Mathematical Sciences: Multivariate Splines: Theory and Application

9303121 Lai为了提高二元样条在计算机辅助几何设计、数值图像处理、偏微分方程数值解等方面的应用效率，研究者研究了光滑度和d度的二元样条，且d值相对于r值较小。多元样条是具有一定光滑性的分段多项式函数。这些碎片不需要像它们都是同一个多项式的一部分那样平滑地组合在一起。粗略地说，分段多项式相对于其平滑度的程度越高，这些片段就越能粗略地组合在一起。由于多项式函数是最容易操作的函数，并且由于现实世界中几乎所有问题本质上都是多变量或多参数的，因此多元样条是对目标函数建模的好工具。虽然二元样条大次与光滑性的理论是可行的，但由于二元样条的次过大，难以在实际中得到有效的应用。也就是说，只有小平滑度的二元样条才能达到计算机辅助几何设计、数值图像处理和偏微分方程数值解所要求的效率和精度。目前，二元样条曲线在少数情况下是可以构造的。该项目承担了这类样条的一般理论的各个方面，着眼于应用。＊＊＊

9303121 Lai The investigator studies bivariate splines of smoothness rand degree d with a small d vs. r to enhance the efficiency of using bivariate splines to solve application problems, and uses new or recent results on multivariate splines to solve some application problems in computer aided geometric design, numerical image processing, and numerical solution of partial differential equations. Multivariate splines are piecewise polynomial functions with certain smoothness. The pieces need not fit together as smoothly as if they all were part of the same polynomial. Roughly speaking, the higher the degree of the piecewise polynomial relative to its smoothness, the more roughly the pieces can fit together. Because polynomial functions are the easiest functions to manipulate and because almost all problems in the real world are multivariable or multiparametric in nature, multivariate splines are a good tool to model the target functions. Although the theory of bivariate splines with large degree versus smoothness is available, bivariate splines have too large a degree to be used in practice efficiently. That is, only bivariate splines with small degree versus smoothness can achieve the efficiency and the accuracy required by computer aided geometric design, numerical image processing, and numerical solution of partial differential equations. At present, those bivariate splines can be constructed for a few cases. The project undertakes aspects of a general theory of such splines, with an eye to the applications. ***