Learning a robust shape parameter for RBF approximation
<|reference_start|>Learning a robust shape parameter for RBF approximation: Radial basis functions (RBFs) play an important role in function interpolation, in particular in an arbitrary set of interpolation nodes. The accuracy of the interpolation depends on a parameter called the shape parameter. There are many approaches in literature on how to appropriately choose it as to increase the accuracy of interpolation while avoiding instability issues. However, finding the optimal shape parameter value in general remains a challenge. We present a novel approach to determine the shape parameter in RBFs: we introduce a data-driven method that controls the condition of the interpolation matrix to avoid numerically unstable interpolations, while keeping a very good accuracy. In addition, we formulate a fall-back procedure that enforces a strict upper bound on the condition number of the generated interpolation matrix. We present numerical test cases to assess the performance of the proposed methods in interpolation tasks and in a RBF based finite difference (RBF-FD) method, in one and two-space dimensions.<|reference_end|>
@article{veiga2024learning,
    title={Learning a robust shape parameter for RBF approximation},
    author={Maria Han Veiga and Faezeh Nassajian Mojarrad and Fatemeh Nassajian
  Mojarrad},
    journal={arXiv preprint arXiv:2408.05081},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.05081},
    primaryClass={math.NA cs.NA}
}