Tikhonov regularization in Hilbert scales under conditional stability assumptions
Tikhonov regularization in Hilbert scales under conditional stability assumptions
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DOI：
10.1088/1361-6420/aadef4
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发表时间：
2018-07
期刊：
影响因子：
2.1
通讯作者：
H. Egger;B. Hofmann
中科院分区：
文献类型：
--
作者：
H. Egger;B. Hofmann
条件稳定性估计允许我们表征许多逆问题的病态程度，但如果没有进一步的假设，它们对于存在数据扰动的稳定解是不够的。本文用希尔伯特尺度的Tikhonov正则化方法研究满足条件稳定性估计的非线性逆问题的稳定解。建立了先验和后验参数选择策略的阶最优收敛率。研究了隐源条件的作用，并阐述了其与希尔伯特尺度正则化结果的关系。讨论了所得结果对某些模型问题的适用性，并通过数值试验对理论结果进行了验证。
Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a priori and a posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.