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：本文所依据的一个关键意见是,非线性不良运算方程式正规化方法趋同的差幅性条件 -- -- 如边界观测中部分差异等同系数的确定 -- -- 往往可以通过扩大参数的界限来达到,使参数能够取决于其他变量。这显然抵消了参数的独特可识别性。本文的第二个关键想法是现在通过惩罚来恢复参数原有的有限依赖性。这显示可导致变式(Tikhonov类型)和迭代(Newton类型)正规化方法的趋同。我们把抽象的趋同性分析纳入一个框架,这个框架典型的参数是:在精减和全自动设置中确定参数的参数。这又通过从流和时间依赖PDE的边界观察中找出系数的三个例子进一步说明。