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

本研究试图对微分几何、流体力学、材料科学、宇宙学等领域的平均曲率流、逆平均曲率流、高斯曲率流、表面扩散流、Willmore流、Ricci流等几何流进行广泛而全面的数值研究，并解决几何流计算中出现的理论和数值难题。该项目的目标是开发准确，稳健，高效的自适应有限元离散化方法，并行迭代求解算法和计算机代码，用于计算基于水平集和相场公式的几何流。研究者的目标是通过强调定性分析和定量计算的结合，对几何流动进行平衡的数值研究，从而为发现和分析几何流动的精细特性提供可靠的计算工具，例如奇点的动力学，这些特性通常很难甚至可能无法用分析方法来预测和表征。本研究提出的方法和算法具有精度高、稳定性强、成本低、效率高等特点。此外，所提出的方法不仅能够准确、有效地逼近奇点发生前和奇点发生后的几何流动。由于流体力学、宇宙学和材料科学的关键应用与几何流动的解决方案直接相关，因此预期所提出的研究的成功完成不仅会提出解决潜在数学问题的新方法，而且还会为理解这些应用提供见解，从而有可能对这些应用科学产生重大影响。此外，待开发的方法将在细胞生物学、地球物理学、图像处理和计算机视觉等其他领域得到应用。该项目的教育部分包括研究生课程开发、培训和指导研究生和本科生。

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.