Geometrics Algorithms in Statistics, Meshing, and Parametric Optimization

本研究的目标是通过在相关几何空间中可视化问题数据来解决应用领域的算法问题，利用这种可视化来开发对这些问题的改进的数学理解，并将这种理解与计算几何技术相结合，以开发有效的算法来解决这些问题。本项目针对的具体问题领域包括：(1)有限元法非结构化四边形和长方体网格的生成；(2)将超平面或其他几何形状拟合到数据点集的鲁棒方法；(3)自适应用户界面问题，例如调整车辆路线系统中距离、旅行时间或风景值等参数的相对权重，使其产生更符合驾驶员偏好的路线。所有这些领域都涉及到通过曲线或曲面的排列来划分空间的基本几何问题。在提出的研究中，进一步的连接主题是使用拓扑方法将全局形状属性从局部几何属性中分离出来，并为讨论这些属性提供精确的语言。研究者将有意识地关注计算拓扑来加速几何计算的进展。

The goal of this research is to solve algorithmic problems from applicationareas by visualizing the problem data in terms of a related geometricspace, using this visualization to develop an improved mathematicalunderstanding of these problems, and combining this understanding with thetechniques of computational geometry to develop efficient algorithms forthe problems. Particular problem domains targeted by this project include(1) generation of unstructured quadrilateral and cuboid meshes for thefinite element method; (2) robust methods for fitting hyperplanes or othergeometric shapes to sets of data points; and (3) adaptive user interfaceproblems, for example tuning the relative weights of parameters such asdistance, travel time, or scenic value in a vehicle routing system, to makeit produce routes that more closely match driver preferences. All of theseareas relate to fundamental geometric problems of partitioning spaces byarrangements of curves or surfaces. A further connecting theme in theproposed research is the use of topological methods to separate globalshape properties from local geometric attributes, and to provide a preciselanguage for discussing these properties. The investigator will apply aconscious focus on computational topology to accelerate progress ingeometric computing.