散乱数据带自然边界条件二元样条光顺及数值微分

考虑一种新的散乱数据带自然边界二元样条光顺问题.根据样条变分理论和Hilbert空间样条函数方法,构造出了显式的二元带自然边界光顺样条解,其表达式简单且系数可以由系数矩阵对称正定的线性方程组确定.证明了解的存在和唯一性,讨论了收敛性和误差估计.并由此得到一种新的基于散乱数据上的正则化二元数值微分的方法.最后,给出了一些数值例子对方法进行了验证.

BIVARIATE SPLINE SMOOTHING WITH NATURAL BOUNDARY CONDITIONS AND NUMERICAL DIFFERENTIATION FOR SCATTERED DATA

Consider a new bivariate spline smoothing problem with natural boundary conditions. By the variational theory of spline and the spline function methods of Hilbert space, the explicit bivariate smoothing spline solution with natural boundary conditions is constructed. Its expression is so simple and the coefficients can be decided by a linear system with symmetry and positive definite coefficient matrix. The existence and uniqueness of the regularized solution are proved. The convergence and error estimates are also provided. On the basis of this, a regularization method for bivariate numerical differentiation of scattered data is found. Some numerical examples are presented to demonstrate our methods.