Application of a Deep Neural Network to Phase Retrieval in Inverse Medium Scattering Problems

We address the inverse medium scattering problem with phaseless data motivated by nondestructive testing for optical fibers. As the phase information of the data is unknown, this problem may be regarded as a standard phase retrieval problem that consists of identifying the phase from the amplitude of data and the structure of the related operator. This problem has been studied intensively due to its wide applications in physics and engineering. However, the uniqueness of the inverse problem with phaseless data is still open and the problem itself is severely ill-posed. In this work, we construct a model to approximate the solution operator in finite-dimensional spaces by a deep neural network assuming that the refractive index is radially symmetric. We are then able to recover the refractive index from the phaseless data. Numerical experiments are presented to illustrate the effectiveness of the proposed model.

Convexity of a discrete Carleman weighted objective functional in an inverse medium scattering problem

We investigate a globally convergent method for solving a one-dimensional inverse medium scattering problem using backscattering data at a finite number of frequencies. The proposed method is based on the minimization of a discrete Carleman weighted objective functional. The global convexity of this objective functional is proved.

Solving a 1-D inverse medium scattering problem using a new multi-frequency globally strictly convex objective functional

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved. We also prove the global convergence of the gradient projection algorithm and derive an error estimate. Numerical examples are presented to illustrate the performance of the proposed algorithm.