A Bayesian Approach to Estimating Background Flows from a Passive Scalar
Jeff Borggaard, Nathan Glatt-Holtz, Justin Krometis · SIAM/ASA Journal on Uncertainty Quantification · 2020
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 regularize it by laying out a functional analytic and Bayesian framework for approaching this problem. In doing so, we leverage substantial recent advances in statistical inference and adjoint methods for infinite-dimensional problems. We then 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.