Multivariate Splines: Theory, Computation and Applications

这个项目是关于多元样条函数的研究。为了提高它们在计算机辅助几何设计（CAGD）和偏微分方程（PDE）数值解中的应用所必需的计算效率，我们将更仔细地研究光滑度为r和d的二元和三元样条。PI将在二元设置中为这些应用识别最佳样条空间，比较固定平滑度的所有样条空间的维度，其底层三角剖分的三角形数量及其逼近能力。PI将在r=1和r=2时实现这些最佳样条空间，用于CAGD和PDE的数值解的典型应用，例如散射数据拟合，填充多边形孔，线性和非线性双调和方程的数值解。PI将研究在H1范数下使用二元样条构造紧支撑标准正交小波，从而可以在不反转线性系统的情况下求解标准椭圆方程的数值解。此外，PI还将研究三元样条空间，并确定最佳的应用于三维偏微分方程的数值解，特别是三维Navier-Stokes方程。一旦确定了r=1的最佳样条空间，PI将在三元设置中实现PDE的数值解。在对这些样条曲线进行计算时，为了提高计算性能，将采用多层次和区域分解方法。

9870178 Lai This project concerns investigation of multivariate spline functions. Bivariate and trivariate splines of smoothness r and degree d, will be studied more carefully to enhance their computational efficiency which is essential for applications in computer aided geometric design (CAGD) and numerical solutions of partial differential equations (PDE). The PI will identify the best spline spaces for these applications in the bivariate setting, comparing the dimension of all spline spaces of a fixed smoothness, the number of triangles of their underlying triangulations and their approximation power. The PI will implement these best spline spaces when r=1 and r=2 for typical applications in CAGD and numerical solution of PDE's, e.g., scattered data fitting, filling polygonal holes, numerical solutions of linear and nonlinear biharmonic equations. The PI will study the construction of compactly supported orthonormal wavelets under H1 norm using bivariate splines so that the numerical solution of the standard elliptic equations can be solved without inverting the linear systems. Furthermore, the PI will also study the trivariate spline spaces and identify the best one for the application in numerical solutions of 3D partial differential equations, in particular 3D Navier-Stokes equations. Once the best spline space for r=1 is identified, the PI will implement it for numerical solution of PDE's in the trivariate setting. When computing with these splines, multi-level and domain decomposition methods will be employed to improve the performance of the computation.