Adaptive Error Modelling in MCMC Sampling for Large Scale Inverse Problems

Abstract

We present a new adaptive delayed-acceptance Metropolis-Hastings algorithm (ADAMH) that adapts to the error in a reduced order model to enable efficient sampling from the posterior distribution arising in complex inverse problems. This use of adaptivity differs from existing algorithms that tune proposals of the Metropolis-Hastings algorithm (MH), though ADAMH also implements that strategy. We build on the recent simplified conditions given by Roberts and Rosenthal (2007) to give practical constructions that are provably convergent to the correct target distribution. The main components of ADAMH are the delayed acceptance scheme of Christen and Fox (2005), the enhanced error model introduced by Kaipio and Somersalo (2007) as well as recent advances in adaptive MCMC (Haario et al., 2001; Roberts and Rosenthal, 2007). We developed this algorithm for automatic calibration of large-scale numerical models of geothermal reservoirs. ADAMH shows good computational and statistical efficiencies on measured data sets. This algorithm could allow significant improvement in computational efficiency when implementing sample-based inference in other large-scale inverse problems.