The Research Of Regularization Parameter In Regularization Method

This paper gives the generic concept of the mathematics and physics inverse problem from material examples. It shows that the mathematics and physics inverse problems are always nonlinear and ill-posed. For solving inverse problems efficiently and conquering ill-posed problem, we have discussed Tikhonov regularization method, it basic idea is that: using a series well posed problem which is appropriate to the original problem to approach the solution of the original problem. Using this method in Hilbert space is easy to get a theoretical analysis, and we have obtained some results and error estimation of Tikhonov regularization method in Hilbert space. This paper also gives another method to solve inverse problem: dispersed regularization method, which using projection method to approximate infinite dimensional inverse problem in finite dimension and gain a ill-posed system firstly, and then at last using the regularization method to solve the system. From analysis we know that the key to the regularization method and theory is how to construct "appropriate problem to gain regular operator and regularization parameter, and how to decide parameter which is suited to the error level of source material, and how to get the numerical implementation of the above work.The problem of how to choice the regularization parameter is not only important but also attractive. To this problem we provide the idea of using genetic algorithms to calculate regularization parameter for the first time. Genetic algorithms follows the organic evolution rules, through operations selection,crossover mutation to finish the survival of the fittest, and repeats the same operation from generation to generation, obtains optimum solution or closes to optimum solution at last. Genetic algorithms has strong adaptability when deals with complex date and nonlinear calculation. This paper has provided specific algorithm and worked out calculation program, which makes use of genetic algorithms together with the rules how to fix on the regularization parameter. The numerical result turns out that the regularization parameter has high precision which gains by using genetic algorithms. This method displays some superiority especially to large-scale ill-posed problem, which conquers the shortage of traditional iteration when deal with large-scale problem or the problems have not had a proper initial- value choice.