A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations -- such as coefficient identification in partial differential equiations (PDE)s from boundary observations -- can often be achieved by extending the seached for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs.

翻译：本文的核心发现是：对于非线性不适定算子方程（例如从边界观测中识别偏微分方程系数）的正则化方法收敛所需的范围不变性条件，通常可以通过允许待求参数依赖额外变量来扩展搜索空间而实现。然而，这显然会削弱参数的唯一可辨识性。本文的第二个关键思想是通过惩罚项恢复原始参数受限的依赖关系，并证明这能引导变分（Tikhonov型）与迭代（牛顿型）正则化方法收敛。我们在偏微分方程参数识别的典型框架中（分别采用简化与全耦合设定）具体化了上述抽象收敛性分析。通过椭圆型与时间相关偏微分方程中三个从边界观测识别系数的实例进一步阐释了这一理论。