Fast and stable sampling methods for solving inverse source and scattering problems

Abstract

This work focuses on fast and stable numerical methods for solving inverse source and scattering problems, utilizing sampling-type methods and deep learning. Our objective is to accurately reconstruct unknown sources or objects using measurements of outgoing waves, focusing on two primary models: acoustic waves governed by Helmholtz equations and electromagnetic waves described by Maxwell’s equations. We first propose a robust and computationally inexpensive sampling-type method for reconstructing small acoustic sources from boundary Cauchy data at a single frequency. This method efficiently identifies the location and intensity of sources, supported by a theoretical foundation using the Green representation formula and an asymptotic expansion for small-volume sources. We validate this approach through 2D numerical examples with synthetic data. We then expand this technique to address the acoustic inverse scattering problem, applying a versatile imaging function suitable for both near-field and far-field data. We provide a rigorous analysis to establish the decay rate of the imaging function. Integration with a deep neural network featuring a U-net architecture allows us to effectively solve the problem. This combined method accelerates the training process, provides accurate reconstruction, and excels in handling noisy and one-incident wave data. Numerical examples in 2D using synthetic data are provided. Finally, we modify the orthogonality sampling method to accommodate a wider range of polarization vectors associated with data for electromagnetic inverse scattering problems. This method is analyzed using the factorization analysis for the far-field operator and the Funk-Hecke formula. We validate its performance against 3D experimental data from Fresnel, comparing it to both its original formulation and traditional factorization methods. Numerical studies demonstrate that our method performs better than these approaches on this specific sparse and limited-aperture real dataset in a fast and simple manner.