An elastic flow for nonlinear spline interpolations in ℝⁿ

Lin, Chun-Chi; Schwetlick, Hartmut; Tran, Dung

doi:10.1090/tran/8639

An elastic flow for nonlinear spline interpolations in $\mathbb {R}^n$
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by Chun-Chi Lin, Hartmut R. Schwetlick and Dung The Tran PDF

Trans. Amer. Math. Soc. 375 (2022), 4893-4942 Request permission

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.

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