Convergence orders in length estimation for exponential parameterization and ε -uniform samplings

R. Kozera, Lyle Noakes, P. Szmielew

    Research output: Chapter in Book/Conference paperConference paper

    Abstract

    © 2015 AIP Publishing LLC. We discuss the problem of approximating the length of a curve γ in arbitrary euclidean space En, from ε-uniformly sampled reduced data Qm={qi}i=0m, where γ(ti) = qi. The interpolation knots {ti}i=0m are assumed in this paper to be unknown (the so-called non-parametric interpolation). We fit Qm with the piecewise-quadratic interpolant γ^2 based on the so-called exponential parameterization (depending on parameter λ ∈ [0, 1]) which estimates the missing knots {ti}i=0m?{t^i}i=0m.The asymptotic orders βε (λ) for length estimation d{γ}?d{γ^2} in case of λ = 0 (uniformly guessed knots) read as βε (0) = min{4,4ε} (for ε>0) - see [1]. On the other hand λ = 1 (cumulative chords) yields βε (1) = min{4,3+ε} (see [2]). This paper establishes a more general result holding for all λ ∈ [0,1] and ε-uniform samplings - see Th. 5. More specifically the respective convergence orders amount to βε (λ) = min{4,4ε} for λ ∈ [0,1). Consequently βε (λ) are independent of λ ∈ [0,1) and the discontinuity in asymptotic orders βε (λ) at λ = 1 occurs, for all ε ∈ (0,1). The full proof of Th. 5 is presented in ICNAAM'14 post-conference journal publication together with more exhaustive relevant experimentation.
    Original languageEnglish
    Title of host publicationAIP Conference Proceedings: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014
    Place of PublicationGreece
    PublisherICNAAM
    Pages660015-1 - 660015-4
    Volume1648
    ISBN (Print)9780735412873
    DOIs
    Publication statusPublished - 2015
    EventInternational Conference on Numerical Analysis and Applied Mathematics 2014, ICNAAM 2014 - Rhodes, Greece
    Duration: 22 Sep 201428 Sep 2014

    Conference

    ConferenceInternational Conference on Numerical Analysis and Applied Mathematics 2014, ICNAAM 2014
    CountryGreece
    CityRhodes
    Period22/09/1428/09/14

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