Executive Summary : | The proposed project focuses on the analysis of Partial Differential Equations (PDEs) and their applications in Inverse Problems. Inverse Problems aim to understand the inner structure of an object or system by observing it from outside. They have numerous applications in fields such as Medical Imaging, Geophysics, Oceanography, and Non-destructive Testing. A toy model for inverse problems is considered, where an unknown medium is probed with acoustic waves of different frequencies and measured as the Cauchy data outside the medium. The goal is to determine the unknown density and/or stiffness parameters from the Cauchy data. An analogous problem is also investigated, involving the system of elastodynamic wave equations or the system of elasticity. The inverse problem for acoustic waves is useful in brain imaging, while the second model for elastic waves is of immense interest in Geophysics. To make these models useful, an inhomogeneous medium with possible discontinuities in material parameters is considered. These sets of discontinuities are called interfaces, and for brain imaging, the skull plays a role as an interface. Mathematical developments in this direction are promising, but there are still many open questions in the vast generality. The project aims to investigate this phenomenon for acoustic and elastic waves and establish determination of discontinuous or layered material coefficients of an inhomogeneous medium from Cauchy data. |