  This paper introduces a computational framework to incorporate flexible
regularization techniques in ensemble Kalman methods for nonlinear inverse
problems. The proposed methodology approximates the maximum a posteriori (MAP)
estimate of a hierarchical Bayesian model characterized by a conditionally
Gaussian prior and generalized gamma hyperpriors. Suitable choices of
hyperparameters yield sparsity-promoting regularization. We propose an
iterative algorithm for MAP estimation, which alternates between updating the
unknown with an ensemble Kalman method and updating the hyperparameters in the
regularization to promote sparsity. The effectiveness of our methodology is
demonstrated in several computed examples, including compressed sensing and
subsurface flow inverse problems.
