Mathematical Sciences: Theory and Applications of Spline Functions

本研究的目的是为多元样条法的理论和应用做出贡献。要解决的理论问题包括多元样条基空间的维度，这些空间的局部基的构造，以及它们的逼近能力的研究。除了定义在三角剖分上的传统空间外，还将研究定义在球体或其他3D对象的三角剖分上的新空间。这些应用包括使用模拟退火法构造最优三角剖分，保形插值法和数据拟合方法的设计，径向基函数用于惩罚最小二乘拟合，以及多变量函数拟合的自适应方法。

The aim of this research is to contribute both the theory and applications of multivariate splines. The theoretical questions to be addressed involve dimensions of multivariate spline spaces, construction of local bases for these spaces, and study of their approximation power. In addition to the traditional spaces defined on triangulations, new spaces defined on triangulations of the sphere or other 3D objects will also be investigated. The applications involve the use of simulated annealing to construct optimal triangulations, the design of shape preserving interpolation and data fitting methods, the use of radial basis functions for penalized least squares fitting, and adaptive methods for fitting functions of many variables.