A posteriori stochastic correction of reduced models in delayed acceptance MCMC, with application to multiphase subsurface inverse problems

TL;DR

This paper introduces a stochastic correction method for reduced models in delayed acceptance MCMC, significantly improving computational efficiency and accuracy in Bayesian inverse problems like geothermal reservoir calibration.

Contribution

It presents a novel stochastic correction approach that enhances reduced model accuracy during MCMC sampling, ensuring convergence and efficiency in complex inverse problems.

Findings

Reduces computational cost in Bayesian inference for geosciences.

Maintains statistical efficiency comparable to full-model sampling.

Demonstrates effectiveness in geothermal reservoir calibration.

Abstract

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.