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We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Bayesianinverse problems with "Besov random tree prior". These priors are given by awavelet series with stochastic coefficients, and certain terms in the expansionvanishing at random, according to the law of so-called Galton-Watson trees.This allows to incorporate random fractal structures and large deviations inthe log-diffusion, which occur naturally in many applications from geophysicsor medical imaging. This framework entails two main difficulties: First, theassociated diffusion coefficient does not satisfy a uniform ellipticitycondition, which leads to non-integrable terms and thus divergence of standardmultilevel estimators. Secondly, the associated space of parameters is Polish,but not a normed linear space. We address the first point by introducingcut-off functions in the estimator to compensate for the non-integrable terms,while the second issue is resolved by employing an independenceMetropolis-Hastings sampler. The resulting algorithm converges in themean-square sense with essentially optimal asymptotic complexity, anddimension-independent acceptance probabilities.

Multilevel Markov Chain Monte Carlo for Bayesian Elliptic Inverse Problems with Besov Random Tree Priors