Geometrics Algorithms in Statistics, Meshing, and Parametric Optimization

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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.