Lp-空间中的变分源条件
Variational source conditions in Lp-spaces
摘要：
针对巴拿赫空间中一般不适定的算子方程，提出并分析了带Lp范数罚的Tikhonov正则化方法的变分源条件（VSC）.我们的分析是基于使用著名的Littlewood-Paley理论和（Rademacher）R-有界性的概念。在这两个分析工具的基础上，我们验证了所提出的变结构控制下的条件稳定性估计和真正的解决方案的Triebel-Lizorkin型空间的正则性要求。在本文的最后一部分中，开发的理论被应用到一个反椭圆问题的测量数据重建可能无界的扩散系数的Lp设置。利用变结构控制，得到了相应的带Lp-范数惩罚的Tikhonov正则化的收敛速度.
英文摘要：
We propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with Lp-norm penalties for a general ill-posed operator equation in a Banach space. Our analysis is based on the use of the celebrated Littlewood-Paley theory and the concept of (Rademacher) R-boundedness. On the basis of these two analytical tools, we validate the proposed VSC under a conditional stability estimate and a regularity requirement of the true solution in terms of Triebel-Lizorkin-type spaces. In the final part of the paper, the developed theory is applied to an inverse elliptic problem with measure data for the reconstruction of possibly unbounded diffusion coefficients in the Lp-setting. By means of VSC, convergence rates for the associated Tikhonov regularization with Lp-norm penalties are obtained.
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