带约束和参数的多变量逼近的理论与方法研究

综合运用数值逼近、计算几何、优化、代数和微分方程的理论和方法，研究带形状约束的多变量逼近；研究带参数的多变量样条；研究基于偏微分方程模型的多变量逼近和插值；研究带约束的双变量Hermite插值；研究带约束的三角样条的构造和Hermite三角多项式插值；研究散乱点Hermite插值；研究散乱数据逼近的光滑性和稳定计算问题；研究带形状约束的曲面和曲面上的曲线表示；研究特征保持细分和非线性Hermite细分方法；研究网格生成的边界约束方法；研究流形样条和流形曲面。较系统地提出带约束和参数的多变量逼近的理论与方法，有效地提出具有特征边界曲线的曲面逼近和插值方法。本项目的研究对完善和丰富多变量逼近和计算机辅助几何设计中相关数学问题的理论与方法、促进数值逼近和几何计算的理论和应用的发展，都具有重要的理论和应用意义。

By comprehensive utilization on the theory and methods of numerical approximation, computational geometry, optimization, algebra and differential equations, we studied the problems on multivariate approximation with shape constraints, multivariate splines with parameters, multivariate approximation and interpolation based on partial differential equation models, bivariate Hermite interpolation with constraints, the construction of trigonometric splines with constraints, trigonometric Hermite interpolation, scattered points Hermite interpolation, the smoothness and stable computing of scattered data approximation, surfaces with shape constraints, curves representation on the surfaces, the representation and computing of interpolation and approximation with geometric constraints, characteristic preserving subdivision and nonlinear Hermite subdivision, boundary constraint methods on gird generation, manifold splines and surfaces. Based on the study of this project, the theory and methods for multivariate approximation with constraints and parameters will be presented overall. The approximation and interpolation methods for surfaces with characteristic boundaries will be presented perfectly. The study on this project is of theory and applied significant on enriching the theory and methods of the multivariate approximation and computer aided geometric design, and promoting the development of theory and application of numerical approximation and geometric computing.