In the following paper, we discuss how to design an ensemble of experiments through the use of compressed sensing. Specifically, we show how to conduct a small number of physical experiments and then use compressed sensing to reconstruct a larger set of data. In order to accomplish this, we organize our results into four sections. We begin by extending the theory of compressed sensing to a finite product of Hilbert spaces. Then, we show how these results apply to experiment design. Next, we develop an efficient reconstruction algorithm that allows us to reconstruct experimental data projected onto a finite element basis. Finally, we verify our approach with two computational experiments.
Over the last several years, there has been considerable growth in camera based observation systems for a variety of safety, scientific, and recreational applications. In order to improve the effectiveness of these systems, we frequently desire the ability to increase the number of observed objects, but solving this problem is not as simple as adding more cameras. Quite often, there are economic or physical restrictions that prevent us from adding additional cameras to the system. As a result, we require methods that coordinate the tracking of objects between multiple cameras in an optimal way. In order to accomplish this goal, we present a new cooperative control algorithm for a camera based observational system. Specifically, we present a receding horizon control where we model the underlying optimal control problem as a mixed integer linear program. The benefit of this design is that we can coordinate the actions between each camera while simultaneously respecting its kinematics. In addition, we further improve the quality of our solution by coupling our algorithm with a Kalman filter. Through this integration, we not only add a predictive component to our control, but we use the uncertainty estimates provided by the filter to encourage the system to periodically observe any outliers in the observed area. This combined approach allows us to intelligently observe the entire region of interest in an effective and thorough manner.