Stability properties for a class of inverse problems

Abstract

Abstract We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator A m \mathcal{A}_{m} depending nonlinearly 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 A m \mathcal{A}_{m} , 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.