In this paper we present a holistic framework for full waveform inversion (FWI) in the visco-acoustic regime. FWI entails the reconstruction of material parameters (such as
density and sound speed) from measurements of reflected wave fields (seismograms). We derive a discontinuous Galerkin (DG) solver for the visco-acoustic wave equation and incorporate it into an inverse solver. For the DG discretization we provide a block diagonal preconditioner for the efficient computation of the time steps by GMRES which yields a convergence estimate in space and time. Numerical tests illustrate these results. Furthermore, we set up an inverse solver of well established Newton-CG type, and we express the required Fréchet derivative and its adjoint in the DG setting. Reconstructions from simulated cross-well seismograms highlight the challenges of FWI and demonstrate the performance of the scheme. Some of the inversion experiments use seismograms generated by an independent FDTD forward solver to avoid an inverse crime.