Computational Challenges in Fluid Transport and Imaging

拟议的研究强调，计算的巨大进步必须由对数值方法背后理论的基本理解驱动。 该提案的主要创新之一是制定中央计划。虽然中心格式被认为是求解时间相关偏微分方程的最强大的数值方法之一，特别是求解输运方程，但如果要充分发挥其潜力，仍然有几个基本问题需要解决。该建议还集中在调查的各向异性的能力水平集方法，隐式方法的实施，和多尺度技术的数据同化的力量。大多数真实的世界的问题是由计算机解决。许多这样的问题是如此复杂，现有的计算方法是不足以提供所需的精度。 这种情况不能通过简单地制造更快的计算机来解决。 事实上，对更高分辨率和更精确模型的追求远远超过了计算能力的提高。 此外，正如过去经常证明的那样，计算机算法和软件的创新比计算能力的进步带来更大的红利。 该项目的研究旨在在计算机程序（所谓的数值算法）背后的数学方面取得创新性进展，这将大大加快计算速度，从而解决该国面临的许多科学和工程问题，例如跟踪沿海地区的污染物或开发快速准确的医学成像传感器。

The proposed research emphasizes that dramatic advances in computation must be driven by a fundamental understanding of the theory behind numerical methods. One of the main innovations in this proposal is the development of central schemes. While central schemes are considered to be among the most powerful numerical methods for time dependent PDEs, in particular for solving transport equations, there are still several fundamental issues to be resolved if they are to reach their full potential. This proposal also concentrates on the investigation of the power of anisotropy of level set methods, the implementation of implicit methods, and the power of multi-scale techniques in data assimilation.Most real world problems are solved by computers. Many such problems are so complicated that the existing computational methods are insufficient to provide the accuracy needed. This situation cannot be solved by simply building faster computers. Indeed, the quest for finer resolution and more accurate models far outstrips gains in computational power. Moreover, as has been demonstrated often in the past, innovations in computer algorithms and software pay larger dividends than advances in computational power. The research in this project seeks to make innovative advances in the mathematics behind computer programs (so-called numerical algorithms) which will have the effect of significantly speeding up computation and thereby solving many scientific and engineering problems facing this country such as tracking pollutants in coastal areas or developing fast and accurate sensors for medical imaging.