Higher-order Riemannian spline interpolation problems: a unified approach via gradient flows
Year: 2026
Type: article
Abstract: This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks, providing a novel framework for tackling these challenges. This method notably extends to the variational spline interpolation problem on Lie groups, which is frequently encountered in mechanical optimal control theory. As a result, our work contributes to both geometric control theory and statistical shape data analysis. We rigorously prove the existence of global solutions in Hölder spaces for the gradient flow and demonstrate that the asymptotic limits of these solutions validate the existence of solutions to the variational spline interpolation problem. This constructive proof also offers insights into potential numerical schemes for finding such solutions, reinforcing the practical applicability of our approach.
Source: Mathematische Annalen
Authors: Chun-Chi Lin, Dung The Tran
Institution: National Taiwan Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Awards: 數學學門重點計畫: Research on Geometric Measure Theory, Partial Differential Equations, and Scientific Computing - Almgren’S Multiple-Valued Functions, Singular Minimal Submanifolds, and Geometric Flows( II ), Research on Geometric Measure Theory, Partial Differential Equations, and Scientific Computing – Higher Multiplicity in Geometric Measure Theory( II )
OpenAlex ID: W7130363283