  Optimization-based samplers such as randomize-then-optimize (RTO) [2] provide
an efficient and parallellizable approach to solving large-scale Bayesian
inverse problems. These methods solve randomly perturbed optimization problems
to draw samples from an approximate posterior distribution. "Correcting" these
samples, either by Metropolization or importance sampling, enables
characterization of the original posterior distribution. This paper focuses on
the scalability of RTO to problems with high- or infinite-dimensional
parameters. We introduce a new subspace acceleration strategy that makes the
computational complexity of RTO scale linearly with the parameter dimension.
This subspace perspective suggests a natural extension of RTO to a function
space setting. We thus formalize a function space version of RTO and establish
sufficient conditions for it to produce a valid Metropolis-Hastings proposal,
yielding dimension-independent sampling performance. Numerical examples
corroborate the dimension-independence of RTO and demonstrate sampling
performance that is also robust to small observational noise.
