Tikhonov regularization of nonlinear III-posed problems in hilbert scales
Tikhonov regularization of nonlinear III-posed problems in hilbert scales
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DOI：
10.1080/00036819208840111
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发表时间：
1992-08
影响因子：
1.1
通讯作者：
A. Neubauer
中科院分区：
文献类型：
--
作者：
A. Neubauer
本文考虑非线性不适定问题F（x）= y0，其中x和y0分别是Hilbert空间X和Y的元素.我们解决这些问题的Tikhonov正则化在希尔伯特尺度。这意味着正则化规范比X中的规范更强。光滑性条件，保证收敛速度相对于原始规范中的数据噪声在X。我们还提出了一个变种的吉洪诺夫正则化，产生这些利率，而不需要的光滑条件的知识。在这个变体中，允许F仅近似已知，并且X可以由有限维子空间近似。最后，我们说明了一个简单的参数估计问题的正则化Sobolev空间所需的条件。
In this paper we consider nonlinear ill-posed problems F(x) = y 0, where x and y 0 are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.