  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.
