  We consider the statistical inverse problem of estimating a background flow
field (e.g., of air or water) from the partial and noisy observation of a
passive scalar (e.g., the concentration of a solute), a common experimental
approach to visualizing complex fluid flows. Here the unknown is a vector field
that is specified by a large or infinite number of degrees of freedom. Since
the inverse problem is ill-posed, i.e., there may be many or no background
flows that match a given set of observations, we adopt a Bayesian approach to
regularize it. In doing so, we leverage frameworks developed in recent years
for infinite-dimensional Bayesian inference. The contributions in this work are
threefold. First, we lay out a functional analytic and Bayesian framework for
approaching this problem. Second, we define an adjoint method for efficient
computation of the gradient of the log likelihood, a key ingredient in many
numerical methods. Finally, we identify interesting example problems that
exhibit posterior measures with simple and complex structure. We use these
examples to conduct a large-scale benchmark of Markov Chain Monte Carlo methods
developed in recent years for infinite-dimensional settings. Our results
indicate that these methods are capable of resolving complex multimodal
posteriors in high dimensions.
