Dynamic Inverse Problems for Wave Phenomena

Gerken, Thies
Universität Bremen: Informatik/Mathematik
inverse problems, evolution equations, wave equation, dynamic inverse problems, time-dependent parameters, elastic wave equation, electrodynamics, ill-posedness
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.
Dynamic Inverse Problems for Wave Phenomena