This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \\ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schr\"odinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT.

翻译：本文致力于在变分源条件下证明变分和迭代正则化方法的收敛速率，这些源条件适用于线性化满足范围不变性条件的反问题。为此，通常需要找到问题的适当松弛形式，这种松弛基于对未知量集合的扩充，并导致反问题具有特定结构的重构形式。我们分析了利用该结构的三种方法：一种变分方案和一种牛顿型方案（其无速率的收敛性已在《rangeinvar》中建立）；此外，我们提出了一种分裂最小化方法，并证明其满足相同的收敛速率结果。范围不变性条件已针对多种从边界观测中识别偏微分方程系数的反问题得到验证，这些反问题在多种层析成像模态中具有实际意义。我们的研究动机尤其来自经典的电阻抗成像反问题，并同时研究了其扩散型方程的原形式与薛定谔方程的重构形式。对于这两种形式，我们找到了可证明满足范围不变性条件的松弛形式。结合《Diss-Weidling》中关于变分源条件的结果与上述三种方法的抽象框架，我们推导出电阻抗成像中变分法、分裂最小化法和牛顿型方法的收敛速率结果。