We introduce a two-phase approximation method designed to resolve singularities in three-dimensional harmonic Dirichlet problems. The approach utilizes the classical Green's function representation, decomposing the function into its singular and regular components. The singular phase employs Green's formula with the singular part, for which we show that it induces the necessary singularities in the solution. The regular phase then introduces a smooth correction to recover the remaining regular part of the solution. The construction employs high-order quadrature rules in the first phase, followed by collocation with a suitable harmonic basis in the second.

Given a finite number of samples of a continuous set-valued function $F$, mapping an interval to nonempty compact subsets of $\mathbb{R}<^>{d}$, $F: [a,b] o K(\mathbb{R}<^>{d})$, we discuss the problem of computing good approximations of $F$. We also discuss algorithms for a direct high-order evaluation of the graph of $F$, namely, the set $Graph(F)=\{(t,y)\ | \ y\in F(t),\ t\in [a,b]\}\in K(\mathbb{R}<^>{d+1})$. A set-valued function can be continuous and yet have points where the topology of the image sets changes. The main challenge in set-valued function approximation is to derive high-order approximations near these points. In a previous paper, together with Q. Muzaffar, we presented an algorithm for approximating set-valued functions with one-dimensional sets ($d=1$) as images, achieving a high approximation order near points of topology change. Here, we build upon the results and algorithms for the case $d=1$, first in more detail for the important case $d=2$, and later for approximating set-valued functions and their graphs in higher dimensions.

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 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 consider the problem of reconstructing missing data on a smooth manifold from incomplete and nonuniform samples. While classical methods for manifold approximation typically assume quasi-uniform data, their performance deteriorates significantly in the presence of large gaps or holes. We propose a unified framework for manifold data imputation that reduces the problem to function reconstruction on locally defined tangent spaces. The approach combines two complementary strategies. The first is a Fourier-based method that determines missing values by prescribing a decay rate of the discrete Fourier coefficients, thereby enforcing high-order smoothness through a global spectral criterion. The second is a local variational method based on minimizing high-order central differences, leading to sparse least-squares systems with favorable stability and conditioning properties. We establish a discrete inverse estimate linking decay of Fourier coefficients to uniform bounds on high-order divided differences, providing a theoretical foundation for the spectral approach. For the variational method, we analyze existence, uniqueness, and scaling behavior, showing that conditioning depends primarily on the geometry of the missing region. These functional reconstruction techniques are integrated with a moving least-squares projection framework to yield a practical algorithm for manifold completion. Numerical experiments, including reconstruction on surfaces with significant missing regions, demonstrate accurate and stable recovery without requiring a global parameterization. The proposed framework provides a flexible and effective approach to manifold data imputation in challenging settings with incomplete data.

This paper presents a novel two-stage neural network approach to approximate 2D functions with jump discontinuities from data across varying noise conditions, from noise-free to highly noisy scenarios. While traditional methods struggle particularly with discontinuity detection in high-noise environments, our methodology demonstrates consistent performance across a spectrum of noise levels in synthetic testing. Our approach combines a specialized U-Net architecture for discontinuity detection with a Dual UNet architecture for function reconstruction. The first stage uses a custom U-Net specifically designed to detect discontinuities in function data, demonstrating improved performance compared to general-purpose segmentation models on this task. The second stage introduces a Dual UNet architecture that processes probabilistic region classifications through multi-modal input fusion and detail-preserving reconstruction. Testing on synthetic data shows competitive performance compared to existing spline-based methods across the tested range of noise conditions. Our approach handles both open and closed curve discontinuities. Current scope and limitations: The method is designed for functions with a single jump discontinuity on gridded data. As a learning-based approach, it provides empirical performance without theoretical approximation guarantees. Within these constraints, the methodology demonstrates practical value for noisy environments and may have potential applications in domains such as medical imaging, signal processing, and computational fluid dynamics where robustness to noise is important.

Global optimization of black-box functions from noisy samples is a fundamental challenge in machine learning and scientific computing. Traditional methods such as Bayesian Optimization often converge to local minima on multi-modal functions, while gradient-free methods require many function evaluations. We present a novel neural approach that learns to find global minima through iterative refinement. Our model takes noisy function samples and their fitted spline representation as input, then iteratively refines an initial guess toward the true global minimum. Trained on randomly generated functions with ground truth global minima obtained via exhaustive search, our method achieves a mean error of 8.05 percent on challenging multi-modal test functions, compared to 36.24 percent for the spline initialization, a 28.18 percent improvement. The model successfully finds global minima in 72 percent of test cases with error below 10 percent, demonstrating learned optimization principles rather than mere curve fitting. Our architecture combines encoding of multiple modalities including function values, derivatives, and spline coefficients with iterative position updates, enabling robust global optimization without requiring derivative information or multiple restarts.

The evaluation of slowly converging series and infinite integrals is a key challenge in numerical analysis and computational mathematics. In their influential 1981 paper, the author and Avram Sidi introduced two effective nonlinear transformations, the d-transformation for series and the D-transformation for infinite integrals, aimed at speeding up their convergence. This review summarizes, contextualizes, and evaluates their contributions, highlighting the mathematical basis, practical significance, and legacy of their work.

. 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.

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.

We address the problem of approximating a set-valued function F, where F:[a,b]→K(Rd) given its samples {F(a+ih)}i=0N, with h=(b−a)/N. We revisit an existing method that approximates set-valued functions by interpolating signed-distance functions. This method provides a high-order approximation for general topologies but loses accuracy near points where F undergoes topological changes. To address this, we introduce new techniques that enhance efficiency and maintain high-order accuracy across [a,b]. Building on the foundation of previous publication, we introduce new techniques to improve the method’s efficiency and extend its high-order approximation accuracy throughout the entire interval [a,b]. Particular focus is placed on identifying and analyzing the behavior of F near topological transition points. To address this, two algorithms are introduced. The first algorithm employs signed-distance quasi-interpolation, incorporating specialized adjustments to effectively handle singularities at points of topological change. The second algorithm leverages an implicit function representation of Graph(F), offering an alternative and robust approach to its approximation. These enhancements improve accuracy and stability in handling set-valued functions with changing topologies.

In this study, we consider the question of repairing and recovering a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Given a noisy point cloud sampled from a low-dimensional manifold, suppose that part of the scattered data is missing, which results in holes. In these settings, the main goal is to accurately and efficiently reconstruct the information within these gaps. While in three-dimensions the problem has been extensively studied, the challenge of reinstating missing information for high-dimensional manifolds remains open. In this paper, we propose a new approach named Manifold Repairing via Locally Optimal Projection (R-MLOP). The method is defined as a minimization problem with three terms. First, we leverage the spatial proximity to the holes to balance between denoising the data and preserving geometric continuity among points situated along the hole’s boundaries. In addition, a penalty term is added to guarantee a quasi-uniform sampling of the unknown manifold. We prove that the suggested solution recovers missing information inside the hole, with an approximation order that is controlled by the density of the given scattered data as well as the size of the amended hole. The effectiveness of our approach is demonstrated by considering different manifold topologies, for single and multiple-hole repairing, in low and high dimensions.

Subdivision schemes are a very useful tool in computer graphics and geometric modelling, allowing for the generation of curves and smooth surfaces starting from discrete data. Even though linear subdivision schemes are used profusely, non-linear schemes offer more flexibility, allowing for the handling of data with irregularities and making the shape preservation easier. In addition, these schemes are useful to deal with subdivision on manifolds, to correct Gibbs oscillations around singularities, and, in general, to try to tackle problems where linear approaches do not provide satisfactory results. This article reviews 25 years of contributions related to the construction, analysis, and use, in different applications, of non-linear subdivision schemes.

We address the challenge of efficiently approximating piecewise smooth functions, particularly those with jump discontinuities. Given function values on a uniform grid over a domain Ω in Rd, we present a novel B-spline-based approximation framework, using new adaptable quasi-interpolation operators. This approach integrates discontinuity detection techniques, allowing the quasi-interpolation operator to selectively use points from only one side of a discontinuity in both one- and two-dimensional cases. Among a range of candidate operators, the most suitable quasi-interpolation scheme is chosen to ensure high approximation accuracy and efficiency, while effectively suppressing spurious oscillations in the vicinity of discontinuities.

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 paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise.

Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve.

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 form a partition of unity, instead of 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 extensive battery of numerical experiments in one, two, and three dimensions that support our conclusions.