Projects per year
Abstract
In many hierarchical inverse problems, not only do we want to estimate high or infinitedimensional model parameters in the parametertoobservable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of highdimensionality, nonlinear dependence, and nonconcave structures in the joint posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we develop scalable optimizationbased Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parametertoobservable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomizethenoptimize (RTO) method [J. M. Bardsley et al., SIAM J. Sci. Comput., 42 (2016), pp. A1317A1347] for exploring the high or infinitedimensional parameter space. We first extend the RTO machinery to the Poisson likelihood and discuss the implementation of RTO in the hierarchical setting. Then, by using RTO either as a proposal distribution in a MetropoliswithinGibbs update or as a biasing distribution in the pseudomarginal MCMC [C. Andrieu and G. O. Roberts, Ann. Statist., 37 (2009), pp. 697725], we present efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudomarginal MCMC has sampling performance robust to model parameter dimensions. Numerical examples in PDEconstrained inverse problems and positron emission tomography are used to demonstrate the performance of our methods.
Original language  English 

Pages (fromto)  2964 
Number of pages  36 
Journal  SIAM/ASA Journal on Uncertainty Quantification 
Volume  9 
Issue number  1 
DOIs  
Publication status  Published  2021 
Keywords
 Hierarchical Bayes
 Inverse problems
 Markov chain Monte Carlo
 Poisson likelihood
 Positron emission tomography
 Pseudomarginalization
Projects
 1 Finished

ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights
Hall, P., Bartlett, P., Bean, N., Burrage, K., DeGier, J., Delaigle, A., Forrester, P., Geweke, J., Kohn, R., Kroese, D., Mengersen, K. L., Pettit, A., Pollett, P., Roughan, M., Ryan, L. M., Taylor, P., Turner, I., Wand, M., Garoni, T., SmithMiles, K. A., Caley, M., Churches, T., Elazar, D., Gupta, A., Harch, B., Tam, S., Weegberg, K., Willinger, W. & Hyndman, R.
Australian Research Council (ARC), Monash University – Internal Department Contribution, University of Melbourne, Queensland University of Technology (QUT), University of Adelaide, University of New South Wales (UNSW), University of Queensland , University of Technology (UTS) Sydney, Monash University – Internal University Contribution, Monash University – Internal Faculty Contribution, Monash University – Internal School Contribution, Roads Corporation (trading as VicRoads) (Victoria)
1/01/17 → 31/12/21
Project: Research