An observer for an occluded reaction-diffusion system with spatially varying parameters.

Spatially dependent parameters of a two-component chaotic reaction-diffusion partial differential equation (PDE) model describing ocean ecology are observed by sampling a single species. We estimate the model parameters and the other species in the system by autosynchronization, where quantities of interest are evolved according to misfit between model and observations, to only partially observed data. Our motivating example comes from oceanic ecology as viewed by remote sensing data, but where noisy occluded data are realized in the form of cloud cover. We demonstrate a method to learn a large-scale coupled synchronizing system that represents the spatio-temporal dynamics and apply a network approach to analyze manifold stability.

Spatially dependent parameter estimation and nonlinear data assimilation by autosynchronization of a system of partial differential equations.

Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a partial differential equation (PDE) model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modeling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing are discussed.

A framework for estimating potential fluid flow from digital imagery.

Given image data of a fluid flow, the flow field, , governing the evolution of the system can be estimated using a variational approach to optical flow. Assuming that the flow field governing the advection is the symplectic gradient of a stream function or the gradient of a potential function-both falling under the category of a potential flow-it is natural to re-frame the optical flow problem to reconstruct the stream or potential function directly rather than the components of the flow individually. There are several advantages to this framework. Minimizing a functional based on the stream or potential function rather than based on the components of the flow will ensure that the computed flow is a potential flow. Next, this approach allows a more natural method for imposing scientific priors on the computed flow, via regularization of the optical flow functional. Also, this paradigm shift gives a framework--rather than an algorithm--and can be applied to nearly any existing variational optical flow technique. In this work, we develop the mathematical formulation of the potential optical flow framework and demonstrate the technique on synthetic flows that represent important dynamics for mass transport in fluid flows, as well as a flow generated by a satellite data-verified ocean model of temperature transport.