一类含有约束的变分问题中权重因子的最优选取

Optimal Selection for the Weighted Coefficients of the Constrained Variational Problems

研究了一类含有线性对流约束的变分问题中权重因子的最优选取· 在变分问题中,泛函权重因子选取的适当与否将影响数值计算的结果· 针对目前权重因子选取的相对随意性,在对观测场和理想场合理假设的条件下,分别讨论了带有弱约束和强约束的变分问题,通过求解相应的Euler方程,运用矩阵理论和偏微分方程的差分方法,得到了在分析场与理想场之间方差最小意义下的客观权重因子· 推证结果表明,若将带约束的变分问题的Euler方程离散成差分形式,且满足根据实际问题提出的合理假设以及差分方程稳定性条件,那么目标泛函中的权重因子在分析场与理想场的最小方差意义下存在最优选取· 它们在理论上更客观可信,可以实现权重因子与数值模式。

The aim is to put forward the optimal selecting of weights in variational problem in which the linear advection equation is used as constraint. The selection of the functional weight coefficients ( FWC) is one of the key problems for the relevant research. It was arbitrary and subjective to some extent presently. To overcome this difficulty, the reasonable assumptions were given for the observation field and analyzed field, variational problems with ' weak constraints' and ' strong constraints' were considered separately. By solving Euler' s equation with the matrix theory and the finite difference method of partial differential equation, the objective weight coefficients were obtained in the minimum variance of the difference between the analyzed field and ideal field. Deduction results show that theoretically the optimal selection indeed exists in the weighting factors of the cost function in the means of the minimal variance between the analysis and ideal field in terms of the matrix theory and partial differential ( corresponding difference ) equation, if the reasonable assumption from the actual problem is valid and the differnece equation is stable. It may realize the coordination among the weight factors, numerical models and the observational data. With its theoretical basis as well as its prospects of applications, this objective selecting method is probably a way towards the finding of the optimal weighting factors in the variational problem.