Equispaced Nodes Research Articles

A new fast algorithm for computing the mock-Chebyshev nodes

Interpolation by polynomials on equispaced points is not always convergent due to the Runge phenomenon, and also, the interpolation process is exponentially ill-conditioned. By taking advantage of the optimality of the interpolation processes on the Chebyshev-Lobatto nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev nodes for polynomial interpolation. Mock-Chebyshev nodes asymptotically follow the Chebyshev distribution, and they are selected from a sufficiently large set of equispaced nodes. However, there are few studies in the literature regarding the computation of these points.In a recent paper [1], we have introduced a fast algorithm for computing the mock-Chebyshev nodes for a given set of (n+1) Chebyshev-Lobatto points using the distance between each pair of consecutive points. In this study, we propose a modification of the algorithm by changing the function to compute the quotient of the distance and show that this modified algorithm is also fast and stable; and gives a more accurate grid satisfying the conditions of a mock-Chebyshev grid with the complexity being O(n). Some numerical experiments using the points obtained by this modified algorithm are given to show its effectiveness and numerical results are also provided. A bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points is discussed.

Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

We consider approximating analytic functions on the interval [-1,1] from their values at a set of m+1 equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this ‘impossibility’ theorem. Our ‘possibility’ theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance epsilon > 0, which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions. It uses algebraic polynomials of degree n on an extended interval [-gamma ,gamma ], gamma > 1, to construct an approximation on [-1,1] via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree n on [-1,1] that is simultaneously bounded by one at a set of m+1 equispaced nodes in [-1,1] and 1/epsilon on the extended interval [-gamma ,gamma ]. We show that linear oversampling, i.e. m = c n log (1/epsilon ) / sqrt{gamma ^2-1}, is sufficient for uniform boundedness of any such polynomial on [-1,1]. This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

A linear barycentric rational interpolant on starlike domains

When an approximant is accurate on an interval, it is only natural to try to extend it to multi-dimensional domains. In the present article we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several-times differentiable functions to interpolate them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions–up to a logarithmic factor–and with an order limited only by the order of differentiability for real functions (provided that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit one to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.

Improving the Accuracy of the Trapezoidal Rule

The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in nonperiodic cases. Commonly used improvements, such as Simpson\textquoteright s rule and the Newton--Cotes formulas, are not much (if at all) better than the even more classical quadrature formulas described by James Gregory in 1670. For increasing orders of accuracy, these methods all suffer from the Runge phenomenon (the fact that polynomial interpolants on equispaced grids become violently oscillatory as their degree increases). In the context of quadrature methods on equispaced nodes, and for orders of accuracy around 10 or higher, this leads to weights of oscillating signs and large magnitudes. This article develops further a recently discovered approach for avoiding these adverse effects.

Numerical quadrature over smooth surfaces with boundaries

This paper describes a high order accurate method to calculate integrals over curved surfaces with boundaries. Given data locations that are arbitrarily distributed over the surface, together with some functional description of the surface and its boundary, the algorithm produces matching quadrature weights. This extends on the authors' earlier methods for integrating over the surface of a sphere and over arbitrarily shaped smooth closed surfaces by also considering domain boundaries. The core approach consists again of combining RBF-FD (radial basis function-generated finite difference) approximations for curved surface triangles, which together make up the full surface. The provided examples include both curved and flat domains. In the highly special case of equi-spaced nodes over a regular interval in 1-D, the method provides a new opportunity for improving on the classical Gregory enhancements of the trapezoidal rule.

Stability of Radial Basis Function Methods for Convection Problems on the Circle and Sphere

This paper investigates the stability of the radial basis function (RBF) collocation method for convective problems on the circle and sphere. We prove that the RBF method is Lax-stable for problems on the circle when the collocation points are equispaced and the transport speed is constant. We also show that the eigenvalues of discretization matrices are purely imaginary in the case of variable coefficients and equispaced nodes. By studying the $$\epsilon $$∈-pseudospectra of these matrices we argue that approximations are also Lax-stable in the latter case. Based on these results, we conjecture that the discretization of transport operators on the sphere present a similar behavior. We provide strong evidence that the method is Lax-stable on the sphere when the collocation points come from certain polyhedra. In both geometries, we demonstrate that eigenvalues of the differentiation matrix deviate from the imaginary axis linearly with perturbations off the set of ideal collocation points. When the ideal set is impractical or unavailable, we propose a least-squares method and present numerical evidence suggesting that it can substantially improve stability without any increase to computational cost and with only a minor cost to accuracy.

The Linear Barycentric Rational Quadrature Method for Volterra Integral Equations

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a composite version of this quadrature rule, loses one order of convergence but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision.

An Extension of the Floater–Hormann Family of Barycentric Rational Interpolants

The barycentric rational interpolants introduced by Floater and Hormann in 2007 are “blends” of polynomial interpolants of fixed degree d. In some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants, which, e.g., are ill-conditioned and lead to Runge’s phenomenon if the interpolation nodes are equispaced. For such nodes, however, the condition of Floater–Hormann interpolation deteriorates exponentially with increasing d. In this paper, an extension of the Floater–Hormann family with improved condition at equispaced nodes is presented and investigated. The efficiency of its applications such as the approximation of derivatives, integrals and primitives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants. Math Subject Classification: 65D05, 65L12, 65D32, 41A05, 41A20, 41A25

Convergence of Linear Barycentric Rational Interpolation for Analytic Functions

Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by $d$, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions for how to choos...

Phase reconstruction by a multilevel iteratively regularized Gauss–Newton method

In this paper we consider the numerical solution of a phase retrieval problem for a compactly supported, linear spline with the Fourier transform , where values of |f| and at finitely many equispaced nodes are given. The unknown phases of complex spline coefficients fulfil a well-structured system of nonlinear equations. Thus the phase reconstruction leads to a nonlinear inverse problem, which is solved by a multilevel strategy and iterative Tikhonov regularization. The multilevel strategy concentrates the main effort of the solution of the phase retrieval problem in the coarse, less expensive levels and provides convenient initial guesses at the next finer level. On each level, the corresponding nonlinear system is solved by an iteratively regularized Gauss–Newton method. The multilevel strategy is motivated by convergence results of IRGN. This method is applicable to a wide range of examples as shown in several numerical tests for noiseless and noisy data.

Quasi-optimal model of the acoustic source

The optimal model of the acoustic source is built based on the Tchebichef nodes by using the boundary element method. The Tchebichef nodes are not equispaced. Then it is difficult to distribute them on the source surface and consequently to measure the nodal values of the source boundary. In order to alleviate these difficulties, a new quasi-optimal model (QOM) is worked out. The method is based on an initially elaborated model with equispaced nodes. In the paper, QOM is constructed for an axisymmetric circular membrane placed in an infinite baffle. The quality of QOM is compared to the qualities of other already known models. The appropriate numerical results are presented.

Uniformly Accurate Finite Difference Schemes forp-Refinement

The accuracy deficiency of high-order conventional finite difference schemes in the high-frequency range is attributed to the approximation property of a power polynomial interpolation over equispaced nodes. To improve the numerical representation of high-frequency solutions, a new family of high-order finite difference schemes is constructed by trigonometric polynomial interpolation. As expected, these new difference schemes can resolve a wider frequency range than conventional finite difference schemes with the same stencil. In particular, they remove the possibility of Runge's phenomenon for uniform mesh. The recently proposed optimized difference schemes are indeed types of mixed difference schemes which make a compromise between power and trigonometric polynomial interpolation.