Analysis and Numerical Simulations of Time Reversal and Multiple Scattering


This thesis contributes to the mathematical analysis and numerical simulation of some problems arising in multiple scattering. This document is divided into two parts. The first part deals with some inverse problems related to the detection and localization of targets using a Time Reversal Mirror (TRM). These devices are able to back-propagate a signal into a medium on the source that emitted it. We first study the DORT method, which is an empirical technique used to focus waves selectively on small unknown scatterers. In the context of acoustic scattering, a numerical investigation of the mathematical results obtained by C. Hazard and K. Ramdani is proposed. These results are then mathematically extended to the electromagnetic case. To conclude the first part, we study numerically the reconstruction of an acoustic point source with a TRM. Here, we are mainly interested in the super-resolution phenomena, that is, the enhancement of the quality of focusing in an heterogeneous medium instead of an homogeneous one. By using a deterministic model, we numerically solve the Helmholtz equation and give examples of numerical simulations that illustrate this phenomena. The second part is devoted to the numerical solution of acoustic multiple scattering problems using integral equations. Here, multiple scattering means that there are at least two scatterers in the medium unlike single scattering where only one obstacle is considered. For circular scatterers, the Fourier coefficients of the four classical boundary integral operators are computed analytically. This allows us to propose an efficient and robust numerical method to solve the multiple scattering problem with many disks. This strategy involves a sparse storage of the matrix of the linear system, which is then solved using the GMRES iterative solver combined with a preconditioner based on single scattering contributions. It appears that all the so-preconditioned integral equations are identical, up to an invertible operator. This result is first proved for circular scatterers and then extended to any arbitrarily shaped regular obstacle. Finally, deriving the Fourier coefficients of the single-layer boundary integral operator gives us the opportunity to study the spectrum of this operator in the low frequency regime. We consider first the single scattering case and then two multiple scattering regimes: a diluted medium where obstacles are far from each other and a dense medium where the scatterers are close.

Université Henri Poincaré - Nancy 1