Efficient Algorithms for Interface Motion and Wave Propagation

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Propagating interfaces and related problems in wave propagation appear in a vast collection of scientific, engineering, and industrial problems. These include problems in breaking water waves, seismic wave propagation through the Earth, semiconductor fabrication, the design of ink jet plotters, and biological aspects of tumor growth and cell membrane boundaries.Under previous NSF-funded work, we have built robust algorithms in everyday use to tackle these problems; two of these include Level Set Methods and Fast Marching Methods. They are used throughout the semiconductor industry to analyze how to make chips smaller and more efficient, in the ink jet process to simulate industrial printing on ever smaller scales, and in the petroleum industry to help locate oil reserves. We propose in this work to extend these computational techniques to significantly more difficult problems, again with a focus on applying them directly to challenging industrial and scientific problems. Here, our focus will go beyond ''first arrival'' solutions which characterize the above problems, and will aim to design interface techniques that will produce the far more complex multiple arrivals solutions to intricate seismic and wave imaging problems, derivative-dependent control theory problems that appear in aircraft avoidance and control problems, and highly refined multiple scale problems in robotics and etching and deposition in semiconductor devices.