A Reduced Order Model Approach to Inverse Scattering in Lossy Layered Media

• DOI: 10.1007/s10915-021-01616-7
• Publication Date: 2021-08-16T04:02:31Z
• Authors: Liliana Borcea; Vladimir Druskin; Jörn Zimmerling
• Keywords: FOS: Mathematics; Mathematics - Numerical Analysis; Dynamical Systems (math.DS); Numerical Analysis (math.NA); Mathematics - Dynamical Systems; 0101 mathematics; 01 natural sciences; 37N30, 65N21, 65L09, 86A22

We introduce a reduced order model (ROM) methodology for inverse electromagnetic wave scattering in layered lossy media, using data gathered by an antenna which generates a probing wave and measures the time resolved reflected wave. We recast the wave propagation problem as a passive infinite-dimensional dynamical system, whose transfer function is expressed in terms of the measurements at the antenna. The ROM is a low-dimensional dynamical system that approximates this transfer function. While there are many possible ROM realizations, we are interested in one that preserves passivity and in addition is: (1) data driven (i.e., is constructed only from the measurements) and (2) it consists of a matrix with special sparse algebraic structure, whose entries contain spatially localized information about the unknown dielectric permittivity and electrical conductivity of the layered medium. Localized means in the intervals of a special finite difference grid. The main result of the paper is to show with analysis and numerical simulations that these unknowns can be extracted efficiently from the ROM.