Mathematical Sciences: Smooth Multivariate Piecewise Polynomials; Subdivision Algorithms

该奖项将支持对逼近理论和细分算法问题的数学研究。关于逼近的工作将处理二元和三元光滑分段多项式函数空间的逼近能力和局部结构。直接的目标是精确地限制近似空间的底层分区或网格的平滑要求和选择对近似能力的限制。潜在的结构相对简单，尽管问题很困难，而且不是由一维理论驱动的。区域被细分，并且函数被定义为等于细分的每一块上的某个多项式(多项式可以随每一块而变化)。施加在函数上的每一个整体光滑度都改变了它的逼近能力。多项式分段之间的相互依赖关系使得很难提出有效的逼近过程。例如，在二维中，使用高度对称的分割(三角形)导致了这样一个定理，即紧支撑的函数已经具有最佳可能的跨度。要确定这是否是一个孤立的现象，还需要做一些工作。细分算法为生成平滑曲线和曲面的标准方法提供了一种有趣的替代方法。这样的算法得到一条曲线或曲面，作为一系列连续细化的分段线性元素的极限，每个分段线性元素都是通过将其前身的片断“削”下来获得的。它们是计算机辅助几何设计领域的基本工具。例如，可以通过控制初始折线的选择来控制极限曲线的形状。因此，人们可以通过这些算法来设计特定的形状并达到指定的设计特征。不完全清楚的是，细分算法是否总是收敛。不出所料，与曲面有关的问题比与曲线有关的问题更为困难。然而，即使在后一种情况下，例子也表明，(在显示器上)所有外观都是平滑的极限曲线实际上是分形图。

This award will support mathematical research on problems of approximation theory and subdivision algorithms. Work on approximations will deal with the approximation power and local structure of spaces of smooth piecewise polynomial functions of two and three variables. The immediate goal is to make precise the limitations put on approximation power by smoothness demands and choice of the underlying partition or mesh of the approximating space. The underlying structure is relatively simple although the problems are difficult and not motivated by one- dimensional theory. A domain is subdivided and a function is defined to be equal to some polynomial on each piece of the subdivision (the polynomials can change with each piece). Each degree of overall smoothness forced on the function changes its approximating power. The interdependency of the polynomial pieces makes it difficult to come up with effective approximation procedures. In two-dimensions for example, use of highly symmetrical partitions (triangles) results in the theorem that the compactly supported functions already have best-possible span. Work is to be done in establishing whether or not this is an isolated phenomenon. Subdivision algorithms provide an intriguing alternative to standard ways of generating smooth curves and surfaces. Such algorithms obtain a curve or surface as the limit of a sequence of successively refined piecewise linear elements, each obtained by "whittling" pieces off its predecessor. They are fundamental tools in the area of computer aided geometric design. One can control the shape of the limiting curve by controlling the choice of initial broken lines, for example. Thus one may be able to design specific shapes and arrive at specified design characteristics through these algorithms. What is not completely clear is whether or not the subdivision algorithms always converge. As might be expected, the problems concerning surfaces are more difficult than those for curves. Yet even in the latter case, examples show that limiting curves which for all appearances are smooth (on a monitor) are actually fractals.