Integral Operator Approaches for Scattered Data Fitting on Spheres
CoRR(2024)
摘要
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.
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