Efficient Algorithms for Interface Motion and Wave Propagation

传播界面和波传播中的相关问题出现在大量的科学、工程和工业问题中。这些问题包括破碎水波、地震波在地球上的传播、半导体制造、喷墨绘图仪的设计以及肿瘤生长和细胞膜边界的生物学方面。在之前由NSF资助的工作中，我们建立了日常使用的健壮算法来解决这些问题；其中两个包括水平集方法和快速行进方法。它们在整个半导体行业都被用来分析如何使芯片变得更小和更高效，在喷墨过程中用来模拟规模越来越小的工业打印，并在石油行业用来帮助确定石油储量。在这项工作中，我们建议将这些计算技术扩展到更困难的问题，再次专注于将它们直接应用于具有挑战性的工业和科学问题。在这里，我们的重点将超越上述问题的“首次到达”解决方案，并将致力于设计接口技术，以产生复杂得多的多到达解决方案，以解决复杂的地震和波成像问题、飞机回避和控制问题中出现的依赖导数的控制理论问题，以及机器人和半导体设备中高度精细的多尺度问题。

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.