A Bayesian Approach to Estimating Background Flows from a Passive Scalar

Abstract

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.

Mixing Rates for Hamiltonian Monte Carlo Algorithms in Finite and Infinite Dimensions

“…Having formulated our mixing result for the exact HMC algorithm associated with (1.1) we would like to be able to demonstrate that the conditions (1.6)-(1.7) which we impose on the potential U can be verified in concrete examples specifically as would apply to the Bayesian approach to PDE inverse problems. Here, as an illustrative example, we consider the problem of recovering a divergence free fluid flow q from the sparse and noisy observation of a passive solute θ(q) as was recently studied in [BGHK18,BGHK19].…”

“…In order to place (1.11) in a rigorous functional setting we adapt some results from [BGHK19,BGHK18]. In view of (8.3), (8.4) we consider a slightly more general version of (1.9) where we include an external forcing term f : [0, T ] × T 2 → R, namely,…”

“…Note however that the examples considered in [BGHK18] involved a covariance C with exponentially decaying spectrum so that (8.45) applies for any s ≥ 0 and (2.6) for any 0 ≤ γ < 1/2. Remark 8.6 (Improved bounds in the time independent case).…”

“…Finally, the second part of the paper is concerned with the application of the theoretical mixing result to the PDE inverse problem of determining a background flow from partial observations of a passive scalar that is advected by the flow. A careful analysis of this inverse problem within a Bayesian framework is carried out in [BGHK18], where the authors also provide numerical simulations showing the effectiveness of the infinite-dimensional HMC algorithm from [BPSSS11] in approximating the target distribution in this case. Here our task is to show that this example, for suitable observations of the passive scalar, satisfies all the conditions needed for our theoretical mixing result to hold, thus complementing the numerical experiments in [BGHK18] with rigorous mixing rates.…”

“…A careful analysis of this inverse problem within a Bayesian framework is carried out in [BGHK18], where the authors also provide numerical simulations showing the effectiveness of the infinite-dimensional HMC algorithm from [BPSSS11] in approximating the target distribution in this case. Here our task is to show that this example, for suitable observations of the passive scalar, satisfies all the conditions needed for our theoretical mixing result to hold, thus complementing the numerical experiments in [BGHK18] with rigorous mixing rates. In the sequel we provide a more detailed summary of the results obtained in the bulk of this manuscript.…”