Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity. (3rd June 2016) - Kirsch, Andreas; Rieder, Andreas
Abstract: It is common knowledge—mainly based on experience—that parameter identification problems in partial differential equations are ill-posed. Yet, a mathematical sound argumentation is missing, except for some special cases. We present a general theory for inverse problems related to abstract evolution equations which explains not only their local ill-posedness but also provides the Fréchet derivative and its adjoint of the corresponding parameter-to-solution map which are needed, e.g., in Newton-like solvers. Our abstract results are applied to inverse problems related to the following first order hyperbolic systems: Maxwell's equation (electromagnetic scattering in conducting media) and elastic wave equation (seismic imaging).