中文摘要：

考虑到传统的几何迭代法仅有一阶的收敛性,提出一个二阶可导的能量函数来刻画当前曲线与目标点集之间的差异.首先根据初始的控制顶点和相应的基函数生成初始的样条曲线,然后求差异函数关于各个控制顶点的梯度,最后采用L-BFGS算法快速寻找最优的插值或者逼近曲线.实验结果表明,文中算法具有超线性的收敛速度,在同样的精度要求下比原来的几何迭代法快出数十倍甚至上百倍;既可用于插值问题,也可用于逼近问题;甚至也能适用于数据点参数可变的情形.

英文摘要：

Considering that the conventional geometric iteration method (GIM) has only 1-order convergence,this paper proposes an energy function with second-order smoothness that characterizes the difference betweenthe up-to-date curve and the given data points. In numerical implementation, we initialize the spline curve accordingto the initial control points and the associated basis functions, then compute the difference function aswell as the gradients with regard to the moveable control points, and finally use the L-BFGS technique to find theoptimal interpolation/approximation curve. Experimental results show that our accelerated algorithm has a super-linear rate of convergence. With the same accuracy requirement, the improved GIM outperforms the originalversion by tens to hundreds of times in terms of efficiency. It can be used to both the interpolation problem andthe approximation problem, and even to the case where the input data points have variable parameters.