Integral Operator Approaches for Scattered Data Fitting on Spheres
This work addresses scattered data fitting problems for researchers in approximation theory and computational mathematics, offering incremental improvements in error analysis and efficiency.
The paper tackles scattered data fitting on spheres by analyzing weighted spectral filter algorithms, deriving optimal Sobolev-type error estimates that avoid saturation and native-space barriers, and proposes a divide-and-conquer scheme to reduce computational burden with optimal error bounds.
This paper focuses on scattered data fitting problems on spheres. We study the approximation performance of a class of weighted spectral filter algorithms, including Tikhonov regularization, Landaweber iteration, spectral cut-off, and iterated Tikhonov, in fitting noisy data with possibly unbounded random noise. For the analysis, we develop an integral operator approach that can be regarded as an extension of the widely used sampling inequality approach and norming set method in the community of scattered data fitting. After providing an equivalence between the operator differences and quadrature rules, we succeed in deriving optimal Sobolev-type error estimates of weighted spectral filter algorithms. Our derived error estimates do not suffer from the saturation phenomenon for Tikhonov regularization in the literature, native-space-barrier for existing error analysis and adapts to different embedding spaces. We also propose a divide-and-conquer scheme to equip weighted spectral filter algorithms to reduce their computational burden and present the optimal approximation error bounds.