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Sparsity has become a key concept for solving of high-dimensional inverseproblems using variational regularization techniques. Recently, using similarsparsity-constraints in the Bayesian framework for inverse problems by encodingthem in the prior distribution has attracted attention. Important questionsabout the relation between regularization theory and Bayesian inference stillneed to be addressed when using sparsity promoting inversion. A practicalobstacle for these examinations is the lack of fast posterior samplingalgorithms for sparse, high-dimensional Bayesian inversion: Accessing the fullrange of Bayesian inference methods requires being able to draw samples fromthe posterior probability distribution in a fast and efficient way. This isusually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In thisarticle, we develop and examine a new implementation of a single componentGibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate thatthe efficiency of our Gibbs sampler increases when the level of sparsity or thedimension of the unknowns is increased. This property is contrary to theproperties of the most commonly applied Metropolis-Hastings (MH) samplingschemes: We demonstrate that the efficiency of MH schemes for L1-type priorsdramatically decreases when the level of sparsity or the dimension of theunknowns is increased. Practically, Bayesian inversion for L1-type priors usingMH samplers is not feasible at all. As this is commonly believed to be anintrinsic feature of MCMC sampling, the performance of our Gibbs sampler alsochallenges common beliefs about the applicability of sample based Bayesianinference.

Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors