Optimal selection for the weighted coefficients of the constrained variational problems

Abstract

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 difference 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.