Optimization-Based MCMC Methods for Nonlinear Hierarchical Statistical Inverse Problems

• URL: http://arxiv.org/abs/2002.06358v1
• Date: Sat, 15 Feb 2020 10:19:42 GMT
• Title: Optimization-Based MCMC Methods for Nonlinear Hierarchical Statistical Inverse Problems
• Authors: Johnathan Bardsley, Tiangang Cui
• Abstract summary: In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyper parameters. In this work, we aim to develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems.
• Score: 0.6091702876917279
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and non-concave structures in the joint posterior posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we aim to develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parameter-to-observable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomize-then-optimize (RTO) method [4] for exploring the high- or infinite-dimensional model parameter space. By using RTO either as a proposal distribution in a Metropolis-within-Gibbs update or as a biasing distribution in the pseudo-marginal MCMC [2], we are able to design efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudo-marginal MCMC has sampling performance robust to model parameter dimensions. We also extend our methods to nonlinear inverse problems with Poisson-distributed measurements. Numerical examples in PDE-constrained inverse problems and positron emission tomography (PET) are used to demonstrate the performance of our methods.

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