arXiv subjects

Juan Ruiz-Alvarez

Publications and source records attributed to Juan Ruiz-Alvarez.

7 recordsLinked to original sources

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.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2022

Numerical integration rules with improved accuracy close to singularities

Sometimes it is necessary to obtain a numerical integration using only discretised data. In some cases, the data contains singularities which position is known but does not coincide with a discretisation point, and the jumps in the function and its derivatives are available at these positions. The motivation of this paper is to use the previous information to obtain numerical quadrature formulas that allow approximating the integral of the discrete data over certain intervals accurately. This work is devoted to the construction and analysis of a new nonlinear technique that allows to obtain accurate numerical integrations of any order using data that contains singularities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depends on the size of the jumps of the function and its derivatives at the singularities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas even close to the singularities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. The numerical experiments performed allow us to confirm the theoretical conclusions reached in this paper.

math.NA↗

arXiv2022

Global and explicit approximation of piecewise smooth 2D functions from cell-average data

Given cell-average data values of a piecewise smooth bivariate function $f$ within a domain $Ω$, we look for a piecewise adaptive approximation to $f$. We are interested in an explicit and global (smooth) approach. Bivariate approximation techniques, as trigonometric or splines approximations, achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. Whereas the boundary of $Ω$ is assumed to be known, the subdivision of $Ω$ to subdomains on which $f$ is smooth is unknown. The first challenge of the proposed approximation algorithm would be to find a good approximation to the curves separating the smooth subdomains of $f$. In the second stage, we simultaneously look for approximations to the different smooth segments of $f$, where on each segment we approximate the function by a linear combination of basis functions $\{p_i\}_{i=1}^M$, considering the corresponding cell-averages. A discrete Laplacian operator applied to the given cell-average data intensifies the structure of the singularity of the data across the curves separating the smooth subdomains of $f$. We refer to these derived values as the signature of the data, and we use it for both approximating the singularity curves separating the different smooth regions of $f$. The main contributions here are improved convergence rates to both the approximation of the singularity curves and the approximation of $f$, an explicit and global formula, and, in particular, the derivation of a piecewise smooth high order approximation to the function.

math.NA↗

arXiv2021

Explicit multivariate approximations from cell-average data

Given gridded cell-average data of a smooth multivariate function, we present a constructive explicit procedure for generating a high-order global approximation of the function. One contribution is the derivation of high order approximations to point-values of the function directly from the cell-average data. The second contribution is the development of univariate B-spline based high order quasi-interpolation operators using cell-average data. Multivariate spline quasi-interpolation approximation operators are obtained by tensor products of the univariate operators.

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