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Bayesian modelling and computational inference by Markov chain Monte Carlo(MCMC) is a principled framework for large-scale uncertainty quantification,though is limited in practice by computational cost when implemented in thesimplest form that requires simulating an accurate computer model at eachiteration of the MCMC. The delayed acceptance Metropolis--Hastings MCMCleverages a reduced model for the forward map to lower the compute cost periteration, though necessarily reduces statistical efficiency that can, withoutcare, lead to no reduction in the computational cost of computing estimates toa desired accuracy. Randomizing the reduced model for the forward map candramatically improve computational efficiency, by maintaining the low cost periteration but also avoiding appreciable loss of statistical efficiency.Randomized maps are constructed by a posteriori adaptive tuning of a randomizedand locally-corrected deterministic reduced model. Equivalently, theapproximated posterior distribution may be viewed as induced by a modifiedlikelihood function for use with the reduced map, with parameters tuned tooptimize the quality of the approximation to the correct posteriordistribution. Conditions for adaptive MCMC algorithms allow practicalapproximations and algorithms that have guaranteed ergodicity for the targetdistribution. Good statistical and computational efficiencies are demonstratedin examples of calibration of large-scale numerical models of geothermalreservoirs and electrical capacitance tomography.

Randomized Reduced Forward Models for Efficient Metropolis--Hastings MCMC, with Application to Subsurface Fluid Flow and Capacitance Tomography