Mathematical Sciences: Stable Qualitatively Correct Interpolants for Highly Accurate Discrete Data

普鲁斯 研究人员开发和分析稳定的方法，用于生成高精度的全局近似值，以基本上由高精度插值数据定义的函数。 几种用于估计常微分方程解的数值算法可以在有限点集上产生高精度的近似。 通常需要根据这些数据构造一个封闭形式的近似;然而，明显的分段多项式插值经常不起作用，特别是对于高度不均匀的网格。 基于局部配点的插值似乎是稳定的，但需要进一步的分析和算法。 一个相关的问题是，找到插值精确的离散数据Sturm-Liouville特征函数：它是可取的，以保持准确性与插值具有正确的振荡特性。 要研究的问题是以前的工作，导致了广泛使用的软件包的自然产物。 在理论和计算机实现的进展寻求。 应用程序是表示曲线出现在计算机图形，并产生图形表示的解决方案中出现的问题，在量子化学和物理。

Pruess The investigator develops and analyzes stable methods for generating approximants of high global accuracy to functions defined essentially by highly accurate interpolatory data. Several numerical algorithms for estimating the solution of an ordinary differential equation can produce highly accurate approximations at a finite set of points. It is often desirable to construct a closed form approximation based on this data; however, the obvious piecewise polynomial interpolants frequently do not work, especially for highly nonuniform meshes. An interpolant based on local collocation seems to be stable but further analysis and algorithms are needed. A related problem is that of finding interpolants given accurate discrete data for Sturm-Liouville eigenfunctions: it is desirable to retain the accuracy with an interpolant having the correct oscillation properties. The problems to be studied are a natural outgrowth of prior work that resulted in a widely used software package. Advances in both theory and computer implementation are sought. Applications are to representation of curves arising in computer graphics, and to producing graphical representations for solutions to problems arising in quantum chemistry and physics.