A projection algorithm on the set of polynomials with two bounds

International audienceThe motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Després, Numer. Algorithms, 76(3), (2017)] and [B. Després and M. Herda, Numer. Algorithms, 77(1), (2018)] where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation

Stability analysis of spectral methods for hyperbolic initial-boundary value systems

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations

A Constructive Method for Approximate Solution to Scalar Wiener-Hopf Equations

This paper presents a novel method of approximating the scalar Wiener-Hopf equation; and therefore constructing an approximate solution. The advantages of this method over the existing methods are reliability and explicit error bounds. Additionally the degrees of the polynomials in the rational approximation are considerably smaller than in other approaches. The need for a numerical solution is motivated by difficulties in computation of the exact solution. The approximation developed in this paper is with a view of generalisation to matrix Wiener-Hopf for which the exact solution, in general, is not known. The first part of the paper develops error bounds in Lp for 1<p<\infty. These indicate how accurately the solution is approximated in terms of how accurate the equation is approximated. The second part of the paper describes the approach of approximately solving the Wiener-Hopf equation that employs the Rational Caratheodory-Fejer Approximation. The method is adapted by constructing a mapping of the real line to the unit interval. Numerical examples to demonstrate the use of the proposed technique are included (performed on Chebfun), yielding error as small as 10^{-12} on the whole real line.Comment: AMS-LaTeX, 19 pages, 10 figures in EPS fil

Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing

17 pagesInternational audiencePerforming numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models

Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues

In this paper we propose a new method for approximating the nonstationary moment dynamics of one dimensional Markovian birth-death processes. By expanding the transition probabilities of the Markov process in terms of Poisson-Charlier polynomials, we are able to estimate any moment of the Markov process even though the system of moment equations may not be closed. Using new weighted discrete Sobolev spaces, we derive explicit error bounds of the transition probabilities and new weak a priori estimates for approximating the moments of the Markov processs using a truncated form of the expansion. Using our error bounds and estimates, we are able to show that our approximations converge to the true stochastic process as we add more terms to the expansion and give explicit bounds on the truncation error. As a result, we are the first paper in the queueing literature to provide error bounds and estimates on the performance of a moment closure approximation. Lastly, we perform several numerical experiments for some important models in the queueing theory literature and show that our expansion techniques are accurate at estimating the moment dynamics of these Markov process with only a few terms of the expansion

On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw-Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n^{-s-1}) for some s > 0, Clenshaw-Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.Comment: 7 pages, the figure of the revision has an unsymmetric example, to appear in SIAM J. Numer. Ana