Research On Parameter Identification Problems Based On Multi-Scale Methods

With the rapid development of science and technology,the problem of parameter identification has become more and more in-depth.The problem of parameter identification has covered many fields such as science and engineering.Its numerical solutions have been developed from the initial pulse spectrum method to the nonlinear optimization method.These numerical solutions keep improving the computational efficiency and the accuracy of the solution.This thesis mainly explored the inversion problem of two-dimensional wave equation in the parameter identification problem,proposed an improved single-scale gradient method,and then combined this method with the multi-scale method to solve the inversion problem of two-dimensional wave equation,and verify its feasibility with numerical simulation analysis.According to the data results,it is proved that the improved method has a good effect on solving the two-dimensional wave equation inversion problem.First of all,this thesis made a basic introduction of the parameter identification problem,summarized the current research status of wave equation and multi-scale methods at home and abroad,and reviewed some of the cutting-edge knowledge useful for the research of this thesis.Secondly,this thesis introduced mathematical model of two-dimensional wave equation.Since inversion and forward modeling are inseparable,the forward modeling method is helpful to solve the inversion problem.Therefore,the forward modeling process of the two-dimensional wave equation was also introduced in this thesis.Because the current nonlinear optimization method has some shortcomings in dealing with the inversion problem,an improved single-scale gradient method was proposed in order to overcome such shortcomings.The basic idea was based on the gradient method,combined with the secant method and the method of advance and retreat,and made improvement in choosing the initial solution and searching step size respectively.Thirdly,this thesis combined the improved single-scale gradient method with the multi-scale method.On the premise of designing a two-layer computational grid,firstly,a certain gradient method was used as a smoothing method to iterate on the coarse grid,in order to obtain a better initial solution as smooth as possible.Secondly,the obtained initial solution was interpolated to the fine grid and was regarded as the initial value on the fine grid,and then executed several iterations to smooth the solution.Since the main part is the iteration on the coarse grid,this thesis reduced the calculation amount so as to improve the calculation efficiency.Finally,this thesis performed numerical simulations based on these two algorithms,and the simulation results proved the feasibility of the single-scale gradient method and the multi-scale gradient method,and the improvement in computational efficiency compared with other methods.In short,the gradient inversion method proposed in this thesis made a contribution to solve the two-dimensional wave equation inversion problem in a certain degree,and can be extended to parameter identification problems.