偏微分方程初始条件的核估计方法

Kernel Estimation of Initial Data of Partial Differential Equation

对于一个适定的偏微分方程组广义初值问题,该文利用非参数回归分析中的核估计方法,对在不同时间和不同空间记录下的数据进行整合,估计出未知函数在初始曲面上的值.对于空间维数为n的问题,此估计受到n(n+1)/2个参数的控制,在一定的最优准则下,可以得到初始数据的最优估计.最后给出了一个海流浅水模式初始资料的估计实例,与大气或海洋数值预报中的其它常用同化方法相比,计算量相对较小.

For a well-posed initial value problem of PDE, this paper estimates the values of its unknown functions on the initial surface with the kemel method in nonparametric regression analysis to deal with the data collected at different time and different positions. In the case of n-dimensional space, this estimation is govemed by n （n ＋ 1 ）/2 parameters which are optimized under certain optimal criteria. As an example, estimation of initial data for oceanic shallow water model is presented. Comparing with other similar methods used in atmospheric or oceanic numerical forecasting, the proposed method requires relatively less calculation.