Dynamic Inverse Problems for Wave Phenomena

Abstract

In this work, we deal with second-order hyperbolic partial differential equations that include time- and space-dependent coefficients, and the inverse problems of identifying these coefficients based on their effect on the equationa s solution. We present the needed theory for such equations, including some regularity results for their solution. This allows to state and analyze the inverse problems, even in an abstract setting where time-dependent operators are sought. Subsequently, we show how these results can be applied to actual partial differential equations. We give a detailed demonstration in the context of the acoustic wave equation. Our results allow the identification of a time- and space-dependent wave speed and mass density in such a setting, and we give an extensive numerical analysis for this case. We also outline how the abstract framework can be applied to other equations, like simple models for electromagnetic waves.