Optimization-Based MCMC Methods for Nonlinear Hierarchical Statistical Inverse Problems
This work addresses computational bottlenecks for researchers in statistical inverse problems, though it appears incremental as it builds on existing RTO and pseudo-marginal MCMC methods.
The authors tackled the computational challenge of solving hierarchical Bayesian inverse problems with high-dimensional, nonlinear, and non-concave structures by developing scalable optimization-based MCMC methods, demonstrating performance in PDE-constrained and PET examples.
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.