiments confirm the effectiveness of the proposed approach in improving approximation accuracy, demonstrating its potential for a wide range of applications.

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arXiv2025

Nonconforming approximation methods for function reconstruction on general polygonal meshes via orthogonal polynomials

In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials and can reproduce higher-degree polynomials. This setting naturally arises in applications where pointwise evaluations are unavailable and only integral measurements over subdomains are accessible. We develop a unisolvence theory and derive necessary and sufficient conditions for the associated approximation spaces to be unisolvent. Specifically, it is shown that unisolvence depends on the parity of the product of the polynomial degree~$m$ and the number of polygon edges~$N$. When this condition is not satisfied, we introduce an enrichment strategy involving an additional linear functional and a suitably designed enrichment function to ensure unisolvence. Numerical experiments confirm the accuracy of the proposed method.

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arXiv2025

A novel interpolation-regression approach for function approximation on the disk and its application to cubature formulas

The interpolation-regression approximation is a powerful tool in numerical analysis for reconstructing functions defined on square or triangular domains from their evaluations at a regular set of nodes. The importance of this technique lies in its ability to avoid the Runge phenomenon. In this paper, we present a polynomial approximation method based on an interpolation-regression approach for reconstructing functions defined on disk domains from their evaluations at a general set of sampling points. Special attention is devoted to the selection of interpolation nodes to ensure numerical stability, particularly in the context of Zernike polynomials. As an application, the proposed method is used to derive accurate cubature formulas for numerical integration over the disk.

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arXiv2025

$C^{\infty}$ rational approximation and quasi-histopolation of functions with jumps through multinode Shepard functions

Histopolation, or interpolation on segments, is a mathematical technique used to approximate a function $f$ over a given interval $I=[a,b]$ by exploiting integral information over a set of subintervals of $I$. Unlike classical polynomial interpolation, which is based on pointwise function evaluations, histopolation reconstructs a function using integral data. However, similar to classical polynomial interpolation, histopolation suffers from the well-known Runge phenomenon when integral data are based on a grid with many equispaced nodes, as well as the Gibbs phenomenon when approximating discontinuous functions. In contrast, quasi-histopolation is designed to relax the strict requirement of passing through all the given data points. This inherent flexibility can reduce the likelihood of oscillatory behavior using, for example, rational approximation operators. In this work, we introduce a $C^{\infty}$ rational quasi-histopolation operator, for bounded (integrable) functions, which reconstruct a function by defeating both the Runge and Gibbs phenomena. A key element of our approach is to blend local histopolation polynomials on a few nodes using multinode Shepard functions as blending functions. Several numerical experiments demonstrate the accuracy of our method.

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arXiv2025

Bivariate polynomial histopolation techniques on Padua, Fekete and Leja triangles

This paper explores the reconstruction of a real-valued function $f$ defined over a domain $Ω\subset \mathbb{R}^2$ using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of $f$ over a given triangulation $\mathcal{T}_N$ of $Ω$ are available and seek a bivariate polynomial that approximates $f$ using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles $\mathcal{T}_M \subset \mathcal{T}_N$ for histopolation, ensuring both the solvability and the well-conditioning of the problem. The remaining triangles can potentially be used to enhance the accuracy of the polynomial approximation through a simultaneous regression. We will introduce histopolation and combined histopolation-regression methods using the Padua points, discrete Leja sequences, and approximate Fekete nodes. The proposed algorithms are implemented and evaluated through numerical experiments that demonstrate their effectiveness in function approximation.

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arXiv2024

Constrained mock-Chebyshev least squares approximation for Hermite interpolation

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 approach leverages both function and derivative evaluations, resulting in more accurate approximations. Numerical experiments are implemented in order to illustrate the effectiveness of the proposed method.

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arXiv2024

Polynomial histopolation on mock-Chebyshev segments

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-Chebyshev subset interpolation and the constrained mock-Chebyshev least-squares approximation. The high accuracy and the numerical stability achieved by these techniques motivate us to extend these methods to histopolation, a polynomial interpolation method based on segmental function averages. While classical polynomial interpolation relies on function evaluations at specific nodes, histopolation leverages averages of the function over subintervals. In this work, we introduce three types of mock-Chebyshev approaches for segmental interpolation and theoretically analyse the stability of their Lebesgue constants, which measure the numerical conditioning of the histopolation problem under small perturbations of the segments. We demonstrate that these segmental mock-Chebyshev approaches yield a quasi-optimal logarithmic growth of the Lebesgue constant in relevant scenarios. Additionally, we compare the performance of these new approximation techniques through various numerical experiments.

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arXiv2024

Two one-parameter families of nonconforming enrichments of the Crouzeix-Raviart finite element

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix--Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.

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