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The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as r (x) = ∑ i = 0 n a i (x) f i / ∑ j = 0 m b j (x) in terms of data values f i and some functions a i and b j for i = 0 , ... , n and j = 0 , ⋯ , m with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with n, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of n. We support our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in P r - Λ k (T) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from "good" sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case. [ABSTRACT FROM AUTHOR]

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1 ≤ p ≤ + ∞ . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε. [ABSTRACT FROM AUTHOR]

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function f ∈ C (I) , by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence { f m (x) } m = 1 ∞ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence { { f m , n (q , α) (x) } n = 1 ∞ } m = 1 ∞ of (q , α) -fractal functions that converges uniformly to f. [ABSTRACT FROM AUTHOR]

Polynomial interpolation or regression models are an important tool in Derivative-free Optimization, acting as surrogates of the real function. In this work, we propose the use of these models in the multiobjective framework of directional direct search, namely the one of Direct Multisearch. Previously evaluated points are used to build quadratic polynomial models, which are minimized in an attempt of generating nondominated points of the true function, defining a search step for the algorithm. Numerical results state the competitiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell' Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. [ABSTRACT FROM AUTHOR]

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for 0 < p ≤ 2 , we study the mean convergence of the Lagrange interpolation polynomials. [ABSTRACT FROM AUTHOR]

In this note, we show that the degree of the interpolation polynomial for equidistant base points is characterized by the regularity of matrices of combinatorical type. [ABSTRACT FROM AUTHOR]

We describe how to perform the backward error analysis for the approximation of exp(A)v by p(s-1A)sv, for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja-Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples. [ABSTRACT FROM AUTHOR]

We discuss matrix polynomials expressed in a Newton basis, and the associated polynomial eigenvalue problems. Properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces. Also, we show how the family of Fiedler pencils can be adapted to matrix polynomials expressed in a Newton basis. These new Newton–Fiedler pencils are shown to be strong linearizations, and some computational aspects related to them are discussed. [ABSTRACT FROM AUTHOR]

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators. [ABSTRACT FROM AUTHOR]

In this work, we present a simple, robust and fast method to the perspective- n -point (P n P) problem for determining the position and orientation of a calibrated camera from known reference points. Our method transfers the pose estimation problem into an optimal problem, and only needs to solve a seventh-order and a fourth-order univariate polynomial, respectively, which makes the processes more easily understood and significantly improves the performance. Additionally, the number of solutions of the proposed method is substantially smaller than existing methods. Experiment results show that the proposed method can stably handle all 3D point configurations, including the ordinary 3D case, the quasi-singular case, and the planar case, and it offers accuracy comparable or better than that of the state-of-art methods, but at much lower computational cost. [ABSTRACT FROM AUTHOR]

In this paper, firstly, based on new recursive algorithms of non-tensor-product-typed bivariate divided differences, scattered data interpolation schemes are constructed in the cases of odd and even interpolating nodes, respectively. Moreover, the corresponding error estimation is worked out, and equivalent formulae are obtained between bivariate high-order non-tensor-product-typed divided differences and high-order partial derivatives. Furthermore, the operation count for the addition/subtractions, multiplication, and divisions approximates O ( n 2 ) in the computation of the interpolating polynomials presented, while the operation count approximates O ( n 3 ) in the case of radial basis functions for sufficiently large n . Finally, several numerical examples show that it is valid for the recursive interpolating polynomial schemes, and these interpolating polynomials change as the order of the interpolating nodes, although the node collection is the same. [ABSTRACT FROM AUTHOR]

Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind. [ABSTRACT FROM AUTHOR]

One of the open problems in hypercomplex analysis is the interpolation of monogenic functions by monogenic polynomials. We consider the case of monogenic functions defined in a domain ofwith values in the algebra of quaternions. The idea is to interpolate these functions by a special system of monogenic polynomials, the so-called pseudo complex polynomials. Quaternionic analysis shows a lot of analogies to complex analysis and the monogenic functions play the role of holomorphic functions. Due to the non-commutative multiplication a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. So, the main goal of this paper is to show that the interpolation problem is solvable for arbitrarily given interpolation nodes and general interpolation data. [ABSTRACT FROM AUTHOR]

The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from univariate primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators and provide promising preliminary results for the bivariate case. [ABSTRACT FROM AUTHOR]

High-order finite difference discontinuity detectors are essential for the location of discontinuities on discretized functions, especially in the application of high-order numerical methods for high-speed compressible flows for shock detection. The detectors are used mainly for switching between numerical schemes in regions of discontinuity to include artificial dissipation and avoid spurious oscillations. In this work a discontinuity detector is analysed by the construction of a piecewise polynomial function that incorporates jump discontinuities present on the function or its derivatives (up to third order) and the discussion on the selection of a cut-off value required by the detector. The detector function is also compared with other discontinuity detectors through numerical examples. [ABSTRACT FROM PUBLISHER]

As a complement to Xu (2012), minimal cubature rules of degree 4 m + 1 for the weight functions W α , β , ± 1 2 ( x , y ) = | x + y | 2 α + 1 | x − y | 2 β + 1 ( ( 1 − x 2 ) ( 1 − y 2 ) ) ± 1 2 on [ − 1 , 1 ] 2 are shown to exist and near minimal cubature rules of the same degree with one node more than minimal are constructed explicitly. The Lagrange interpolation polynomials on the nodes of the near minimal cubature rules are also studied. [ABSTRACT FROM AUTHOR]

We analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation. These errors are particularly relevant for the first barycentric formula with the Chebyshev points of the second kind. Here, we propose a method for reducing them. [ABSTRACT FROM AUTHOR]

We show that the best degree reduction of a given polynomial P from degree n to m with respect to the discrete $$L_2$$ -norm is equivalent to the best Euclidean distance of the vector of h-Bézier coefficients of P from the vector of degree raised h-Bézier coefficients of polynomials of degree m. Moreover, we demonstrate the adequacy of h-Bézier curves for approaching the problem of weighted discrete least squares approximation. Applications to discrete orthogonal polynomials are also presented. [ABSTRACT FROM AUTHOR]

An algorithm for computing an analytic function of a matrixis described. The algorithm is intended for the case wherehas some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations. [ABSTRACT FROM AUTHOR]

In this paper, we construct a numerical method based on cubic splines in tension for solving regularized long wave equation. The truncation error is analysed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes. Numerical stability of the method has been studied by using a linearized stability analysis. Test problems are dealt with. The numerical simulations can validate and demonstrate the advantages of the method. [ABSTRACT FROM AUTHOR]

We obtain by elementary methods necessary and sufficient conditions for a k -dimensional cubature formula to hold for all polynomials of degree up to 2 m − 1 when the nodes of the formula have Lagrange polynomials of degree at most m . The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m − 1 evaluated at the node plus an orthogonal polynomial of degree m . Stronger conditions are given for the case where the cubature formula holds for all polynomials of degree up to 2 m . This result is applied in one dimension to obtain a quadrature formula where the nodes are the roots of a quasi-orthogonal polynomial of order 2. In two dimensions the result is applied to obtain constructive proofs of cubature formulas of degree 2 m − 1 for the Geronimus and the Morrow–Patterson classes of nodes. A cubature formula of degree 2 m is obtained for a subclass of Morrow–Patterson nodes. Our discussion gives new proofs of previous theorems for the Chebyshev points and the Padua points, which are special cases. [ABSTRACT FROM AUTHOR]

Conic sections have many applications in industrial design, however, they cannot be exactly represented in polynomial form. Hence approximating conic sections with polynomials is a challenging problem. In this paper, we use the monomial form of Delgado and Peña (DP) curves and present a matrix representation for them. Using the matrix form and the least squares method, we propose a simple and efficient algorithm for approximating conic sections by DP curves of arbitrary degree with endpoint interpolation. Finally, we test and compare the proposed algorithm on some numerical examples which validates and confirms efficiency of it. [ABSTRACT FROM AUTHOR]

By using the polarization identity, we propose a family of quasi-interpolants based on bivariate $${\fancyscript{C}}^1$$ cubic super splines defined on triangulations with a Powell-Sabin refinement. Their spline coefficients only depend on a set of local function values. The quasi-interpolants reproduce cubic polynomials and have an optimal approximation order. [ABSTRACT FROM AUTHOR]

This article shows in detail how to construct in a simple and ordered way a set of rational functions defined on the pyramid topology. The set of functions is parameterized by an integer p . It is shown that these functions, defined in a hierarchical way, constitute a basis for a complete polynomial interpolation space of degree p on the pyramid domain. In order to help this definition we use a denumerable sequence of orthogonal polynomials defined on an interval of the real line. A priori, any increasing sequence of polynomials in one variable can be used. The bases are constructed to be used in the class C 0 method of finite elements. The rational functions thus defined can be combined to represent any polynomial of degree p . Thus, given an arbitrary number p , one defines a finite element whose geometry is a pyramid that has associated a complete interpolation space of degree p . Moreover, this element is adequate to be used with the p -adaptive technique on heterogeneous meshes of finite elements hierarchical. [ABSTRACT FROM AUTHOR]

We construct norming meshes with cardinality 𝒪(ns),s = 3, for polynomials of total degree at mostnon the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensionalC2star-like domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4). [ABSTRACT FROM AUTHOR]

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49-63, ). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical. [ABSTRACT FROM AUTHOR]

We consider the problem of interpolating a multivariable function defined on a bounded domain using its traces on parametric hypersurfaces. Our approach is based on the theory of operator polynomial interpolation. We construct the corresponding operator interpolation polynomial for a given function and analyse in detail particular two- and three-dimensional cases. Numerical examples presented in the paper show the flexibility of the proposed approach. [ABSTRACT FROM PUBLISHER]

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253-280, ) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots. [ABSTRACT FROM AUTHOR]

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr's formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples. [ABSTRACT FROM AUTHOR]

We study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols. These algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties. We illustrate our results with several examples. [ABSTRACT FROM AUTHOR]

Bivariate polynomials defined on the unit disk are used to reconstruct a wavefront from a data sample. We analyze the interpolation problem arising in critical sampling, that is, using a minimal sample. The interpolant is expressed as a linear combination of Zernike polynomials, whose coefficients represent relevant optical features of the wavefront. We study the propagation of errors of the polynomial values and their coefficients, obtaining bounds for the Lebesgue constants and condition numbers. A node distribution leading to low Lebesgue constants and condition numbers for degrees up to 20 is proposed. [ABSTRACT FROM AUTHOR]

Computing the roots of a univariate polynomial can be reduced to computing the eigenvalues of an associated companion matrix. For the monomial basis, these computations have been shown to be numerically stable under certain conditions. However, for certain applications, polynomials are more naturally expressed in other bases, such as the Lagrange basis or orthogonal polynomial bases. For the Lagrange basis, the equivalent stability results have not been published. We show that computing the roots of a polynomial expressed in barycentric form via the eigenvalues of an associated companion matrix pair is numerically stable, and give a bound for the backward error. Numerical experiments show that the error bound is approximately an order of magnitude larger than the backward error. We also discuss the matter of scaling and balancing the companion matrix to bring it closer to a normal pair. With balancing, we are able to produce roots with small backward error. [ABSTRACT FROM AUTHOR]

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes. [ABSTRACT FROM AUTHOR]

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation. [ABSTRACT FROM PUBLISHER]

We prove that the ENO reconstruction and ENO interpolation procedures are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity of the piecewise polynomial ENO procedure. [ABSTRACT FROM AUTHOR]

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness. [ABSTRACT FROM AUTHOR]

We construct trivariate $$C^r$$ macro-elements for any $$r\ge 1$$ over the Worsey-Farin refinement of any tetrahedral partition. This extends the construction of $$C^1$$ cubic Worsey-Farin elements and $$C^2$$ elements of degree nine to the $$C^r$$ situation with $$r>2$$. We also show that the degree of polynomials used for our macro-elements is the lowest. [ABSTRACT FROM AUTHOR]

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1-10, ), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ϵ. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed. [ABSTRACT FROM AUTHOR]

We prove that any fat, subanalytic compact subset of $\mathbb R^N$ possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389-398, ).) We also show that an infinitely differentiable map f from a compact set Q in $\mathbb R^N$ onto a Markov compact set K in $\mathbb C^l$ ( l ≤ N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389-398, ) for analytic maps in case Q is a subset of $\mathbb R^N$. Versions for $\mathcal C^k$ maps with sufficiently large k are also given. [ABSTRACT FROM AUTHOR]

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss-Jacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost. [ABSTRACT FROM AUTHOR]

Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse structure in the Hessian matrix. To be able to exploit Hessian sparsity, existing optimization approaches require the knowledge of the sparsity structure. The goal of this paper is to develop and analyze a method where the sparse models are constructed automatically. The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements. Such a recovery is achieved by minimizing the ℓ-norm of a vector subject to the measurements constraints. We suggest an approach for building sparse quadratic polynomial interpolation models by minimizing the ℓ-norm of the entries of the model Hessian subject to the interpolation conditions. We show that this procedure recovers accurate models when the function Hessian is sparse, using relatively few randomly selected sample points. Motivated by this result, we developed a practical interpolation-based trust-region method using deterministic sample sets and minimum ℓ-norm quadratic models. Our computational results show that the new approach exhibits a promising numerical performance both in the general case and in the sparse one. [ABSTRACT FROM AUTHOR]

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut's rational interpolant at equidistant nodes to the family of Floater-Hormann interpolants, which includes the former as a special case. [ABSTRACT FROM AUTHOR]

Given a function f defined on a bounded polygonal domain $${\Omega \subset \mathbb{R}^2}$$ and a number N > 0, we study the properties of the triangulation $${\mathcal{T}_N}$$ that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the W semi-norm for 1 ≤ p < ∞, and we consider Lagrange finite elements of arbitrary polynomial order m − 1. We establish sharp asymptotic error estimates as N → +∞ when the optimal anisotropic triangulation is used. A similar problem has been studied in Babenko et al. (East J Approx. 12(1):71-101, ), Cao (J Numer Anal. 45(6):2368-2391, ), Chen et al. (Math Comput. 76:179-204, ), Cohen (Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin, ), Mirebeau (Constr Approx. 32(2):339-383, ), but with the error measured in the L norm. The extension of this analysis to the W norm is required in order to match more closely the needs of numerical PDE analysis, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the L error norm. Our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. [ABSTRACT FROM AUTHOR]

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots-that are not considered by the authors of the corresponding spline basis-thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data. [ABSTRACT FROM AUTHOR]

In this paper, a spline collocation method is applied to solve a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The presented method is dependent on four collocation points to be satisfied by four parameters θ j ∈(0, 1], j=1(1) 4 in each subinterval. It turns out that the proposed method when applied to the concerned system is a fourth-order convergent method and gives numerical results which are better than those produced by other spline methods [E.A. Al-Said and M.A. Noor, Finite difference method for solving fourth-order obstacle problems, Int. J. Comput. Math. 81(6) (2004), pp. 741–748; F. Geng and Y. Lin, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Appl. Math. Comput. 200 (2008), pp. 231–241; J. Rashidinia, R. Mohammadi, R. Jalilian, and M. Ghasemi, Convergence of cubic-spline approach to the solution of a system of boundary-value problems, Appl. Math. Comput. 192 (2007), pp. 319–331; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using non polynomial spline technique, Appl. Math. Comput. 185 (2007), pp. 128–135; S.S. Siddiqi and G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Appl. Math. Comput. 190(1) (2007), pp. 652–661; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using cubic spline, Appl. Math. Comput. 187(2) (2007), pp. 1219–1227; Siraj-ul-Islam, I.A. Tirmizi, F. Haq, and S.K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1448–1460]. Moreover, the absolute stability properties appear that the method is A-stable. Two numerical examples (one for each case of boundary conditions) are given to illustrate practical usefulness of the method developed. [ABSTRACT FROM PUBLISHER]