Computational Challenges in Geometrical Flows: Numerical Methods and Analysis, Algorithmic Development and Software Engineering

几何流中的计算挑战：数值方法和分析、算法开发和软件工程

• 批准号: 0410266
• 负责人: Xiaobing Feng
• 金额: --
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410266&HistoricalAwards=false
• 关键词: Computational; Challenges; Geometrical; Flows; Numerical

This research is an attempt to conduct an extensive and comprehensive numerical study of geometrical flows, such as the mean curvature flow, the inverse mean curvature flow, the Gauss curvature flow, the surface diffusion flow, the Willmore flow, and the Ricci flow, from differential geometry, fluid mechanics, materials science, and cosmology, and to address theoretically and numerically challenging issues arising from geometrical flow computations. The goal of this project is to develop accurate, robust, and efficient adaptive finite element discretization methods, parallel iterative solution algorithms and computer codes for computing geometrical flows based on both level set and phase field formulations. The investigator aims to carry out a balanced numerical study for the geometrical flows by emphasizing both qualitative analysis and quantitative computation, and thus to provide reliable computational tools for discovering and analyzing fine properties such as dynamics of the singularities of the geometrical flows, which often are difficult and even may not be possible to predicate and characterize by analytical means. The methods and algorithms resulting from this research will have the following attractive features: high accuracy, strong stability, low cost, and high efficiency. In addition, the proposed methods are also capable of accurately and efficiently approximating the geometrical flows not only before but also beyond the onset of singularities.As critical applications from fluid mechanics, cosmology, and materials science are directly tied to the solutions of geometrical flows, it is expected that successful completion of the proposed research has the potential to significantly impact these applied sciences not only by presenting new methods for solving underlying mathematical problems but also providing insights for the understanding of each of these applications. Furthermore, the methods to be developed will find applications in other fields such as cell biology, geophysics, image processing, and computer vision. The educational component of the project consists of graduate graduate course development, training and mentoring both graduate and undergraduate students through the project.

这项研究是一种尝试对几何流动的广泛而全面的数值研究，例如平均曲率流，平均曲率流，高斯曲率流，表面扩散流，willmore流量以及RICCI流，以及来自差异几何，流体机械，材料机械，材料科学和整体学和数字的问题，以及质疑的问题。该项目的目的是开发准确，健壮和有效的自适应有限元离散方法，基于级别集合和相位场公式的计算几何流量的平行迭代解决方案算法和计算机代码。该研究者的目的是通过强调定性分析和定量计算，进行几何流量的平衡数值研究，从而提供可靠的计算工具来发现和分析良好的良好特性，例如几何流量的奇异性动态，甚至可能难以通过分析和分析来培养和表征分析。这项研究产生的方法和算法将具有以下吸引人的特征：高精度，强稳定性，低成本和高效率。此外，所提出的方法还能够准确，有效地近似几何流动不仅在之前而且超出了奇异之处。作为流动力学，宇宙学和材料科学的关键应用，直接与几何流动的解决方案直接相关，而不仅可以通过拟议的研究来实现这些方法，从而可以实现良好的方法来实现数学的成功，从而实现了数学的成功，而求解了求解的方法。而且还为理解这些应用程序中的每一个提供了见解。此外，要开发的方法将在其他领域找到应用，例如细胞生物学，地球物理，图像处理和计算机视觉。该项目的教育部分包括研究生课程的发展，培训和指导研究生和本科生。