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Multi-Level Markov Chain Monte Carlo for Bayesian Elliptic Inverse Problems with Besov Random Tree Priors

by A. Stein and V.H. Hoang

(Report number 2023-06)

Abstract
We propose a multilevel Markov Chain Monte Carlo-FEM algorithm to solve elliptic Bayesian inverse problems with "Besov random tree prior". These priors are given by a wavelet series with stochastic coefficients, and certain terms in the expansion vanishing at random, according to the law of so-called Galton-Watson trees. This allows to incorporate random fractal structures and large deviations in the log-diffusion, which occur naturally in many applications from geophysics or medical imaging. This framework entails two main difficulties: First, the associated diffusion coefficient does not satisfy a uniform ellipticity condition, which leads to non-integrable terms and thus divergence of standard multilevel ML estimators. Secondly, the associated space of parameters is Polish, but not a normed linear space, and thus prevents random walk or preconditioned Crank-Nicolson proposals for the Markov chains. We address the first point by introducing cut-off functions in the estimator to compensate for the non-integrable terms, while the second issue is resolved by employing an independence Metropolis-Hastings sampler. The resulting algorithm converges in the mean-square sense with essentially optimal asymptotic complexity, and dimension-independent acceptance probabilities.

Keywords: Bayesian inverse problem, Besov prior, Galton-Watson tree, Markov Chain Monte Carlo, multilevel Monte Carlo, parametric PDE

@Techreport{SH23_1043,
  author = {A. Stein and V.H.  Hoang},
  title = {Multi-Level Markov Chain Monte Carlo for Bayesian Elliptic Inverse Problems with Besov Random Tree Priors},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-06.pdf },
  year = {2023}
}

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