Reconstruction of a Time-dependent Potential from Wave Measurements
This work tackles a challenging inverse problem for time-dependent parameters in wave equations, but the results are incremental, extending static parameter reconstruction methods to the dynamic case.
The paper addresses the reconstruction of a time-dependent potential in the inhomogeneous wave equation from wave field measurements, providing existence and uniqueness results and a Fréchet derivative that satisfies the tangential cone condition. Numerical examples demonstrate feasibility and efficiency, though no concrete performance numbers are given.
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.