  Sample-based Bayesian inference provides a route to uncertainty
quantification in the geosciences, and inverse problems in general, though is
very computationally demanding in the naive form that requires simulating an
accurate computer model at each iteration. We present a new approach that
constructs a stochastic correction to the error induced by a reduced model,
with the correction improving as the algorithm proceeds. This enables sampling
from the correct target distribution at reduced computational cost per
iteration, as in existing delayed-acceptance schemes, while avoiding
appreciable loss of statistical efficiency that necessarily occurs when using a
reduced model. Use of the stochastic correction significantly reduces the
computational cost of estimating quantities of interest within desired
uncertainty bounds. In contrast, existing schemes that use a reduced model
directly as a surrogate do not actually improve computational efficiency in our
target applications. We build on recent simplified conditions for adaptive
Markov chain Monte Carlo algorithms to give practical approximation schemes and
algorithms with guaranteed convergence. The efficacy of this new approach is
demonstrated in 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.
