Some Theoretical and Numerical Studies for Inverse Problems in Wave Propagations 

Description

This project is concerned with some theoretical and numerical studies for several  inverse problems associated with electromagnetic (EM) and elastic wave propagations.  In the first part of this project, motived by applications from brain imaging such  as electroencephalography (EEG) and magnetoencephalography (MEG), we consider an  inverse problem associated with the Maxwell system in simultaneously recovering an unknown  electric source term and the surrounding medium parameters inside a bounded  body by the corresponding boundary measurements. We shall be mainly concerned with  the theoretical identifiability study; that is, given the measurement data, what kind of  unknowns one can recover. We aim to establish sufficient conditions on the sources and  the medium parameters, under which all of them can be simultaneously identified. The  results that we propose to establish are new to the literature. On the one hand, the theoretical  identifiability results provide the “faith” in recovering certain unknowns; and on  the other hand, it points out the possible directions on how to recover more general unknowns  by either approximating them with the ones that can be identified, or by devising  different measurement methods.  We also propose to extend the simultaneous identifiability results for the Maxwell  system to the Lamé system, governing the propagation of elastic waves. The corresponding  theoretical identifiability results may be of practical importance to earthquake research,  where one intends to infer knowledge on both the source of the earthquake and the interior  structure of the earth by the corresponding seismic wave measurements. The second part of the project is devoted to the mathematical design of novel input/  instruction devices using EM wave detections. Nowadays, the majority of input and  instruction technology is based on text user interfaces (TUIs) or graphical user interfaces  (GUIs), using devices such as keyboards, mouse or touch-screens. The input/instruction  device bridges the computer and the human being who wants to interact or communicate  with the computer. There are some emerging new technologies using body language for  the input and instruction. The body language could be motion trajectories or gestures.  In this project, we propose two novel devices of using inverse EM scattering techniques to  understand the body language. We shall construct the mathematical modelling and develop  the corresponding mathematical and computational strategies to overcome several  challenging issues involved in the conceptual designs.