Sequential estimation of velocity fields using episodic proper orthogonal decomposition

In this paper, the problem of approximating velocity fields at future and past times based on information available at the current time is addressed. A novel method called “episodic POD” is described and developed that enables us to achieve our objective. Application of episodic POD to an ensemble of flow data results in a set of spatio-temporal eigenfunctions and a set of coefficients associated with the eigenfunctions. From these eigenfunctions, we develop two models called the “forward model” and “inverse model” that enable us to approximate the velocity fields at future and past times, based on information provided at the current time. A second set of models, the forward and inverse sequential models are also developed that enable the dynamic update of approximated velocity fields when new information is made available, making these models more adept at on-line estimation. The various properties associated with these models are described in detail, and four different examples are used to validate the models and show the different properties associated with episodic POD. It is shown through numerical validation, that the episodic POD model has a form that is dynamically consistent with the original system. It is also shown that episodic POD outperforms linear Kalman filters in the presence of noise.