OPTIMAL RATES FOR LAVRENTIEV REGULARIZATION WITH ADJOINT SOURCE CONDITIONS
OPTIMAL RATES FOR LAVRENTIEV REGULARIZATION WITH ADJOINT SOURCE CONDITIONS
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DOI：
10.1090/mcom/3237
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发表时间：
2018-03-01
影响因子：
2
通讯作者：
Hofmann, Bernd
中科院分区：
文献类型：
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作者：
Plato, Robert;Mathe, Peter;Hofmann, Bernd
希尔伯特空间中的不适定算子方程有多种正则化方法。如果基础算子是增生的，那么拉夫伦蒂耶夫正则化(奇异摄动)是一个直接的选择。正则化误差的相应收敛速度取决于给定的光滑性假设，并且对于一般的增生算子，这些收敛速度可以是关于算子或其伴随的。以前的分析显示了不同的收敛速度，它们的最佳性尚不清楚，特别是对于伴随源条件。基于T.Kato的基础研究[J.Math.SoC。Japan 13(1961)，no.3,247-274]，我们建立了这种情况下的幂型收敛速度。通过用极限阶度量这类算子的最优性，我们证明了一般增生算子在直接源条件和伴随源条件下收敛速度的最优性，也证明了正半定自伴算子子类的收敛速度的最优性.
There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive, then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato [J. Math. Soc. Japan 13(1961), no. 3, 247-274], we establish power type convergence rates for this case. By measuring the optimality of such rates in terms of limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of positive semidefinite selfadjoint operators.