AN ELASTIC FLOW FOR NONLINEAR SPLINE INTERPOLATIONS IN ℝn

Chun Chi Lin; Hartmut R. Schwetlick; Dung The Tran

doi:10.1090/tran/8639

AN ELASTIC FLOW FOR NONLINEAR SPLINE INTERPOLATIONS IN ℝⁿ

Chun Chi Lin, Hartmut R. Schwetlick, Dung The Tran

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Abstract

In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in n-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.

Original language	English
Pages (from-to)	4893-4942
Number of pages	50
Journal	Transactions of the American Mathematical Society
Volume	375
Issue number	7
Early online date	4 May 2022
DOIs	https://doi.org/10.1090/tran/8639
Publication status	Published - 1 Jul 2022

Keywords

curve fitting
elastic spline
Fourth-order geometric flow
spline interpolation

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.1090/tran/8639

4th.order.Flow.Splines-27.11.2021.PDFAccepted author manuscript, 622 KB

Cite this

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title = "AN ELASTIC FLOW FOR NONLINEAR SPLINE INTERPOLATIONS IN ℝn",

abstract = "In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in n-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable H{\"o}lder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.",

keywords = "curve fitting, elastic spline, Fourth-order geometric flow, spline interpolation",

author = "Lin, {Chun Chi} and Schwetlick, {Hartmut R.} and Tran, {Dung The}",

note = "This work was partially supported by the research grant of the National Science Council of Taiwan (NSC-100-2115-M-003-003), the National Center for Theoretical Sciences at Taipei, and the Max-Planck-Institut f{\"u}r Mathematik in den Naturwissenschaften in Leipzig. The third author received financial support from Taiwan MoST 108-2115-M-003-003-MY2. ",

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AB - In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in n-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.

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