Accelerating MCMC via Kriging-based adaptive independent proposals and delayed rejection

Abstract

Markov Chain Monte Carlo (MCMC) simulation has a considerable computational burden when the target probability density function (PDF) evaluation involves a black-box, potentially computationally-expensive, numerical model. A novel framework to accelerate MCMC is developed here for such applications. It leverages a Kriging surrogate model approximation to the target PDF to improve computational efficiency, while preserves convergence properties to the exact target PDF, avoiding potential accuracy problems introduced through the surrogate model error. The approach relies on the delayed-rejection (DR) scheme and combines the rapid exploration characteristics of global (independent) proposals with the local search robustness of random walk proposals under the MCMC setting. The global proposal is chosen as the Kriging-based target PDF approximation. This proposal may resemble, depending on the surrogate model error, the actual target PDF very well, and therefore can provide significant computational benefits, including fast convergence of the Markov chain to its stationary distribution and, more importantly, low correlation between samples. At each MCMC step a candidate sample is generated from the independent proposal. If rejected, DR allows an extra random walk around the current sample. The DR step guarantees convergence to the actual target PDF, circumventing robustness problems that may arise when the Kriging-based independent proposal offers a poor approximation to (i.e., underestimates) the actual target PDF. The overall computational efficiency is further improved through an adaptive updating of the Kriging surrogate model during the MCMC sampling phase by extracting information from candidate samples whose inclusion in the training database can substantially enhance Kriging’s accuracy. The computational efficiency and robustness of the established algorithm, termed Adaptive Kriging Delayed Rejection Adaptive Metropolis algorithm(AK-DRAM), are demonstrated in two analytical benchmark problems and two engineering problems focusing on conditional failure sample simulation and Bayesian inference.