正则化最小二乘法在球面上噪声光滑函数近似中的参数选择策略
Parameter choice strategies for least-squares approximation of noisy smooth functions on the sphere
AI总结：
本文研究了利用正则化最小二乘法对球面上噪声光滑函数进行多项式重构，分析了正则化参数和惩罚参数的选择策略，并通过数值实验验证了理论结果。
摘要：
我们考虑了通过正则化最小二乘法对单位球面上的光滑函数进行多项式重构，该方法基于An等人在SIAM J. Numer. Anal. 50 (2012)中的方法。我们使用精确计算所有球面多项式至2M次的正权立方公式节点来获取函数值。我们首先获得了重构误差与正则化参数和正则化算子惩罚参数之间的界。然后讨论了先验和后验策略用于选择这些参数。最后，我们通过数值实验验证了理论结果。
英文摘要：
We consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method of An et al., SIAM J. Numer. Anal. 50 (2012), 1513--1534. As nodes we use the points of a positive-weight cubature formula that is exact for all spherical polynomials of degree up to $2M$, where $M$ is the degree of the reconstructing polynomial. We first obtain a reconstruction error bound in terms of the regularization parameter and the penalization parameters in the regularization operator. Then we discuss a priori and a posteriori strategies for choosing these parameters. Finally, we give numerical examples illustrating the theoretical results.
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