Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:382

Article

Hesse, Kerstin¹  Sloan, Ian H.²  Womersley, Robert S.² 

[1] Paderborn Univ, Inst Math, D-33098 Paderborn, Germany

[2] Univ New South Wales, Sch Math & Stat, Sydney, NSW 2052, Australia

关键词: Local L-2 error estimates;    Smoothing parameter strategies;    Penalized least-squares;    Radial basis functions;    Sobolev extension theorem;    Sphere;   

DOI  :  10.1016/j.cam.2020.113061

来源: Elsevier

【 摘 要 】

In this paper we derive local L-2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere S-d subset of 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 Omega subset of S-d with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise l(2)-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 S-d. 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 L-2(Omega) error estimates for each strategy. As auxiliary tools for proving the local L-2 error estimates we develop both a local L-2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

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