Research On Tikhonov Regularization Method Based On Particile Swarm Optimization

In recent years, with the rapid development of human society and the improvement of scientific research, the inverse problem is widely applied in many areas. Therefore, many experts pay more attentions to study inverse problems and inversion methods. The greatest difficulty in this field is the ill-posedness in the process of solving, which is the intrinsic property of inverse problem. A lot of experts studied in this problem deeply and proposed many inversion methods, which have their own features. However, the most effective and applicable method is Tikhonov regularization method for calculating the inversion of the ill-posed problem. However, the treatment of smoothing functional is complex for traditional Tikhonov regularization method. In many cases, we could not obtain the satisfied solution. In this paper, a modified Tikhonov regularization method is proposed based on the particle swarm optimization. Smoothing function is optimized by the PSO algorithm and better results are obtained.This paper mainly introduces the theory of the Tikhonov regularization method and particle swarm optimization algorithm in detail and analyzes the advantages and disadvantages of them. Based on the advantages and disadvantages of these two algorithms, a hybrid approach is presented, which can get more accurate solutions. The particle swarm algorithm is used to solve the extreme value of the flattening functional, which is an essential procedure of the Tikhonov regularization method. By doing this, a better global solution can achieved. Based on the modified method presented in this paper, we firstly take two kinds of integral equations of(first kind Fredholm integral equation and first kind Voltorra integral equation) as an example. The modified Tikhonov regularization method and traditional Tikhonov regularization method are applied to solve these two kinds of integral equations, respectively. Compared with the numerical results of the two methods, the modified method has more accurate result than that of traditional Tikhonov regularization method. And we adopt different methods to discretize integral equations into the system of linear equations in the solving process. Additional, we use two different parameter regularization selection strategies L-curve method and GCV method. For these two methods, we can find that the accuracy of the results obtained by the modified method are much better than that of the traditional Tikhonov regularization method. Finally, modified method is also applied to the parameters inversion of the fractional order anomalous diffusion equation. In this paper, we not only consider the influence of regularization parameter the selection method on the solution, but also carry out the sensitivity analysis of three parameters, including learning factor, the number of the population and the number of iterations. By comparing the solutions obtained with different parameters, we find that the learning factor and the number of the population have an impact on the accuracy of the inversion results.