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Scattered manifold-valued data approximation

Content Provider	Springer Nature Link
Author	Yu, Thomas Sprecher, Markus Grohs, Philipp
Copyright Year	2016
Abstract	We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples $$(f(\xi _i))_{i\in \mathcal {I}}$$ , where the data sites $$(\xi _i)_{i\in \mathcal {I}}$$ are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.
Ending Page	1010
Page Count	24
Starting Page	987
File Format	PDF
ISSN	0029599X
e-ISSN	09453245
Journal	Numerische Mathematik
Issue Number	4
Volume Number	135
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-07-08
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	ApplicationMathematics/Computational Methods of Engineering Approximation Splines Local Riemannian geometry Theoretical, Mathematical and Computational Physics Riemannian data Mathematics Manifold-valued function Scattered data Algorithms for functional approximation Mathematical Methods in Physics Numerical Analysis Model reduction Numerical and Computational Physics, Simulation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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