Adaptive Approximation Error Models for Efficient Uncertainty Quantification with Application to Multiphase Subsurface Fluid Flow

Sample-based Bayesian inference provides a route to uncertainty quantification in the geosciences, though is very computationally demanding in the na\"ive form that requires simulating an accurate computer model at each iteration. We present a new approach that adaptively builds a stochastic model for the error induced by a reduced model. This enables sampling from the correct target distribution at reduced computational cost, while avoiding appreciable loss of statistical efficiency. We build on recent simplified conditions for adaptive Markov chain Monte Carlo algorithms to give practical approximation schemes and algorithms with guaranteed convergence. We demonstrate the efficacy of our new approach on two computational examples, including calibration of a large-scale numerical model of a real geothermal reservoir, that show good computational and statistical efficiencies on both synthetic and measured data sets.