多维向量场特征参数的几何代数统一计算方法

以几何代数多维统一运算为基础, 探讨了多维向量场的多重向量表达与基本运算, 给出了向量场微分的模板卷积计算方法. 利用几何积对内外积的统一表达, 建立散度和旋度的统一计算方法并进行算法实现. 该方法不仅再现了向量场散度和旋度参数的几何关联性及其微分特性, 更实现了多维统一与坐标无关的计算. 基于北美风场的模拟实验显示, 该算法比基于梯度的间接求解算法具有更好的对比度与平滑性特征, 且具有较强的抗噪音能力, 从而可为向量场数据分析提供新的思路与方法基础.

Geometric algebra-based unified characteristic parameters computation method of multi-dimensional vector field

Based on the multi-dimensional unified geometric algebra, this paper discussed the multi-vector expression and computation of multi-dimensional vector filed. The vector differentiations are computed by template convolution. With the help of the unified expression of inner and outer product of geometric product, the unified calculation method of divergence and curl and the implementation algorithm are proposed. This method can not only reproduce the geometric association between vector field divergence and rotation parameters as well as their differential geometry characteristics, but also achieve a unified and multi-dimensional coordinate-free calculation. Simulation experimental results based on the wind fields of North American suggest that our method has much better performance of contrast and smoothness against the gradient-based algorithm. It also has strong ability to deal with vector fields of high noise and provide new insights and method foundations for vector field data analysis.