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.

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.

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. In contrast to classical WENO schemes, the progressive strategy allows the recursive recovery of high-order accuracy even when the largest stencil is affected by a discontinuity. The method achieves high-order accuracy in smooth regions together with 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.

In this paper, we introduce a nonlinear subdivision scheme designed to converge to piecewise smooth functions. The scheme is formulated as a convex nonlinear combination of Chaikin-type linear schemes. In regions of regularity, it attains a high level of smoothness and accuracy, comparable to that of the C2(R) fourth-order linear algorithm that it approximates. Near discontinuities, the scheme successfully avoids Gibbs-type oscillations and does not introduce any artificial intermediate points within the intervals containing the discontinuities. Furthermore, the proposed scheme adaptively responds to discontinuities while utilizing the maximum number of available points within smooth regions, thus ensuring the highest attainable order of accuracy in every case. A set of numerical experiments is provided, illustrating and supporting the theoretical analysis with respect to convergence, stability, accuracy, and regularity across both smooth regions and regions with discontinuities. To the best of our knowledge, this represents the first algorithm capable of simultaneously exhibiting all these desirable properties.

In this article we try to provide a review of recent works that propose numerical solutions of interface problems in the context of interpolation and approximation. We examine works that deal with adapted methods for polynomial interpolation and approximation, multiresolution and subdivision schemes, Hermite splines, approximation by B-splines, and some applications to numerical integration formulas. The discussion considers the applications, benefits, and potential challenges of these techniques when facing the solution of this kind of interface problems. This review is intended to serve as a reference for researchers who are interested in using these methods to approximate interface problems in this context.

In this paper, a novel space-time combined discretization for scalar hyperbolic conservation laws is presented. The resulting scheme is based on obtaining a direct relationship between high order time derivatives and high order space derivatives, in which no mixed derivatives are involved. As a result, schemes of arbitrarily high order in both space and time can be derived, while in turn having a numerical domain of dependence growing linearly with respect to the order, as opposed to classical high order in space and time discretizations, which usually have a numerical domain of dependence that grows quadratically with respect to the order. Finally, in order to tackle discontinuities, WENO reconstructions with efficient weight design and unconditionally optimal accuracy near critical points are used.

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 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.

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.

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 Rn, 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.

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.

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.

Non-linear interpolation techniques have been extensively used in several settings such as image processing, data analysis, finance, economy, and overall in solving numerically partial differential equations. In this context, weighted essentially non-oscillatory (WENO) methods have been developed for different problems. The idea of the non-linear methods is to use the data information to avoid the data points where discontinuities or steep gradients appear. The classic WENO combines some polynomial interpolators with a certain order of accuracy to obtain a new interpolator that is more accurate in smooth zones and, at least, as accurate as the original ones when the data present a discontinuity. The construction of the classic WENO method has been formulated for uniform grids and to approximate a particular point of the grid. Recently, Martí et al. in [J. Sci. Comput. 100 (2024), 6] introduced a WENO version for non-uniform grids and used it for the solution of conservation laws and hyperbolic systems in the context of finite volume methods. In this paper, this method is extended and improved, focusing on three aspects. Its efficiency: a more efficient procedure to iteratively compute the reconstruction polynomials for the different types of reconstructions considered in this work is proposed. Its versatility: the designed method can be used in the context of a non-uniform stencil with the reconstruction point located at an arbitrary position. Its order of accuracy: some theoretical results showing that the order of the proposed scheme is optimal on smooth data and as high as possible near discontinuities are presented. Finally, some numerical experiments are performed to illustrate the theoretical results that are presented.

. This paper is devoted to the construction and analysis of an iterative Moser-secant's method. The principal properties that it presents is that is free of both inverses and jacobians. The order of convergence is the same as the classical secant method. A similar Moser-Steffesensen's method with second order is also proposed. Some interesting applications and some numerical experiments, including the approximation of optimization problems, are presented.

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.

Shepard’s 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 ℝ^n near discontinuities along a hypersurface in ℝ^n-1 is challenging for the Shepard’s method or any other linear technique for sparse data due to the inherent difficulty in accurately capturing sharp transitions. This article is devoted to constructing a non-linear Shepard’s 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 reduce the smearing of the traditional Shepard’s method by incorporating WENO’s adaptive and non-linear weighting mechanism. To address this challenge, we non-linearly modify the weight function in a general Shepard’s method, considering any weight function, rather than relying solely on the inverse of the distance squared. This approach effectively reduces the smearing of discontinuities providing a sharper approximation. The numerical experiments presented demonstrate the superior performance of the new method close to the discontinuities and confirm the theoretical results exposed in this manuscript.

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.

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.

Lane-Riesenfeld's subdivision schemes provide a versatile tool for generating smooth and detailed curves and surfaces, making them valuable assets in computer graphics and geometric modeling. These schemes offer flexibility in controlling the regularity of the limit function by adjusting some parameters. In this paper, we propose a nonlinear modification of these algorithms for piecewise smooth functions. The new algorithm converges to piecewise smooth limit functions maintaining the high regularity of Lane-Riesenfeld's subdivision schemes in smooth zones. The practical results presented show a very good numerical behavior of the proposed algorithms.

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.

This article is devoted to introducing new spline quasi-interpolants for the sharp approximation of data in one and several dimensions. The new construction aims to obtain accurate approximations close to singularities of the function from which the data are obtained. The technique relies in an accurate knowledge of the position of the singularity, which can be known or approximated, and that allows for the obtention of accurate approximations of the jumps of the function and its derivatives at this point. With this information, it is possible to compute correction terms for the B-spline bases of the spline that are affected by the presence of singularities. The result is a piecewise smooth reconstruction with full accuracy (meaning by full accuracy, the accuracy of the spline at smooth zones). The numerical experiments presented support the theoretical results obtained.