Hierarchical Ensemble Kalman Methods with Sparsity-Promoting Generalized Gamma Hyperpriors

TL;DR

This paper develops a hierarchical ensemble Kalman framework with sparsity-promoting hyperpriors for nonlinear inverse problems, enabling effective regularization and improved solution estimates.

Contribution

It introduces an iterative algorithm combining ensemble Kalman updates with hyperparameter tuning for sparsity, advancing regularization techniques in inverse problem solving.

Findings

Effective in compressed sensing applications

Improves solutions in subsurface flow inverse problems

Demonstrates sparsity promotion through hyperpriors

Abstract

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.