  We present a new nonlinear optimization approach for the sparse
reconstruction of single-photon absorption and two-photon absorption
coefficients in photoacoustic tomography (PAT). This framework comprises of
minimizing an objective functional involving a least squares fit of the
interior pressure field data corresponding to two boundary source functions,
where the absorption coefficients and the photon density are related through a
semi-linear elliptic partial differential equation (PDE) arising in PAT.
Further, the objective functional consists of an $L^1$ regularization term that
promotes sparsity patterns in absorption coefficients. The motivation for this
framework primarily comes from some recent works related to solving inverse
problems in acousto-electric tomography and current density impedance
tomography. We provide a new proof of existence and uniqueness of a solution to
the semi-linear PDE. Further, a proximal method, involving a Picard solver for
the semi-linear PDE and its adjoint, is used to solve the optimization problem.
Several numerical experiments are presented to demonstrate the effectiveness of
the proposed framework.
