Abstract
Energy minimization is one of the properties that make univariate splines so favorable in many problems of approximation and estimation; interpolation in and extrapolation from sparse data sites and smoothing of noisy data in particular. In this paper, we present a novel approach to approximate energy minimization on certain classes of submanifolds that gives rise to new methods for extrapolation and smoothing on submanifolds. To accomplish this, we minimize intrinsic functionals approximately by minimising a suitable extrinsic formulation of the functional augmented by a penalty on the first order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by tensor product B-splines.
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Notes
Note that this stands in sharp contrast to the Euclidean setting: In the Euclidean case, the affine functions remain affine polynomials for connected subdomains with a smooth boundary, whereas now cutting away a portion of a smooth compact manifold may introduce affine functions that are not constants!
Just mere polynomials can serve as an example here, as the restrictions of a basis for a polynomial space of fixed degree in \({\mathbb {R}}^d\) is no longer linearly independent in \({\mathbb {R}}^{d-1}\).
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Maier, LB. Sparse Data Interpolation and Smoothing on Embedded Submanifolds. J Sci Comput 84, 19 (2020). https://doi.org/10.1007/s10915-020-01268-z
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DOI: https://doi.org/10.1007/s10915-020-01268-z