The inference of an incompressible flow from limited pressure data

In a variety of situations in fluid dynamics, we have data from a limited number of sensors and we wish to use these data to deduce more about the flow field. This problem can be posed as an example of elliptic inference: we seek to infer the source field (e.g., vorticity, an obstacle) from an observation field (e.g., pressure, velocity) that depends elliptically on the source. In this talk, we discuss the solution of this problem in a Bayesian ensemble-based setting: we draw a finite number of samples from a prior distribution (our initial guess) and then perform a Kalman update across the ensemble to develop our posterior, with a corresponding improved estimate of the state. We will discuss a new low-rank form of this update, in which the retained modes serve are the most informative directions in the map from the sensor data to the state update. We will demonstrate the low-rank inference in a variety of flow problems.