A note on a bilevel problem for parameter learning for inverse problems with the wave equation

In this paper we consider a bilevel problem for determining the optimal regularization parameter in an inverse problem with the linear wave equation transferring results from Holler, Kunisch, and Barnard (2018), where a general function space setting and applications to (bilinear) elliptic problems have been addressed. We analyze the well-posedness and derive the optimality conditions for the bilevel problem for the wave equation. Moreover, for given noisy data the numerical performance of the approach to find the regularization parameter is compared for different choices of priors in the Tikohonov regularization term of the lower level problem.