论文标题
一类反问题的稳定性
Stability properties for a class of inverse problems
论文作者
论文摘要
我们为一类逆问题建立Lipschitz稳定性。在该类别中，相关的直接问题由积分运算符AM提出，具体取决于非线性的参数m并在函数u上运行。在反转步骤中，u和m都是未知的，但我们只对恢复m感兴趣。我们讨论了弹性方程的这种反问题的示例，以及在电磁理论中的地震学和逆散射问题的应用。假设AM的一些注射率和规律性属性，我们证明具有有限数量数据点的反问题是可以解决的，并且该解决方案在数据中是Lipschitz稳定的。我们显示了一个重建示例，说明了神经网络的使用。
We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator Am depending non-linearly on a parameter m and operating on a function u. In the inversion step both u and m are unknown but we are only interested in recovering m. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for Am, we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.
