CDS&E: Collaborative Research: Least-Squares Finite Element Methods for Data Assimilation in Large-Scale Simulations

摘要#1249858/托马斯·曼特菲尔#1249950/杰弗里·海斯计算机模拟在科学发现中的作用在许多领域继续增长，包括生物、金融、化学和医学。在许多情况下，这些计算机模拟是基于偏微分方程组的解，偏微分方程组是只能在大型计算机上近似求解的数学方程。通过使用更先进的实验技术，科学发现也在继续，这些技术使我们能够获得比以往任何时候都多的数据，并获得以前没有的新数据。科学家现在迫切需要的是一种方法，将计算机模拟与现有的丰富数据结合起来。为了帮助解决这一需求，我们建议开发先进的最小二乘有限元方法，这种方法在解决计算机模拟与实验数据相结合的问题方面具有许多优势。首先，这种方法足够灵活，可以吸收模拟中任何位置的数据。如果要模拟通过血管的血液流动，实验数据可以位于任何位置，包括靠近血管壁或靠近血管中心。其次，该方法可以解释实验数据的准确性。如果血管中心附近的血流数据比管壁附近的血流数据更准确，则模拟将更接近中心附近的准确数据，而不会与管壁附近的数据匹配得那么紧密，因为数据可能包含重大误差。提出的方法的最后一个优点是它在计算上是有效的。它从一开始就被设计成与可伸缩的多级数学技术一起很好地工作，并在现代的多处理器计算机体系结构上很好地工作。这不是一种会被巨大而复杂的问题淹没的方法，但它在今天的计算机和明天的计算机上将是有效的。如果机械师想要评估汽车发动机的状况，他们会打开引擎盖，检查发动机的关键部件。心脏健康状况的评估是一个更具挑战性的问题，因为我们不能轻易且安全地打开引擎盖。另一种方法是心脏病专家将FDA批准的微泡注入血液中，这些微泡比红细胞小，然后可以使用外部超声仪安全地显示这些微泡。这些微泡的运动给出了心脏中血液流动的指示，但需要更多的信息来正确评估心脏的健康状况。具体地说，心脏病学家对心脏的压力变化和心脏的整体效率感兴趣。为了获得这一额外信息，我们可以在计算机上模拟心脏中的血液流动，理想情况下，将运动微泡的数据与计算机模拟相结合，这样我们就可以获得每个人自己心脏的特定信息。在从医学到微生物学再到气象学的各个领域，需要我们将计算机模拟与实验数据相结合的问题正变得越来越普遍。这项提案中描述的工作将为将实验数据与计算机模拟相结合提供一个强大的新工具。这种方法将允许科学家考虑数据的准确性，以便较准确的数据将与模拟紧密匹配，而较不准确的数据也将与模拟不匹配。这种方法在下一代计算机上将是有效的，因为它支持先进的数学技术。总体而言，这种方法的积极影响应该扩展到许多不同的科学领域。该项目将涉及蒙大拿州立大学和科罗拉多大学博尔德分校的研究生和本科生，这些学生将在大学之间进行广泛的互动。该项目还将包括开发新的工程和数学课程内容，并支持科学会议。

Abstract#1249858 / Manteuffel, Thomas#1249950 / Jeffrey Heys The role of computer simulations in scientific discovery continues to grow in many fields, including biology, finance, chemistry, and medicine. In many cases, these computer simulations are based on the solution of partial differential equations, which are mathematical equations that can only be approximately solved on large computers. Scientific discovery is also continuing through the use of more advanced experimental techniques that are enabling us to obtain more data than ever before and obtain new data that was not available previously. A critical need for scientists now is an approach for combining the computer simulations with the abundant data that is now available. To help address this need, we are proposing the development of advanced least-square finite element methods, which have a number of advantages for solving this problem of combining computer simulations with experimental data. First, the approach is flexible enough that it can assimilate data from any location in the simulation. If you are simulating blood flow through a vessel, the experimental data can be located anywhere, including near the wall or near the center of the vessel. Second, the approach can account for the accuracy of the experimental data. If the blood flow data is more accurate near the center of the vessel than near the wall, the simulation will more closely match the accurate data near the center, and it will not match the data near the wall as closely because the data likely contains significant error. A final advantage for the proposed approach is that it is computationally efficient. It has been designed from the beginning to work well with scalable, multilevel mathematical techniques and work well on modern, multiprocessor computer architectures. This is not an approach that will be overwhelmed by large, complex problems, but it will be efficient on today's computers and tomorrow's computers. If a mechanic wishes to assess the condition of a cars engine, they will open up the hood and inspect the critical parts of the engine. The assessment of the health of the heart is a much more challenging problem because we cannot easily and safely open up the hood. An alternative approach is for a cardiologist to inject FDA approved microbubbles, these are bubbles that are smaller than red blood cells, into the blood, and then these microbubbles can be safely visualized using an external ultrasound machine. The movement of these microbubbles gives an indication of the blood flow in the heart, but more information is needed to properly assess the health of the heart. Specifically, cardiologists are interested in pressure changes in the heart and the overall efficiency of the heart. To obtain this additional information, we can simulate the flow of blood in the heart on a computer, and, ideally, combine the data from the moving microbubbles with the computer simulation so that we can obtain information specific to each individuals own heart. Problems that require us to combine a computer simulation with experimental data are becoming increasing common in fields from medicine to microbiology to meteorology. The work described in this proposal will provide a powerful new tool for combining experimental data with computer simulations. The approach will allow scientist to account for the accuracy of the data so that the more accurate data will be matched closely by the simulation and less accurate data will not match the simulation as well. The approach will be efficient on the next generation of computers because it supports advanced mathematic techniques. Overall, the positive impact of this approach should extent to many different scientific fields. The project will involve graduate and undergraduate students at both Montana State University and the University of Colorado-Boulder, and these students will interact extensively between universities. The project will also include the development of new engineering and mathematics course content and support scientific conferences.