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In this article, we study several reconstruction methods for the inversesource problem of photoacoustic tomography (PAT) with spatially variable soundspeed and damping. The backbone of these methods is the adjoint operators,which we thoroughly analyze in both the $L^2$- and $H^1$-settings. They arecasted in the form of a nonstandard wave equation. We derive the well-pawednessof the aforementioned wave equation in a natural functional space, and alsoprove the finite speed of propagation. Under the uniqueness and visibilitycondition, our formulations of the standard iterative reconstruction methods,such as Landweber's and conjugate gradients (CG), achieve a linear rate ofconvergence in either $L^2$- or $H^1$-norm. When the visibility condition isnot satisfied, the problem is severely ill-posed and one must apply aregularization technique to stabilize the solutions. To that end, we study twoclasses of regularization methods: (i) iterative, and (ii) variationalregularization. In the case of full data, our simulations show that the CGmethod works best; it is very fast and robust. In the ill-posed case, the CGmethod behaves unstably. Total variation regularization method (TV), in thiscase, significantly improves the reconstruction quality.

Reconstruction algorithms for photoacoustic tomography in heterogenous damping media