Two classes of state estimation schemes, variational (4DVar) and ensemble Kalman (EnKF), have been developed and used extensively by the weather forecasting community as tractable alternatives to the standard matrix-based Kalman update equations for the estimation of high- dimensional systems. Variational schemes iteratively minimize a cost function with respect to the state estimate, using efficient vector-based gradient descent methods, but fail to capture the moments of the PDF of this estimate. Ensemble Kalman methods represent well the principal moments of the PDF, accounting for the measurements with a sequence of Kalman-like updates with the covariance of the PDF approximated via the ensemble. Here, we first introduce a tractable method for updating an ensemble of estimates in a variational fashion, capturing correctly both the estimate and the leading moments of its PDF. We then extend this variational ensemble framework to facilitate its consistent hybridization with the ensemble Kalman smoother. Finally, it is shown that the resulting Hybrid (variational/Kalman) Ensemble Smoother (HEnS) significantly outperforms the existing 4DVar and EnKF approaches used operationally today for high-dimensional state estimation. The second part of this dissertation examines the best possible interconnect topologies for switchless multiprocessor computer systems. We focus first on hexagonal interconnect graphs and their extension to problems on the sphere. Eight families of efficient tiled layouts have been discovered that make such interconnects trivial to scale to large cluster sizes while incorporating no long wires. In the resulting switchless interconnect designs, the physical proximity of the cells created and the logical proximity of the nodes to which these cells are assigned coincide perfectly, so all communication between physically adjacent cells during the PDE simulation require communication over just a single hop in the computational cluster. Lastly, we attempt to generalize two classes of directed graphs into a unified theory in which the well-known cartesian and butterfly graphs are special cases of a more general class of interconnect that better spans the design parameter space
