Potential flows: a playground for non-local and nonlinear inference problems*

An accurate estimation of the flow field from limited and noisy observations is crucial in many fields of engineering. To tackle these inverse problems, classical localization schemes suppress correlations at long distances. However, these techniques are not well suited for incompressible fluid problems, in which the observations are typically obtained by non-local and nonlinear mappings of the state, e.g. the pressure Poisson equation. Instead these inference problems have low-rank structure in the sense that low-dimensional projections of the observations are most informative of a low-dimensional subspace of the state space. Thus, interactions between state and observation variables are best described as clusters of variables, reminiscent of the fast multipole method (FMM). To drive further research in this area, we show that potential flow problems constitute a nice playground to experiment with  non-local and nonlinear inference problems, while allievating the high-dimensionality of a standard incompressible fluid problem. Indeed, estimating the position and strength of a handful of singularities (e.g. vortices, sources) from pressure observations distills the main features of a realistic inverse problem. To exploit the low-rank structure, we present a systematic procedure to identify these clusters of variables from the sensitivity analysis of the observation operator. We also explore connections between the proposed sensitivity analysis and the FMM. Finally, we present recent advances in nonlinear prior-to-posterior transformations to perform consistent inference with nonlinear state-space models.