球面t-设计在球面最小二乘逼近中的应用研究

Spherical t-designs can be regarded as the nodes of an equal positive weight quadrature rule with algebraic precision t over the sphere, which has a broad application background. How to find a "good" spherical t-design is a hot issue in computational mathematics and optimization fields. In this project, we will first construct "well conditioned" spherical t-designs by solving nonlinear constrained optimization problems. Then we will employ the interval method to verify numerical spherical t-designs, especially for large t. Secondly, we will study the regularized least squares approximation by using obtained spherical t-designs. In particular, we will propose some new regularization operators based on approximation problems to establish new least squares models. We study how to solve regularized least squares problems efficiently. Thirdly, we will explore the choices of regularization parameters for a satisfying approximation. The sharp error bound of regularized least squres approximation will be given. Last but far from the least, we will apply the obtained theoretical result into pratical problems: medical imaging, atomspheric science, satellite data and so on. The spherical approximation problems have been of great concern by international experts. We look forward to establishing the project with the support from NNSF, to carry more favorable theoretic research and practical applications.

球面t-设计可以认为是球面上一具有代数精度为t的等正权数值积分公式的结点，它具有广阔的应用背景。如何找到一个“良好”的球面t-设计是计算数学、最优化领域的一个热点问题。 本项目首先通过求解约束优化问题得到“好条件”的球面t-设计，并且用区间法方法进行可靠性检验，特别是对于较大的t。后对于球面上的正则化最小二乘法进行深入研究，具体包含：针对球面上的逼近问题提出新的正则化算子，建立新的最小二乘法模型； 研究高效求解球面正则化最小二乘法的算法； 探讨正则化参数的选取，使得逼近达到满意效果；给出在选取“良好”的球面t-设计作为结点集合情况下精确误差估计。最后，本项目将理论研究的成果应用到医疗图像、大气科学、卫星数据等实际问题中。球面上的逼近问题已经得到国内外专家的高度关注和研究。我们期待该课题的立项，以便开展理论研究和实际应用。

本课题研究了如下内容：.1.在二维球面上研究了高次球面t-设计的构造，给出了t到100的好条件球面t-设计，并用区间法给出了可靠性检验。.2.在高维球面上研究了球面t-设计的构造方法，在点集合是基本系统的条件下，给出了t和N的关系。并且提出了新的变分问题，可以做为高维球面t-设计的构造方法。.3.与国内学者合作，研究了Bessel变换的数值方法，给出了误差估计，可以作为球面数值积分的基础。.1.2 是在申请书中的拟解决问题。3.是研究过程中发现的新课题。有关球面最小二乘问题及多项式逼近的研究又得到了国家自然基金的连续资助，正在进一步研究中。