  In this paper, we perform non-linear minimization using the Hybridizable
Discontinuous Galerkin method (HDG) for the discretization of the forward
problem, and implement the adjoint-state method for the computation of the
functional derivatives. Compared to continuous and discontinuous Galerkin
discretizations, HDG reduces the computational cost by working with the
numerical traces, hence removing the degrees of freedom that are inside the
cells. It is particularly attractive for large-scale time-harmonic quantitative
inverse problems which make repeated use of the forward discretization as they
rely on an iterative minimization procedure. HDG is based upon two levels of
linear problems: a global system to find the numerical traces, followed by
local systems to construct the volume solution. This technicality requires a
careful derivation of the adjoint-state method, that we address in this paper.
We work with the acoustic wave equations in the frequency domain and illustrate
with a three-dimensional experiment using partial reflection-data, where we
further employ the features of DG-like methods to efficiently handle the
topography with p-adaptivity.
