Approximation of eigenfunctions in kernel-based spaces
Approximation of eigenfunctions in kernel-based spaces
复制标题
基于核的空间中特征函数的逼近
DOI：
--
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发表时间：
2014
影响因子：
1.7
通讯作者：
R. Schaback
中科院分区：
文献类型：
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作者：
G. Santin;R. Schaback
数值分析中基于内核的方法具有在“本机”希尔伯特空间中产生最佳恢复过程的优点 ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathcal {H}$end{document} 在其中复制。紧域上的连续核可以扩展到 ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} 中既是 L2 正交又是正交的特征函数egin{document}$mathcal {H}$end{document} （Mercer 扩展）。本文检查了相应的特征空间，并证明它们在 ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} 的所有其他子空间中具有最优性例如{文档}$mathcal {H}$end{文档}。这些结果与近似理论中的 n 宽度有很强的联系，并且它们确定最佳近似的误差与特征值的衰减密切相关。尽管特征空间和特征值不易获得，但可以使用由核相对于 n 个节点或中心的平移所跨越的标准 n 维子空间来很好地近似。我们通过这样的子空间给出本征系统的数值近似的误差界限。一系列示例表明，我们通过贪婪点选择策略的数值技术可以高精度地计算特征系统。
Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the “native” Hilbert space ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathcal {H}$end{document} in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L2-orthonormal and orthogonal in ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathcal {H}$end{document} (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of ℋdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathcal {H}$end{document}. These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.