On the robustness of inverse scattering for penetrable, homogeneous objects with complicated boundary

The acoustic inverse obstacle scattering problem consists of determining the shape of a domain from measurements of the scattered far field due to some set of incident fields (probes). For a penetrable object with known sound speed, this can be accomplished by treating the boundary alone as an unknown curve. Alternatively, one can treat the entire object as unknown and use a more general volumetric representation, without making use of the known sound speed. Both lead to strongly nonlinear and nonconvex optimization problems for which recursive linearization provides a useful framework for numerical analysis. After extending our shape optimization approach developed earlier for impenetrable bodies, we carry out a systematic study of both methods and compare their performance on a variety of examples. Our findings indicate that the volumetric approach is more robust, even though the number of degrees of freedom is significantly larger. We conclude with a discussion of this phenomenon and potential directions for further research.Comment: 24 pages, 9 figure

Numerical methods for calculating poles of the scattering matrix with applications in grating theory

Waveguide and resonant properties of diffractive structures are often explained through the complex poles of their scattering matrices. Numerical methods for calculating poles of the scattering matrix with applications in grating theory are discussed. A new iterative method for computing the matrix poles is proposed. The method takes account of the scattering matrix form in the pole vicinity and relies upon solving matrix equations with use of matrix decompositions. Using the same mathematical approach, we also describe a Cauchy-integral-based method that allows all the poles in a specified domain to be calculated. Calculation of the modes of a metal-dielectric diffraction grating shows that the iterative method proposed has the high rate of convergence and is numerically stable for large-dimension scattering matrices. An important advantage of the proposed method is that it usually converges to the nearest pole.Comment: 9 pages, 2 figures, 4 table

Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks

This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of classical methods. The architecture is motivated by the filtered back-projection formula in the full aperture regime and with homogeneous background, and it leverages the underlying equivariance of the problem and compressibility of the integral operator. This drastically reduces the number of training parameters, and therefore the computational and sample complexity of the method. In particular, we obtain an architecture whose number of parameters scale sub-linearly with respect to the dimension of the inputs, while its inference complexity scales super-linearly but with very small constants. We provide several numerical tests that show that the current approach results in better reconstruction than optimization-based techniques such as full-waveform inversion, but at a fraction of the cost while being competitive with state-of-the-art machine learning methods.Comment: 21 pages, 9 figures, and 4 table