Accelerating MCMC with Active Subspaces

Carson Kent, Stanford University

The Markov chain Monte Carlo (MCMC) method is the computational workhorse for Bayesian inverse problems. However, MCMC struggles in high-dimensional parameter spaces since its iterates must sequentially explore a high-dimensional space for accurate inference. This struggle is compounded in physical applications when the nonlinear forward model is computationally expensive. One approach to accelerate MCMC is to reduce the dimension of the state space. Active subspaces are an emerging set of tools for dimension reduction. When applied to MCMC, the active subspace separates a low-dimensional subspace, which is informed by the data, from its orthogonal complement, which is constrained by the prior. With this information, one can run the sequential MCMC on the active variables while sampling independently according to the prior on the inactive variables. We use this technique to perform MCMC in a 100-dimensional parameter space for a PDE-based forward model and provide theoretical bounds on the distance between the true posterior and its approximation with the active subspace.

Abstract Author(s): P. Constantine, C. Kent, T. Bui-Thanh