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 equations (PDEs) from boundary observations—can often be achieved by extending the searched 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.

中文翻译：

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收敛保证了偏微分方程中的系数重建，通过变分法和牛顿型方法通过范围不变性从边界测量中重建

本文的一个关键观察结果是，非线性病态算子方程的正则化方法收敛的范围不变性条件（例如从边界观测中识别偏微分方程（PDE）中的系数）通常可以通过扩展搜索范围来实现对于参数来说，允许它依赖于其他变量。但这显然会抵消参数的唯一可识别性。本文的第二个关键思想是通过惩罚来恢复参数的原始受限依赖性。这被证明可以导致变分（吉洪诺夫型）和迭代（牛顿型）正则化方法的收敛。我们在简化且一次性设置的偏微分方程参数识别的典型框架中具体化了抽象收敛分析。通过椭圆偏微分方程和瞬态偏微分方程中边界观测的系数识别的三个例子进一步说明了这一点。