Shixu Meng

Yuyuan Zhou, Shixu Meng

This work proposes a hybrid method (ULR) which integrates a rotation-equivariance-aware neural network and a low-rank structure to solve the two dimensional inverse medium scattering problem. The neural network is to model the data corrector which maps the full data to the Born data, and the low-rank structure is to design an inverse Born solver that finds a regularized solution from the perturbed Born data. The proposed rotation-equivariance-aware neural network naturally incorporates the reciprocity relation and the rotation-equivariance in inverse scattering, while the low-rank structure effectively filters high-frequency noise in the output of the neural network and leads to a regularized method supported by theoretical stability in the Born region. For a comparative study, we replace the low-rank inverse Born solver by another rotation-equvariance-aware neural network to propose a two-step neural network (UU). Furthermore, we extend the proposed methods (ULR and UU) to tackle the more challenging case with only limited aperture data. A variety of numerical experiments are conducted to compare the proposed ULR, UU, and a black-box neural network.

Shixu Meng

In this work we study the theoretical Lipschitz stability and propose a low-rank-assisted numerical method for the inverse medium scattering beyond the Born region. The proposed low-rank structure is based on the disk prolate spheroidal wave functions, which are eigenfunctions of both the Born forward operator and a Sturm-Liouville differential operator. We obtain Lipschitz stability for unknowns in a low-rank space in the fully nonlinear case and characterize the explicit Lipschitz constant in the linearized region. We further propose an ensemble Kalman filter to iteratively update the unknown in the proposed low-rank space whose dimension is intrinsically determined by the wave number. Moreover the ensembles are sampled according to a novel trace class covariance operator motivated by the connection between the proposed low-rank space and the Sturm-Liouville differential operator. Finally numerical examples are provided to illustrate the feasibility of the proposed method.

Liliana Borcea, Fioralba Cakoni, Shixu Meng

We introduce a direct, linear sampling approach to imaging in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and has unknown geometry due to compactly supported wall deformations. The goal of imaging is to determine these deformations and to identify localized scatterers in the waveguide, using a remote array of sensors that emits time harmonic probing waves and records the echoes. We present a theoretical analysis of the imaging approach and illustrate its performance with numerical simulations.