Variable hilbert scales and their interpolation inequalities with applications to tikhonov regularization
Variable hilbert scales and their interpolation inequalities with applications to tikhonov regularization
复制标题
可变希尔伯特尺度及其插值不等式及其在吉洪诺夫正则化中的应用
DOI：
10.1080/00036819508840400
复制
发表时间：
1995
影响因子：
1.1
通讯作者：
M. Hegland
中科院分区：
文献类型：
--
作者：
M. Hegland
利用Hilbert空间中自伴算子的谱理论，构造了可变Hilbert标度。证明了一个嵌入定理和一个内插定理(基于Jenssen不等式)。它们推广了Natterer[Applic]关于“普通”希尔伯特标度的已知结果。《肛门》，18(1984)，第29-37页]。通过应用插值不等式，得到了正则化方法的最佳可能误差和实际误差的界。这些界限扩展了标准界限，特别是包括指数和对数误差定律。类似的结果早些时候由Hegland[SIAM J.Numer]建立。分析，29(1992年)，第1446-14611页，仅适用于紧致运算符。这里，它们被推广到包括无界运算符。Nair等人对Tikhonov正则化的详细讨论。[科技。代表MR8-94，CMA，澳大利亚纳特。Uni.，19941指出，被认为是次优的参数选择策略可以给出比所谓的最优选择更高的收敛速度！这个就是..。
Variable Hilbert scales are constructed using the spectral theory of self-adjoint operators in Hilbert spaces. An embedding and an interpolation theorem (based on Jenssen's inequality) are proved. They generalize known results about “ordinary” Hilbert scales derived by Natterer [Applic. Anal., 18 (1984), pp.29-37]. Bounds on best possible and actual errors for regularization methods are obtained by applying the interpolation inequality. These bounds extend the standard ones, and, in particular, include exponential and logarithmic error laws. Similar results were established earlier by Hegland [SIAM J. Numer. Anal., 29 (1992), pp. 1446-14611 for compact operators only. Here, they are generalized to include unbounded operators. A detailed discussion of Tikhonov regularization by Nair et al. [Tech. Rep. MR8-94, CMA, Aust. Nat. Uni., 19941 indicates that parameter choice strategies, which were thought to be suboptimal, can give substantially higher convergence rates than the so-called optimal choices! This im...