A Bayesian Approach to Estimating Background Flows from a Passive Scalar

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.