  In this paper we consider the inverse problem of identifying the initial data
in a fractionally damped wave equation from time trace measurements on a
surface, as relevant in photoacoustic or thermoacoustic tomography. We derive
and analyze a time stepping method for the numerical solution of the
corresponding forward problem. Moreover, to efficiently obtain reconstructions
by minimizing a Tikhonov regularization functional (or alternatively, by
computing the MAP estimator in a Bayesian approach), we develop an adjoint
based scheme for gradient computation. Numerical reconstructions in two space
dimensions illustrate the performance of the devised methods.
