Geometrics Algorithms in Statistics, Meshing, and Parametric Optimization

这项研究的目标是通过将问题数据可视化到相关的几何空间来解决应用领域的算法问题，使用这种可视化来发展对这些问题的更好的数学理解，并将这种理解与计算几何技术相结合来开发问题的有效算法。该项目针对的特定问题领域包括：(1)用于有限元方法的非结构化四边形和长方体网格的生成；(2)用于将超平面或其他几何形状拟合到数据点集合的稳健方法；以及(3)自适应用户界面问题，例如调整车辆路径系统中诸如距离、行程时间或景观值等参数的相对权重，以使其生成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。在所提出的研究中，另一个相关主题是使用拓扑方法将全局形状属性与局部几何属性分开，并为讨论这些属性提供精确的语言。研究人员将不知不觉地将重点放在计算拓扑上，以加快几何计算的进展。

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.