Applications Of The Four-Dimensional Variation Methods On Parameter Optimization Of Differential Equations

As we all know, the accuracy of numerical weather forecasting is decreasing rapidly with the increasing of the prediction time because of the errors in the parame-ters in governing differential equations, the errors in initial data and the errors caused by numerical methods. For the initial data, meteorologists have done a lot of work about atmospheric motion to objectively describe the state of the atmospheric mo-tion. Thus, they put forward the powerful method of four-dimensional variational data assimilation. The purpose of this thesis is to apply four-dimensional variational data assimilation methods to optimize parameters in differential equations.By the main idea of the four-dimensional variational data assimilation method and follow its procedure of implementation, this thesis propose a method to opti-mize parameters in differential equations by using some given discrete solutions of the equations. We first assume that some discrete solutions of differential equation is known while the parameters are unknown. The cost function with the parameters as its variables is derived by the least square method. Then we use the adjoint method to calculate the gradient direction of the cost function and obtain the parameter val-ues by widely-used optimization methods. In numerical experiments, we use classical Lorenz model and Hamilton equations as numerical models to illustrate successfully the efficiency of the methods.Finally, the thesis also attempts to apply this method to a practical problem in chemical reactions and give some numerical results.