# Dynamic Inverse Wave Problems - Part II: Operator Identification and Applications

## TL;DR

This paper develops a theoretical framework for analyzing inverse wave problems modeled by hyperbolic PDEs, focusing on operator identification, well-posedness, and applications to elasticity and electrodynamics.

## Contribution

It introduces a general approach for analyzing dynamic inverse problems, including differentiability and ill-posedness, and applies it to practical equations in elasticity and electrodynamics.

## Key findings

- Established well-posedness and regularity of the forward operator.
- Proved Fréchet differentiability and local ill-posedness.
- Derived explicit adjoint formulas for practical applications.

## Abstract

We present a framework which enables the analysis of dynamic inverse problems for wave phenomena that are modeled through second-order hyperbolic PDEs. This includes well-posedness and regularity results for the forward operator in an abstract setting, where the operators in an evolution equation represent the unknowns. We also prove Fr\'echet-differentiability and local ill-posedness for this problem. We then demonstrate how to apply this theory to actual problems by two example equations motivated by linear elasticity and electrodynamics. For these problems it is even possible to obtain a simple characterization of the adjoint of the Fr\'echet-derivative of the forward operator, which is of particular interest for the application of regularization schemes.