We present a GPU-based image reconstruction algorithm inspired by sparse Bayesian learning (SBL). Reweighted quadratic penalties are used to promote sparsity in the pixeldifferences domain. In addition to the object estimate, SBL also computes the posterior variances and exploits them for the reweighting of the penalties. This enables to automatically learn the balance between the data-fit and the prior without using any tuning parameters. We consider several methods to accelerate the computation of the posterior variances in large scale problems in order to resolve the main computational bottleneck in SBL. These methods are based on conjugate gradient solvers and rely on fast forward/back-projections which are implemented on a GPU to exploit various levels of parallelism. We also show how the proposed algorithm can be used for adaptive sensing where the measurements are selected sequentially based on information learned from previously available data, leading to a reduction in the total number of measurements.