Title | ||
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Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data |
Abstract | ||
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In this paper we derive local L2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere Sd⊆Rd+1, given noisy, scattered, local data representing an underlying function from a Sobolev space of order s>d∕2 defined on a non-empty connected open set Ω⊆Sd with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ℓ2-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order s on Sd. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive L2(Ω) error estimates for each strategy. As auxiliary tools for proving the local L2 error estimates we develop both a local L2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments. |
Year | DOI | Venue |
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2021 | 10.1016/j.cam.2020.113061 | Journal of Computational and Applied Mathematics |
Keywords | DocType | Volume |
primary,secondary | Journal | 382 |
ISSN | Citations | PageRank |
0377-0427 | 0 | 0.34 |
References | Authors | |
0 | 3 |
Name | Order | Citations | PageRank |
---|---|---|---|
Kerstin Hesse | 1 | 50 | 9.67 |
Ian H. Sloan | 2 | 1180 | 183.02 |
Robert S. Womersley | 3 | 258 | 74.51 |