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.
- Acousto-electric tomography
- Non-linear PDE optimization
- Hybrid imaging
- Non-linear conjugate gradient