Optimality for ill-posed problems under general source conditions
Optimality for ill-posed problems under general source conditions
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DOI：
10.1080/01630569808816834
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发表时间：
1998
影响因子：
1.2
通讯作者：
U. Tautenhahn
中科院分区：
文献类型：
--
作者：
U. Tautenhahn
本文考虑线性不适定问题，其中噪声数据yδ是Hilbert空间X和Y之间的线性算子，而不是y。假设具有适当函数φ的一般源条件，我们研究以下问题：（i）在假设（ii）下，可以获得用于识别x的（最佳可能的）精度，是否存在保证这种最佳可能精度的特殊正则化方法，即，在集合Mδ，E？关于问题（i），我们证明了在一定条件下，存在inf sup，其中“inf”覆盖所有方法，“sup”覆盖所有方法。关于问题（ii），我们证明了一般正则化方法的最优性，并将我们的一般最优性结果指定给Tikhonov型方法和谱方法。以不同函数φ（λ）表征的热传导方程问题为例进行了模型研究。
In this paper we consider linear ill-posed problems where instead of y noisy data yδ are available with and is a linear operator between Hilbert spaces X and Y. Assuming the general source condition with appropriate functions φ we study following questions:(i) which (best possible) accuracy can be obtained for identifying x from under the assumptions (ii) are there special regularization methods which guarantee this best possible accuracy, i.e., which are optimal on the set Mδ,E? Concerning question (i) we prove that under certain conditions there holds inf sup with where the ‘inf’ is taken over all methods and the 'sup' is taken over all .and Concerning question (ii) we prove the optimality of a general class of regularization methods and specify our general optimality results to Tikhonov type methods and to spectral methods. Heat equation problems backward in time which are characterized by different functions φ(λ) serve as model examples.