Topics in Approximation Theory

本研究项目探讨了近似理论和样条函数应用中的一些问题。这些问题被分成四大类：多元逼近、细分算法和细化方程、保形插值和逼近以及用于求解球面几何偏微分方程的调和有限元。了解本研究中所考虑的样条空间的近似性质在各种应用中具有重要的基础意义，例如有限元法。虽然主要的兴趣将是在非均匀分区上定义的空间（如平面和球面三角形上的分段多项式空间），但很大一部分的研究也将致力于移位不变空间的重要特殊情况。所得的一些理论结果将应用于构造适用于求解球面偏微分方程的有限元。例如浅水方程、非发散正压涡度方程和物理大地测量学的测高-重力方程。这些方程的精确解对于模拟和理解大气运动、地幔和外核的演化以及地球磁场的动力学等物理现象至关重要。

This research project addresses a number of problems in applications of approximation theory and spline functions. The problems are grouped into four broad areas: multivariate approximation, subdivision algorithms and refinement equations, shape preserving interpolation and approximation, and harmonic finite elements for solving partial differential equations in spherical geometry. Understanding of approximation properties of the spline spaces to be considered in this research is of fundamental importance in various applications, such as the finite element method. While the primary interest will be in spaces defined on non-uniform partitions (such as spaces of piecewise polynomials on planar and spherical triangulations), a large portion of the research will also be devoted to the important special case of shift-invariant spaces. Some of the obtained theoretical results will be applied in the construction of finite elements suitable for solving partial differential equations whose domain is the sphere. Examples are the shallow water equations, nondivergent barotropic vorticity equation, and altimetry-gravimetry equations of physical geodesy. Accurate solutions of those equations are of vital importance for modeling and understanding physical phenomena such as the motion of the atmosphere, evolution of the Earth's mantle and outer core, and the dynamics of the Earth's magnetic field.