The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis.

翻译：反面问题的特点在于数据扰动方面的不稳定性。为了稳定反向过程,必须制定和采用正规化方法。在这项工作中,我们引入和分析过滤的对角框架分解概念,将标准过滤的单值分解扩展至框架情况。作为通用单质系统的框架可以更好地适应某一类潜在解决方案。在本文中,我们显示过滤的对角框架分解产生一种趋同的正规化方法。此外,我们还在源类型条件下得出趋同率,并证明在假设考虑的框架是Riesz-basis的情况下,符合最优化的顺序。