Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity. (3rd June 2016)

• Record Type: Journal Article
• Authors: Kirsch, Andreas; Rieder, Andreas
• Is Part Of: Inverse problems. Volume 32:Number 8(2016:Aug.)
• Publication Date: 2016-06-03
• Subjects: inverse problems -- ill-posed problems -- evolution equation -- Fréchet-differentiability -- electrodynamic wave equations -- elastic wave equations; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/8/085001
• Languages: English
• ISSNs: 0266-5611

Abstract

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).