论文标题
通过范围不变性通过变异和牛顿类型方法从边界测量中的PDE中的系数重建保证的收敛保证
Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton type methods via range invariance
论文作者
论文摘要
本文基础的一个关键观察结果是，非线性不足操作员方程（例如，在偏差方程中的系数识别（PDE）中的系数识别（PDE）s的范围不变性条件通常可以从边界观察中获得 - 通常可以通过扩展允许IT的含义的参数来实现其他可依赖其他变量来实现。但是，这显然可以抵消参数的唯一可识别性。本文的第二个关键思想现在是通过惩罚来恢复参数的原始限制依赖性。这表明这会导致变异（Tikhonov类型）和迭代（牛顿类型）正则化方法的收敛性。我们将抽象的收敛分析在PDE中典型的参数识别的框架中，以降低和全面设置为特征。从椭圆形和时间依赖性PDE中的边界观测值中的系数鉴定的三个示例进一步说明了这一点。
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.
