Publications

Abstract not provided.

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, non-local diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is non-conforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

We consider the problem of selecting a small set (mosaic) of sensor images (footprints) whose union covers a two-dimensional Region Of Interest (ROI) on Earth. We take the approach of modeling the mosaic problem as a Mixed-Integer Linear Program (MILP). This allows solutions to this subproblem to feed into a larger remote-sensor collection-scheduling MILP. This enables the scheduler to dynamically consider alternative mosaics, without having to perform any new geometric computations. Our approach to set up the optimization problem uses maximal disk sampling and point-in-polygon geometric calculations. Footprints may be of any shape, even non-convex, and we show examples using a variety of shapes that may occur in practice. The general integer optimization problem can become computationally expensive for large problems. In practice, the number of placed footprints is within an order of magnitude of ten, making the time to solve to optimality on the order of minutes. This is fast enough to make the approach relevant for near real-time mission applications. We provide open source software for all our methods, "GeoPlace."

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Remote sensing systems have firmly established a role in providing immense value to commercial industry, scientific exploration, and national security. Continued maturation of sensing technology has reduced the cost of deploying highly-capable sensors while at the same time increased reliance on the information these sensors can provide. The demand for time on these sensors is unlikely to diminish. Coordination of next-generation sensor systems, larger constellations of satellites, unmanned aerial vehicles, ground telescopes, etc. is prohibitively complex for existing heuristics- based scheduling techniques. The project was a two-year collaboration spanning multiple Sandia centers and included a partnership with Texas A&M University. We have developed algorithms and software for collection scheduling, remote sensor field-of-view pointing models, and bandwidth- constrained prioritization of sensor data. Our approach followed best practices from the operations research and computational geometry communities. These models provide several advantages over state of the art techniques. In particular, our approach is more flexible compared to heuristics that tightly couple models and solution techniques. First, our mixed-integer linear models afford a rig- orous analysis so that sensor planners can quantitatively describe a schedule relative to the best possible. Optimal or near-optimal schedules can be produced with commercial solvers in opera- tional run-times. These models can be modified and extended to incorporate different scheduling and resource constraints and objective function definitions. Further, we have extended these mod- els to proactively schedule sensors under weather and ad hoc collection uncertainty. This approach stands in contrast to existing deterministic schedulers which assume a single future weather or ad hoc collection scenario. The field-of-view pointing algorithm produces a mosaic with the fewest number of images required to fully cover a region of interest. The bandwidth-constrained al- gorithms find the highest priority information that can be transmitted. All of these are based on mixed-integer linear programs so that, in the future, collection scheduling, field-of-view, and band- width prioritization can be combined into a single problem. Experiments conducted using the de- veloped models, commercial solvers, and benchmark datasets have demonstrated that proactively scheduling against uncertainty regularly and significantly outperforms deterministic schedulers. Acknowledgement We would like to acknowledge John T. Feddema, Brian N. Post, John H. Ganter, and Swaroop Darbha for providing critical project stewardship and fruitful remote sensing utilization discus- sions. A special thanks to Mohamed S. Ebeida for his contributions to the development of the Maximal Poisson Sampling technique. We would also like to thank Kaarthik Sundar and Jianglei Qin for their significant scheduling algorithm and model development contributions to the project. The authors would like to acknowledge the Sandia LDRD program for their support of this work. Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Cor- poration, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

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