On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems
On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems
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DOI：
10.1080/00036810903208148
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发表时间：
2010-01-01
影响因子：
1.1
通讯作者：
Yamamoto, Masahiro
中科院分区：
文献类型：
--
作者：
Hofmann, Bernd;Yamamoto, Masahiro
在过去的几年中，巴纳赫空间中非线性不适定问题的吉洪诺夫正则化的收敛率结果已经发表，其中源条件的经典概念被某些水平集上的变分不等式所取代。这实质上推进了对前向算子和解决方案的非光滑情况的分析。事实上，这种变分不等式结合了算子非线性的结构条件和解的平滑特性。变分不等式中不同的指数对应于不同水平的收敛速度。在本文中，我们讨论巴拿赫空间设置中出现的指数范围。为了减轻广义源条件、前向算子的非线性程度和相关变分不等式之间的交叉联系，我们研究了希尔伯特空间情况，甚至证明了线性算子的一些逆结果。最后，我们概述了偏微分方程的变分正则化和条件稳定性估计之间相互作用的一些方面。作为一个例子，我们将该理论应用于抛物线方程的特定参数识别问题。
In the past few years, convergence rates results for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces have been published, where the classical concept of source conditions was replaced with variational inequalities holding on some level sets. This essentially advanced the analysis of non-smooth situations with respect to forward operators and solutions. In fact, such variational inequalities combine both structural conditions on the nonlinearity of the operator and smoothness properties of the solution. Varying exponents in the variational inequalities correspond to different levels of convergence rates. In this article, we discuss the range of occurring exponents in the Banach space setting. To lighten the cross-connections between generalized source conditions, degree of nonlinearity of the forward operator and associated variational inequalities we study the Hilbert space situation and even prove some converse result for linear operators. Finally, we outline some aspects for the interplay of variational regularization and conditional stability estimates for partial differential equations. As an example, we apply the theory to a specific parameter identification problem for a parabolic equation.