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Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) by T. Gerstner and M. Griebel) is explored., Comment: 24 pages, 1 figure

It is well known that Lagrange interpolation based on equispaced nodes can yield poor results. Oscillations may appear when using high degree polynomials. For functions of one variable, the most celebrated example has been provided by Carl Runge in 1901, who showed that higher degrees do not always improve interpolation accuracy. His example was then extended to multivariate calculus and in this work we show that it is meaningful, in an appropriate sense, also for Whitney edge elements, namely for differential 1-forms.

In this article, the optimal control for nonlinear SIRC model is studied in fractional order using the Caputo fractional derivative. Graph signal flow is given of the model and simulated by Simulink/Matlab which helps in discussing the topological structure of the model. Dynamics of the system versus certain parameters are studied via bifurcation diagrams, Lyapunov exponents and Poincare maps. The existence of a uniformly stable solution is proved after control. The obtained results display the activeness and suitability of the Mittag Generalized-Leffler function method (MGLFM). The approximate solution of the fractional order SIRC model using MGLFM is explained by giving the figures of solutions before and after control. Also, we plot the 3D relationships with different alpha (fractional order) which display the originality and suitability of the results.

International audience; In [12], the authors provide an evaluation of combinations of hierarchical watersheds, showing that some combinations outperform individual hierarchical watersheds. In this article, we aim to achieve a deeper understanding of those combinations. We study which of the four combinations evaluated in [12] always result in flattened (simplified) hierarchical watersheds. Then, we provide a sufficient condition for a combination to always output flattened hierarchical watersheds, and a new combining function that outputs flattened hierarchical watersheds. We conclude that, among the combinations assessed in [12], only a particular case of combinations with infimum always result in flattened hierarchical watersheds.

In December 2019, a new outbreak in Wuhan, China has attracted world-wide attention, the virus then spread rapidly in most countries of the world, the objective of this paper is to investigate the mathematical modelling and dynamics of a novel coronavirus (COVID-19) with Caputo-Fabrizio fractional derivative in the presence of quarantine and isolation strategies. The existence and uniqueness of the solutions for the fractional model is proved using fixed point iterations, the fractional model are shown to have disease-free and an endemic equilibrium point.We construct a fractional version of the four-steps Adams-Bashforth method as well as the error estimate of this method. We have used this method to determine the numerical scheme of this model and Matlab program to illustrate the evolution of the virus in some countries (Morocco, Qatar, Brazil and Mexico) as well as to support theoretical results. The Least squares fitting is a way to find the best fit curve or line for a set of points, so we apply this method in this paper to construct an algorithm to estimate the parameters of fractional model as well as the fractional order, this model gives an estimate better than that of classical model.

Developing a robust algorithm to diagnose and quantify the severity of the novel coronavirus disease 2019 (COVID-19) using Chest X-ray (CXR) requires a large number of well-curated COVID-19 datasets, which is difficult to collect under the global COVID-19 pandemic. On the other hand, CXR data with other findings are abundant. This situation is ideally suited for the Vision Transformer (ViT) architecture, where a lot of unlabeled data can be used through structural modeling by the self-attention mechanism. However, the use of existing ViT may not be optimal, as the feature embedding by direct patch flattening or ResNet backbone in the standard ViT is not intended for CXR. To address this problem, here we propose a novel Multi-task ViT that leverages low-level CXR feature corpus obtained from a backbone network that extracts common CXR findings. Specifically, the backbone network is first trained with large public datasets to detect common abnormal findings such as consolidation, opacity, edema, etc. Then, the embedded features from the backbone network are used as corpora for a versatile Transformer model for both the diagnosis and the severity quantification of COVID-19. We evaluate our model on various external test datasets from totally different institutions to evaluate the generalization capability. The experimental results confirm that our model can achieve state-of-the-art performance in both diagnosis and severity quantification tasks with outstanding generalization capability, which are sine qua non of widespread deployment., Graphical abstract Image, graphical abstract

This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments., Comment: 279 pages, 136 figures, document produced partially as outcome of the conference Double Higgs Production at Colliders - Fermilab - Chicago (US) 4 - 9 September 2018. Submitted to Review in Physics. The editors can be contacted at the following address: hh-2018-paper-editors@cern.ch

We illustrate a suitable adaptation and modification of classical epidemic evolution models that proves helpful in the study of Covid-19 spread in Italy.

Highlights • COVID-19 pandemic is modeled via the fractional order SIDARTHE mathematical model. • The solution existence of the proposed COVID-19 pandemic fractional order SIDARTHE mathematical model is proved. • Four optimal control plans are produced and implemented for the proposed COVID-19 pandemic fractional order SIDARTHE mathematical model. • Fractional order numerical study is applied for the proposed COVID-19 pandemic fractional order SIDARTHE mathematical model., Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose a high-order ENO method based on radial basis function (RBF) to solve hyperbolic conservation laws on general two-dimensional grids. The radial basis function reconstruction offers a flexible way to deal with ill-conditioned cell constellations. We introduce a smoothness indicator based on RBFs and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting which circumvents the stagnation of the error and keeps the condition number of the reconstruction bounded. We conclude with several challenging numerical examples in two dimensions to show the robustness of the method.

Highlights • A review about the use and evolution of IoT solutions in smart health care is presented. • The work discusses the role of AI in the Internet of Medical Things context. • Particular attention is given to solutions and approaches exploiting Fog and Edge computing for running AI algorithms., In recent times, we assist to an ever growing diffusion of smart medical sensors and Internet of things devices that are heavily changing the way healthcare is approached worldwide. In this context, a combination of Cloud and IoT architectures is often exploited to make smart healthcare systems capable of supporting near realtime applications when processing and performing Artificial Intelligence on the huge amount of data produced by wearable sensor networks. Anyway, the response time and the availability of cloud based systems, together with security and privacy, still represent critical issues that prevents Internet of Medical Things (IoMT) devices and architectures from being a reliable and effective solution to the aim. Lately, there is a growing interest towards architectures and approaches that exploit Edge and Fog computing as an answer to compensate the weaknesses of the cloud. In this paper, we propose a short review about the general use of IoT solutions in health care, starting from early health monitoring solutions from wearable sensors up to a discussion about the latest trends in fog/edge computing for smart health.

Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P estimates the geodesic curvature of C. This result allows us to evaluate the geodesic curvature of discrete curves on surfaces. In particular, we apply such result to planar and spherical 4-point angle-based subdivision schemes. We show that such schemes cannot generate in general $G^2$-continuous curves. We also give a novel example of $G^2$-continuous subdivision scheme on the unit sphere using only points and discrete geodesic curvature called curvature-based 6-point spherical scheme.

Most classical finite element schemes for the (Navier–)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior force term, using ${\boldsymbol{H}}(\operatorname{div})$-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix–Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart–Thomas and Brezzi–Douglas–Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a two- and a three-dimensional test case.

This article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319–333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor–corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately $$4-\alpha $$ , where $$\alpha $$ is the order of time derivative.

We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from $m$ measurements obtained through known linear functionals on $\mathcal{H}$. Problems of this type are well studied \cite{MRW} under an assumption that $u$ belongs to a prescribed model class, e.g. a known compact subset of $\mathcal{H}$. Motivated by reduced modeling for parametric partial differential equations, this paper considers another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$, or more generally, how well $u$ can be approximated by each $k$-dimensional subspace $V_k$ of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$. A recovery algorithm for the one-space formulation, proposed in \cite{MPPY}, is proven here to be optimal and to have a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multi-space case for which it is shown that the set of all $u$ satisfying the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is to identify any point in this intersection which can provide a much better accuracy than in the one-space problem. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem. A detailed analysis of one of them provides a posteriori performance estimates for the iterates, stopping criteria, and convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$., 27 pages

In this paper, a novel high-order numerical method is formulated to solve fractional differential equations. The fractional derivative is described in the Caputo sense due to its applicability to real-world phenomena. First, the fractional differential equation is reduced into a Volterra-type integral equation by applying the Laplace and inverse Laplace transform. Then, the piecewise Lagrange interpolation polynomial of degree five is utilized to approximate unknown function. The truncation error estimates for the novel schemes is derived, and it is theoretically established that the order of convergence of the numerical method is $O(h^6)$. The stability analysis of the novel method is also carefully investigated. Numerical examples are given to show the accuracy, applicability and the effectiveness of the novel method.

In this paper, we propose an interpolation formula for periodic functions. This formula can be regarded as an analog of the Sinc approximation, which is an interpolation formula for functions defined on the entire infinite interval. Theoretical error analysis and numerical examples show that the proposed formula converges exponentially for analytic periodic functions., Comment: 9 pages, 1 figure I have decided to withdraw the paper because the contents of the paper were already proposed by other researchers

In 2012 the ATLAS and CMS collaborations discovered at the LHC the Higgs boson decaying to vector bosons. This discovery has provided a strong indication that the mechanism of Electroweak Symmetry Breaking (EWSB) is similar to the one predicted by Brout-Englert-Higgs (BEH) nearly 50 years before. Since then, one of the priorities of the LHC program, as well as of the majority of the future collider proposals, is to measure directly the parameters of the EWSB potential. The goal is to identify if it has indeed the straightforward quartic shape predicted by BEH or it is more complex, as the result of an unexplored physics nature. The answer to this major scientific question will have a considerable impact on our understanding of vacuum properties and the history of the universe through the EWSB during the Big Bang. The only direct way to probe these couplings is through the measure of the production of multiple Higgs bosons, two being the simplest case. In this paper, we present a comprehensive review of the current searches and the state of the art insights on the topic. In particular, we explain why this ambitious project is even more challenging than the discovery of the Higgs boson itself. Finally, we sketch the plans of the HEP community for how to access the parameters of the BEH mechanism. This review is adapted to a curious reader familiar with particle physics in general or a scientist who wants to have a landscape overview of the topic.

We propose a novel approach to allocating resources for expensive simulations of high fidelity models when used in a multifidelity framework. Allocation decisions that distribute computational resources across several simulation models become extremely important in situations where only a small number of expensive high fidelity simulations can be run. We identify this allocation decision as a problem in optimal subset selection, and subsequently regularize this problem so that solutions can be computed. Our regularized formulation yields a type of group lasso problem that has been studied in the literature to accomplish subset selection. Our numerical results compare performance of algorithms that solve the group lasso problem for algorithmic allocation against a variety of other strategies, including those based on classical linear algebraic pivoting routines and those derived from more modern machine learning-based methods. We demonstrate on well known synthetic problems and more difficult real-world simulations that this group lasso solution to the relaxed optimal subset selection problem performs better than the alternatives., Comment: 27 pages, 10 figures

The Floater--Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the Floater--Hormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice., Comment: 13 pages, 4 figures, 1 table

In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the interval $0, Comment: To appear in Involve-A Journal of Mathematics, 16 pages

We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^4$$\end{document}f∈C4 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{(4)}\ge 0$$\end{document}f(4)≥0 and possible endpoint singularities the algorithm produces an approximation to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _a^bf(x)\,{\mathrm d}x$$\end{document}∫abf(x)dx within a given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε asymptotically as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}ε→0. Moreover, it is optimal among all adaptive Simpson quadratures, i.e., needs the minimal number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n(f,\varepsilon )$$\end{document}n(f,ε) of function evaluations to obtain an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-approximation and runs in time proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n(f,\varepsilon )$$\end{document}n(f,ε).

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multilevel approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example. For the first time, we will also prove error estimates for derivatives and give approximation orders in terms of the fill distance of the finest data set.

The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G^{2} $$\end{document}G2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G^{1} $$\end{document}G1-approximation schemes are introduced to approximate the popular generalized Cornu spirals with the help of the parametric rational cubic trigonometric Bézier curves. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G^{2} $$\end{document}G2-approximation scheme has two free parameters whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G^{1} $$\end{document}G1-approximation scheme has four free parameters. To approximate the generalized Cornu spirals, the values of these free parameters are optimized by the minimization of the maximum relative curvature error of approximation. By comparing the relative curvature errors of approximation schemes, the developed approximation schemes are found less erroneous and more efficient than the existing GCS approximation schemes.

I want to prove that all classical techniques of interpolation and approximation as Lagrange, Taylor, Hermite interpolations Beziers interpolants, Quasi interpolants, Box splines and others (radial splines, simplicial splines) are derived from a \textbf{unique} simple hilbertian scheme. For sake of simplicity, we shall consider only elementary examples which could be easily generalized.

Author(s): Lo, B; Minden, V; Colella, P | Abstract: A new method for solving the transverse part of the free-space Maxwell equations in three dimensions is presented. By taking the Helmholtz decomposition of the electric field and current sources and considering only the divergence-free parts, we obtain an explicit real-space representation for the transverse propagator that explicitly respects finite speed of propagation. Because the propagator involves convolution against a singular distribution, we regularize via convolution with smoothing kernels (B-splines) prior to sampling based on a method due to Beyer and LeVeque (1992). We show that the ultimate discrete convolutional propagator can be constructed to attain an arbitrarily high order of accuracy by using higher-order regularizing kernels and finite difference stencils and that it satisfies von Neumann's stability condition. Furthermore, the propagator is compactly supported and can be applied using Hockney's method (1970) and parallelized using the same observation as made by Vay, Haber, and Godfrey (2013), leading to a method that is computationally efficient.

We study the problem of interpolation by a complete spline of 2n − 1 degree given in B-spline representation. Explicit formulas for the first nand the last ncoefficients of B-spline decomposition are found. It is shown that other B-spline coefficients can be computed as a solution of a banded system of an equitype linear equations.

International audience; A new nonlinear representation of multiresolution decompositions and new thresholding adapted to the presence of discontinuities are presented and analyzed. They are based on a nonlinear modification of the multiresolution details coming from an initial (linear or nonlinear) scheme and on a data dependent thresholding. Stability results are derived. Numerical advantages are demonstrated on various numerical experiments.

We exhibit efficient algorithms to perform the following task: Given a function $f$ defined on a finite subset $E \subset \mathbb R^n$, compute a $C^m$ function $F$ on $\mathbb R^n$, with a controlled $C^m$ norm, that approximates $f$ on the subset $E$.

It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L2 or L ∞ norms of interpolants can be bounded above by discrete ℓ2 and ℓ ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance. peerReviewed

2004-02; International audience; We prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen in Euclidean spaces. Assuming Jensen's hypotheses, our method yields numerical schemes for computing, in euclidean $\mathbb R$, the solution of viscosity of equation $\Delta_\infty=0$ with Dirichlet's condition.

2004-16; International audience; In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented.

A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numerically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface., Comment: 21 pages, 7 figures. Submitted to Appl. Math. Comput

We derive analyticity criteria for explicit error bounds and an exponential rate of convergence of the magic point empirical interpolation method introduced by Barrault et al. (2004). Furthermore, we investigate its application to parametric integration. We find that the method is well-suited to Fourier transforms and has a wide range of applications in such diverse fields as probability and statistics, signal and image processing, physics, chemistry and mathematical finance. To illustrate the method, we apply it to the evaluation of probability densities by parametric Fourier inversion. Our numerical experiments display convergence of exponential order, even in cases where the theoretical results do not apply.

International audience; Spline quasi-interpolants with best approximation orders and small norms are useful in several applications. In this paper, we construct the so-called near-best discrete and integral quasi-interpolants based on H-splines, i.e., B-splines with regular hexagonal supports on the uniform three-directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials, and minimize an upper bound of their infinite norms depending on a finite number of free parameters. We show that this problem has always a solution, but it is not unique in general. Concrete examples of these types of quasi-interpolants are given in the two last sections.

Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires the solution of a dense linear system. Therefore, much attention has been given to iterative methods. In this paper, we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data. Thus, our method applies to the corresponding Toeplitz matrices. The number of iterations required to achieve a given tolerance is independent of the number of variables.

We introduce a method for calculating rational interpolants when some (but not necessarily all) of their poles are prescribed. The algorithm determines the weights in the barycentric representation of the rationals; it simply consists in multiplying each interpolated value by a certain number, computing the weights of a rational interpolant without poles, and finally multiplying the weights by those same numbers. The supplementary cost in comparison with interpolation without poles is about (v + 2)N, where v is the number of poles and N the number of interpolation points. We also give a condition under which the computed rational interpolation really shows the desired poles.

Positive polynomial operator that approximates Urison operator, when integration domain is a "regular triangle" is investigated. We obtain Bernstein Polynomials as a particular case.

For the last almost three decades, since the famous Buchberger-M��ller(BM) algorithm emerged, there has been wide interest in vanishing ideals of points and associated interpolation polynomials. Our paradigm is based on the theory of bivariate polynomial interpolation on cartesian point sets that gives us related degree reducing interpolation monomial and Newton bases directly. Since the bases are involved in the computation process as well as contained in the final output of BM algorithm, our paradigm obviously simplifies the computation and accelerates the BM process. The experiments show that the paradigm is best suited for the computation over finite prime fields that have many applications., 24 pages, 7 figures, submitted to JCAM

In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients whenever a nodes is added to or removed from the interpolation. The second algorithm, which constructs the interpolation coefficients by decomposing the Vandermonde-like matrix iteratively, can not be used to update or downdate an interpolation, yet is more numerically stable than the first algorithm and is more efficient when the coefficients of multiple interpolations are to be computed over the same set of nodes., Comment: 18 pages, submitted to the Journal of Scientific Computing.

Through introducing a new iterative formula for divided differnce using Neville's and Aitken's algorithms,we study new iterative methods for interpolation,numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly or unevanly spaced data. Basic computer algorithms for new methods are given, Comment: 11 pages, comments are invited

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics., Comment: 28 pages

The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form., Comment: This paper has been withdrawn by the author due to a crucial sign error in main formula

A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville elimination. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate., Comment: 13 pages. We have extended the numerical experiments

We prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen in Euclidean spaces. Assuming Jensen's hypotheses, our method yields numerical schemes for computing, in euclidean $\mathbb R$, the solution of viscosity of equation $��_\infty=0$ with Dirichlet's condition., 2004-02

This note proposes a method for pricing high-dimensional American options based on modern methods of multidimensional interpolation. The method allows using sparse grids and thus mitigates the curse of dimensionality. A framework of the pricing algorithm and the corresponding interpolation methods are discussed, and a theorem is demonstrated that suggests that the pricing method is less vulnerable to the curse of dimensionality. The method is illustrated by an application to rainbow options and compared to Least Squares Monte Carlo and other benchmarks., 12 pages, tables omitted

The goal of this article is that of understanding how the oscillation and concentration effects developed by a sequence of functions in $\mathbb{R}^{d} $ are modified by the action of Sampling and Reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high frequency behavior of bounded sequences in $L^{2}(mathbb{R}^{d}) $. We actually present explicit formulas that make possible to compute such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.