Towards a Unified Framework for Inverse Source Problems and Multiscale Modeling in Layered Media

Active manipulation of the wave phenomena has been an emerging research area due to its far-reaching applications. These applications are from military and industrial utilization, such as secure wireless communication and enhanced remote sensing, down to our daily lives, like personal audio synthesis, active noise canceling headphones, and intelligent wireless internet connectivity routers. The active field control problem is treated as an inverse source problem (ISP). The main goal of an inverse source problem is to characterize a surface source from the knowledge of the radiating fields in the exterior regions. This problem differs from the classic inverse source problem in that the patterns to be approximated are mutually different. In addition, the source can be characterized precisely due to the analyticity of fields in regions without sources. In general, the inverse problem is challenging due to its ill-posedness. Therefore, the significance of this thesis is towards the design of numerically stable methods for inverse source problems. More specifically, it presents several constructive schemes for actively manipulating fields that satisfy the scalar Helmholtz equation or vector Maxwell's equations.

This thesis mainly focuses on developing a unified mathematical framework for active field manipulation in acoustics and electromagnetics, especially in layered media. First, the acoustic field control is investigated in two media, free space and a homogeneous ocean with a constant depth. The characterized source can realize single and multiple-region control and simultaneously control the far-field pattern. Then, we extend the numerical framework to actively manipulate the electromagnetic field in both homogeneous and layered media. Furthermore, we discuss the sensitivity of the active scheme (concerning power budget and control accuracy) as a function of the control parameters. A small part of this thesis also investigates the electromagnetic modeling of multiscale problems in layered media. The purpose is to validate the accuracy and robustness of the boundary integral equation (BIE) method, also used in ISP formulations. A detailed theoretical analysis is included in each of these areas. Several numerical examples validate the proposed theory's stability and robustness.