arXiv subjects

Dionisio F. Yáñez

Publications and source records attributed to Dionisio F. Yáñez.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Non-Linear Partition of Unity Methods for Scattered Data Interpolation with Discontinuities

Scattered data approximation with discontinuities is challenging due to the Gibbs phenomenon, which significantly reduces accuracy near interfaces. The recently introduced Non-Linear Partition of Unity Method (NL-PUM) addresses this by combining Radial Basis Function (RBF) interpolation with a non-linear Weighted Essentially Non-Oscillatory (WENO) strategy. While effective, NL-PUM's performance relies heavily on two fixed hyperparameters: the RBF shape parameter and the patch radius. This work extends NL-PUM by adapting both hyperparameters locally using Leave-One-Out Cross-Validation (LOOCV) minimized via Global Optimization with Optimistic Improvement (GOOI). Our main innovation is a smoothness indicator linking a discontinuity-aware shrinkage process to LOOCV-based radius selection: patches in smooth regions remain unchanged, while those near discontinuities automatically shrink to avoid crossing the interface. The resulting method, LOOCV-NL-PUM-GOOI, requires no prior knowledge of interface geometry and introduces no extra cost beyond standard adaptive shape parameter selection. Numerical experiments on synthetic test functions with jump discontinuities and a real-data application to Norwegian Fjords elevation data confirm that this approach substantially reduces approximation error near discontinuities while preserving full accuracy in smooth regions.

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arXiv2026

Data-dependent approximation through RBF

In this article we present a modification of classical Radial Basis Function (RBF) interpolation techniques aimed at reducing oscillations near discontinuities in one and two dimensions. Our approach introduces an adaptive mechanism by varying the shape parameter of the RBFs and making it data-dependent, forcing it to tend to infinity in the vicinity of discontinuities. This modification results in kernel functions that locally resemble %Kronecker delta functions, effectively minimizing spurious oscillations. To detect discontinuities, we employ smoothness indicators: for grid-based data, these are computed as undivided second-order differences squared. For scattered data, we use least squares approximations of the Laplacian multiplied by the square of the mean local separation of the stencil points, and then squared. These indicators guide the adaptive adjustment of the shape parameter. We prove the invertibility of the resulting interpolation matrix and propose a solution strategy that maintains the condition number comparable to that of a system where points near discontinuities are excluded. Numerical experiments in one and two dimensions demonstrate that the proposed method significantly reduces oscillations near discontinuities across various kernel types, whether locally or globally supported. At the same time, the interpolation accuracy and matrix conditioning in smooth regions remain essentially unchanged, as measured by the infinity norm of the error and the condition number.

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arXiv2026

Multivariate Data-dependent Partition of Unity based on Moving Least Squares method

Data approximation is essential in fields such as geometric design, numerical PDEs, and curve modeling. Moving Least Squares (MLS) is a widely used method for data fitting; however, its accuracy degrades in the presence of discontinuities, often resulting in spurious oscillations similar to those associated with the Gibbs phenomenon. This work extends the integration of MLS with the Weighted Essentially Non-Oscillatory (WENO) method and with an innovative partition of unity approach to higher dimensions. We propose a data-dependent operator using the novel Non-Linear Partition of Unity based on Moving Least Squares method in $\mathbb{R}^n$, which improves accuracy near discontinuities and maintains high-order accuracy in smooth regions. We demonstrate some theoretical properties of the method and perform numerical experiments to validate its effectiveness.

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arXiv2026

A Cell-Average Non-Separable Progressive Multivariate WENO Method for Image Processing Applications

Accurate and efficient reconstruction techniques are essential in multiresolution analysis and image compression, particularly when the data are represented as cell averages. In this work, we present a non-separable progressive multivariate Weighted Essentially Non-Oscillatory (WENO) scheme specifically designed for cell-average data, with applications to digital image processing. The proposed method extends Harten's multiresolution framework through a non-linear WENO reconstruction adapted to the cell-average context, achieving high-order accuracy in smooth regions and stable, non-oscillatory behavior near discontinuities. We also establish theoretical results regarding the consistency and approximation properties of the method. Finally, several numerical experiments on piecewise smooth functions and digital images are presented to demonstrate its performance and validate its effectiveness against the linear Lagrange reconstruction of the same order of accuracy.

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arXiv2026

Implicit-explicit all-speed schemes for compressible Cahn-Hilliard-Navier-Stokes equations

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and is specifically designed to handle the challenges posed by the low Mach number limit, where the system approaches to an incompressible behavior. In this regime, standard explicit schemes suffer from severe time-step restrictions due to fourth-order diffusion terms and the stiffness induced by fast acoustic waves. To overcome this, we employ an IMEX strategy which splits the governing equations into stiff and non-stiff components. The stiff terms, arising from pressure, viscous forces and fourth-order Cahn-Hilliard contributions, are treated implicitly, while the remaining are dealt explicitly.

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arXiv2026

Weighted least squares subdivision schemes for noisy data on triangular meshes

This paper presents and analyses a new family of linear subdivision schemes to refine noisy data given on triangular meshes. The subdivision rules consist of locally fitting and evaluating a weighted least squares approximating first-degree polynomial. This type of rules, applicable to any type of triangular grid, including finite grids or grids containing extraordinary vertices, are geometry-dependent which may result in non-uniform schemes. For these new subdivision schemes, we are able to prove reproduction, approximation order, denoising capabilities and, for some special type of grids, convergence as well. Several numerical experiments demonstrate that their performance is similar to advanced local linear regression methods but their subdivision nature makes them suitable for use within a multiresolution context as well as to deal with noisy geometric data as shown with an example.

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arXiv2025

Implicit-explicit schemes for compressible Cahn-Hilliard-Navier-Stokes equations on staggered grids

We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations discretized on staggered (MAC) grids. The scheme is based on finite difference approximations that ensure a stable coupling among the velocity, density and phase field, with symmetric operators acting on the discretized viscosity terms. Standard explicit methods suffer from severe time-step restrictions due to the presence of second to fourth-order diffusion terms introduced by the Cahn-Hilliard and Navier-Stokes operators. To overcome these challenges, we develop an IMEX Runge-Kutta scheme that treats the stiff terms implicitly while the convective terms are dealt with explicitly, with the advantage that only linear systems are solved at each stage. Numerical experiments are performed to verify the stability, accuracy and efficiency of the proposed approach.

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arXiv2025

Linear minimum-variance approximants for noisy data

Inspired by recent developments in subdivision schemes founded on the Weighted Least Squares technique, we construct linear approximants for noisy data in which the weighting strategy minimizes the output variance, thereby establishing a direct correspondence with the Generalized Least Squares and the Minimum-Variance Formulas methodologies. By introducing annihilation-operators for polynomial spaces, we derive usable formulas that are optimal for general correlated non-uniform noise. We show that earlier subdivision rules are optimal for uncorrelated non-uniform noise and, finally, we present numerical evidence to confirm that, in the correlated case, the proposed approximants are better than those currently used in the subdivision literature.

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arXiv2025

Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D

The approximation of data is a fundamental challenge encountered in various fields, including computer-aided geometric design, the numerical solution of partial differential equations, or the design of curves and surfaces. Numerous methods have been developed to address this issue, providing good results when the data is continuous. Among these, the Moving Least Squares (MLS) method has proven to be an effective strategy for fitting data, finding applications in both statistics and applied mathematics. However, the presence of isolated discontinuities in the data can lead to undesirable artifacts, such as the Gibbs phenomenon, which adversely affects the quality of the approximation. In this paper, we propose a novel approach that integrates the Moving Least Squares method with the well-established non-linear Weighted Essentially Non-Oscillatory (WENO) method. This combination aims to construct a non-linear operator that enhances the accuracy of approximations near discontinuities while maintaining the order of accuracy in smooth regions. We investigate the properties of this operator in one dimension, demonstrating its effectiveness through a series of numerical experiments that validate our theoretical findings.

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arXiv2025

A general correction for numerical integration rules over piece-wise continuous functions

This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and the jump in the function and its derivatives at that position are known. Traditional numerical integration rules are exact for polynomials of certain degree. However, they may not provide accurate results for piece-wise polynomials or functions with discontinuities without modifying the location and number of data points in the formula. Our proposed correction terms address this limitation, enabling the integration rule to conserve its accuracy even in the presence of a jump discontinuity. The numerical experiments that we present support the theoretical results obtained.

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arXiv2025

Non-linear Partition of Unity method

This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations, particularly near discontinuities, by adapting weights dynamically. The method is rooted in the Partition of Unity framework, enabling efficient decomposition of large datasets into subproblems while maintaining accuracy. Smoothness indicators and compactly supported functions ensure precision in regions with discontinuities. Error bounds are calculated and validate its effectiveness, showing improved interpolation in discontinuous and smooth regions. Some numerical experiments are performed to check the theoretical results.

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arXiv2024

Weighted Essentially Non-Oscillatory Shepard method

Shepard method is a fast algorithm that has been classically used to interpolate scattered data in several dimensions. This is an important and well-known technique in numerical analysis founded in the main idea that data that is far away from the approximation point should contribute less to the resulting approximation. Approximating piecewise smooth functions in $\mathbb{R}^n$ near discontinuities along a hypersurface in $\mathbb{R}^{n-1}$ is challenging for the Shepard method or any other linear technique for sparse data due to the inherent difficulty in accurately capturing sharp transitions and avoiding oscillations. This letter is devoted to constructing a non-linear Shepard method using the basic ideas that arise from the weighted essentially non-oscillatory interpolation method (WENO). The proposed method aims to enhance the accuracy and stability of the traditional Shepard method by incorporating WENO's adaptive and nonlinear weighting mechanism. To address this challenge, we will nonlinearly modify the weight function in a general Shepard method, considering any weight function, rather than relying solely on the inverse of the distance squared. This approach effectively reduces oscillations near discontinuities and improves the overall interpolation quality. Numerical experiments demonstrate the superior performance of the new method in handling complex datasets, making it a valuable tool for various applications in scientific computing and data analysis.

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arXiv2024

Data dependent Moving Least Squares

In this paper, we address a data dependent modification of the moving least squares (MLS) problem. We propose a novel approach by replacing the traditional weight functions with new functions that assign smaller weights to nodes that are close to discontinuities, while still assigning smaller weights to nodes that are far from the point of approximation. Through this adjustment, we are able to mitigate the undesirable Gibbs phenomenon that appears close to the discontinuities in the classical MLS approach, and reduce the smearing of discontinuities in the final approximation of the original data. The core of our method involves accurately identifying those nodes affected by the presence of discontinuities using smoothness indicators, a concept derived from the data-dependent WENO method. Our formulation results in a data-dependent weighted least squares problem where the weights depend on two factors: the distances between nodes and the point of approximation, and the smoothness of the data in a region of predetermined radius around the nodes. We explore the design of the new data-dependent approximant, analyze its properties including polynomial reproduction, accuracy, and smoothness, and study its impact on diffusion and the Gibbs phenomenon. Numerical experiments are conducted to validate the theoretical findings, and we conclude with some insights and potential directions for future research.

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arXiv2024

Numerical integration rules based on B-spline bases

In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove the order of approximation and extend the results to several dimensions. Finally, some numerical experiments are performed in order to check the theoretical results.

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arXiv2024

A non-separable progressive multivariate WENO-$2r$ point value

The weighted essentially non-oscillatory {technique} using a stencil of $2r$ points (WENO-$2r$) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of $r+1$ nodes. The result is an interpolant of order $2r$ at the smooth parts and order $r+1$ when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.

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arXiv2024

A note on some bounds between cubic spline interpolants depending on the boundary conditions: Application to a monotonicity property

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the first derivative at the endpoints. Albeit a negative answer appears to be intuitive, we have found no results regarding this fact. In this short work we prove that the answer to that question is actually negative.

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arXiv2023

Subdivision schemes based on weighted local polynomial regression. A new technique for the convergence analysis

The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of simple calculations. However, in some applications, the collected data are given with noise and most of schemes are not adequate to process them. In this paper, we present some new families of binary univariate linear subdivision schemes using weighted local polynomial regression. We study their properties, such as convergence, monotonicity, polynomial reproduction and approximation and denoising capabilities. For the convergence study, we develop some new theoretical results. Finally, some examples are presented to confirm the proven properties.

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arXiv2023

Non-linear WENO B-spline based approximation method

In this work we present a new WENO b-spline based quasi-interpolation algorithm. The novelty of this construction resides in the application of the WENO weights to the b-spline functions, that are a partition of unity, instead to the coefficients that multiply the b-spline functions of the spline. The result obtained conserves the smoothness of the original spline and presents adaption to discontinuities in the function. Another new idea that we introduce in this work is the use of different base weight functions from those proposed in classical WENO algorithms. Apart from introducing the construction of the new algorithms, we present theoretical results regarding the order of accuracy obtained at smooth zones and close to the discontinuity, as well as theoretical considerations about how to design the new weight functions. Through a tensor product strategy, we extend our results to several dimensions. In order to check the theoretical results obtained, we present an extended battery of numerical experiments in one, two and tree dimensions that support our conclussions.

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