Mathematical Sciences: Miltivariate Interpolation and Approximation

结合不同领域的技术（代数， 几何学和组合学）的调查员和他的同事 最近确定了三元样条的通用维数 定义在四面体上的（足够高次的）空间 分解 在这个项目中，技术将得到进一步发展， 应用于多变量样条研究，特别是关于 以下问题：精确尺寸公式，改进的上 以及维数的下限，维数对 几何退化，局部解的存在性和构造 基，识别完整的有用子空间和超空间 样条空间，插值问题的可解性，影响 退化对数值稳定性和可解性的影响 插值问题、算法、曲面设计、逼近 定义在曲面上的函数，以及域的研究 除了单纯的分区。 样条是光滑的分段多项式函数。 他们是 普遍用于解决涉及一个函数的问题， 自变量 关于几个独立的 变量，样条（在本文中称为 “多元样条”）的理解要少得多， 结构要复杂得多。 他们有伟大的 潜在的应用，如解决部分 微分方程（例如，模拟流体流动、热 分布、燃烧或放射性衰变），插值 和数据的近似，以及形状的设计（例如，的 车辆）。 它们也是值得被 研究自己的权利。

Combining techniques from disparate areas (Algebra, Geometry,and Combinatorics) the investigator and his coworkers recently determined the generic dimension of trivariate spline spaces (of sufficiently high degree) defined on tetrahedral decompositions. In this project the techniques will be further developed and applied to multivariate spline research, particularly as regards the following problems: exact dimension formulas, improved upper and lower bounds on dimensions, dependence of the dimension on geometric degeneracies, existence and construction of local bases,identification of useful sub- and superspaces of the full spline space, solvability of interpolation problems, effects of degeneracies on numerical stability and solvability of interpolation problems, algorithms, surface design, approximation of functions defined on surfaces, and investigation of domain partitions other than simplicial ones. Splines are smooth piecewise polynomial functions. They are used ubiquitously to solve problems involving functions of one independent variable. Regarding functions of several independent variables, splines (which in this context are called "multivariate splines") are much less understood and their structure is vastly more complicated. Yet they have great potential for applications such as the solution of partial differential equations (e.g., modeling fluid flow, heat distribution, combustion, or radioactive decay), interpolation and approximation of data, and the design of shapes (e.g., of a vehicle). They are also fundamental objects that deserve to be studied in their own right.