Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines.Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D171. Introduction. Contemporary research in atmospheric sciences, geodesy and geophysics requires the use of global data heterogeneously distributed in space around the Earth. For spherical data interpolation/approximation tensor products of univariate splines are not a good choice, since data locations are not usually spaced over a regular grid. Radial basis functions are not good candidates either, since the data values may have no rotational symmetry.Spherical Bernstein-Bézier splines (introduced in [1] and studied in [2] and [7]) are well suited for scattered data interpolation/approximation problems. The spherical spline functions have many properties in common with classical polynomial splines over planar triangulations. Moreover, many spline interpolation and approximation methods for planar scattered data problems have analogs in spherical setting [2].One of the disadvantages of using homogeneous spherical splines is that spline spaces of even and odd degrees have only zero function in common due to homogeneity of the basis. More explicitly, the spline space S
We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error bound for the penalized least squares splines additionally depends on the penalty parameter.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.