Electromagnetic Inverse Wave Scattering via Reduced Order Modeling

Abstract

The inverse wave scattering problem seeks to determine spatially variable coefficients in wave equations, that model an unknown and heterogeneous medium, from recordings of waves generated by probing signals. This inverse problem has a wide range of applications such as radar imaging, medical imaging and geophysical prospecting. It is widely studied, but the existing inversion methodologies remain unsatisfactory.

This thesis introduces a reduced order model (ROM) approach for inverse scattering with electromagnetic waves in two different settings:

1.Non-magnetic, lossy layered media that give strong attenuation of the waves.

2. Two-dimensional, lossless and non-magnetic anisotropic media.

In both cases, the wave propagation is governed by Maxwell's equations. We show how to reformulate these equations as infinite-dimensional dynamical systems. The ROMs are low-dimensional (algebraic) dynamical systems that can be computed non-iteratively using numerical linear algebra methods. This computation uses the measurements of the waves at user controlled antennas. It does not require knowledge of the medium through which the waves propagate, so the ROMs are data driven.

In the case of lossy layered media, we get a so-called port-Hamiltonian (pH) dynamical system with transfer function expressed in terms of the measurements from the antenna. The data-driven ROM consists of a matrix with special algebraic structure that can be interpreted as a finite difference scheme of Maxwell's equations. Its transfer function is a rational approximation of the true transfer function, and the ROM preserves the passivity of the dynamical system i.e., it does not generate energy. We show that the entries of the ROM matrix are determined by local averages of the unknown electrical conductivity and dielectric permittivity on a special staggered grid. This grid can be computed and it can be used to estimate the medium. The result is an efficient inversion algorithm that outperforms the conventional nonlinear least squares data fitting approach.

For inverse scattering in two-dimensional lossless, non-magnetic and anisotropic media, the wave propagation is recast as an exact time stepping scheme i.e., a discrete time dynamical system with states given by the snapshots of the wave on a uniform time grid. The dynamical system is driven by an operator, called the propagator, that contains all the information about the unknown, matrix valued dielectric permittivity. The ROM is a Galerkin projection of this dynamical system on the space spanned by the snapshots. We show that although this space is unknown, the ROM can be computed just from the measurements at the user controlled antennas. We characterize the properties of the ROM and show how it can be used to estimate the unknown snapshots in the medium. These estimates are used in two ways: First, we define an imaging function that is inexpensive to compute and is designed to localize the reflective part of the medium, corresponding mathematically to the jump discontinuities of the components of the dielectric permittivity matrix. Second, we formulate an inversion algorithm for estimating quantitatively this matrix. We motivate and analyze these methods and demonstrate with numerical simulations that they outperform the traditional approaches to imaging and inverse scattering.