Accelerating MCMC with active subspaces

TL;DR

This paper introduces an active subspace approach to accelerate MCMC sampling in high-dimensional Bayesian inverse problems, reducing computational cost while providing bounds on approximation bias.

Contribution

It proposes a novel integration of active subspaces with MCMC, including theoretical bounds on the approximation error and practical demonstrations on complex models.

Findings

Active subspaces can significantly reduce MCMC computational cost.

The method provides a bound on the Hellinger distance between true and approximate posteriors.

Demonstrated effectiveness on high-dimensional PDE-based models.

Abstract

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 the high-dimensional space. 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 part of an emerging set of tools for subspace-based dimension reduction. An active subspace in a given inverse problem indicates a separation between a low-dimensional subspace that is informed by the data and 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. However, this approach to increase efficiency may introduce bias. We provide a bound on the Hellinger distance between the true posterior and its active subspace- exploiting approximation. 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.