散乱数据的多维样条积分
Multidimensional spline integration of scattered data
摘要：
我们介绍了一种数值方法，用于重建一个多维表面上的梯度测量在某些值的坐标。该方法包括定义一个多维样条函数和最小化其导数和测量梯度之间的偏差。与沿着某条路径的多维积分不同，本方法产生连续、光滑的表面，此外，它也适用于非等距且不在矩形网格上对齐的输入数据。函数值、一阶和二阶导数以及积分都很容易计算。该方法还对统计误差和系统误差进行了适当的估计。
英文摘要：
We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.
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