We add a time-dependent potential to the inhomogeneous wave equation and consider the task of reconstructing this potential from measurements of the wave field. This dynamic inverse problem becomes more involved compared to static parameters, as, e.g. the dimensions of the parameter space do considerably increase. We give a specifically tailored existence and uniqueness result for the wave equation and compute the Fréchet derivative of the solution operator, for which also show the tangential cone condition. These results motivate the numerical reconstruction of the potential via successive linearization and regularized Newton-like methods. We present several numerical examples showing feasibility, reconstruction quality, and time efficiency of the resulting algorithm.
For parameter identification problems the Fréchet-derivative of the parameter-to-state map is of particular interest. In many applications, e.g. in seismic tomography, the unknown quantity is modeled as a coefficient in a linear differential equation, therefore computing the derivative of this map involves solving the same equation, but with a different right-hand side. It then remains to show that this right-hand side is regular enough to ensure the existence of a solution. For second-order hyperbolic PDEs with time-dependent parameters the needed results are not as readily available as in the stationary case, especially when working in a variational framework. This complicates for example the reconstruction of a time-dependent density in the wave equation. To overcome this problem we extend the existing regularity results to the time-dependent case.
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 Fréchet-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 Fréchet-derivative of the forward operator, which is of particular interest for the application of regularization schemes.
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