  This paper considers the non-linear inverse problem of reconstructing an
electric conductivity distribution from the interior power density in a bounded
domain. Applications include the novel tomographic method known as
acousto-electric tomography, in which the measurement setup in Electrical
Impedance Tomography is modulated by ultrasonic waves thus giving rise to a
method potentially having both high contrast and high resolution. We formulate
the inverse problem as a regularized non-linear optimization problem, show the
existence of a minimizer, and derive optimality conditions. We propose a
non-linear conjugate gradient scheme for finding a minimizer based on the
optimality conditions. All our numerical experiments are done in
two-dimensions. The experiments reveal new insight into the non-linear effects
in the reconstruction. One of the interesting features we observe is that,
depending on the choice of regularization, there is a trade-off between high
resolution and high contrast in the reconstructed images. Our proposed
non-linear optimization framework can be generalized to other hybrid imaging
modalities.
