  In this article, we revisit iterative methods for solving the inverse problem
of photoacoustic tomography in free space. Recently, there have been
interesting developments on explicit formulations of the adjoint operator,
demonstrating that iterative methods are an attractive choice for photoacoustic
image reconstruction. In this work, we propose several modifications of current
formulations of the adjoint operator which help speed up the convergence and
yield improved error estimates. We establish a stability analysis and show
that, with our choices of the adjoint operator, the iterative methods can
achieve a linear rate of convergence, in the $L^2$-norm as well as in the
$H^1$-norm. In addition, we analyze the normal operator from the microlocal
analysis point of view. This gives insight into the convergence speed of the
iterative methods and choosing proper weights for the mapping spaces. Finally,
we present numerical results using various iterative reconstruction methods for
full as well as limited view data. Our results demonstrate that Nesterov's fast
gradient and the CG methods converge faster than Landweber's and iterative time
reversal methods in the visible as well as the invisible case.
