Accelerating Mcmc with Active Subspaces

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 that is informed by the data from its orthogonal complement that 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 provide a theoretical bound on the Hellinger distance between the true posterior and its approximation with the active subspace. And we demonstrate the active subspace-accelerated MCMC on two computational examples: (i) a two-dimensional parameter space with a quadratic forward model and one-dimensional active subspace and (ii) a 100-dimensional parameter space with a PDE-based forward model and a two-dimensional active subspace.