Homotopy Perturbation Regularization For Solving Nonlinear Inverse Problems

More and more attention has been paid to the inverse problem in modern scientific research,and the effective way to solve the inverse problem is to put forward the corresponding regularization method to change the unstable nonlinear problem into a stable approximate problem,The degree of approximation depends on the selection of regularization parameters.The Tikhonov method depends on the smoothness of the solution in parameter selection,and the global minimum of the target functional is also affected by the initial value.But in practice,it is difficult to capture the priori information of the effective solution.In this paper,the iterative method is applied to solve nonlinear ill-posed problems to obtain stable approximations.In this paper,on the basis of Euler equation,homotopy perturbation regularization method is constructed,which not only reduces the selection of initial value,but also obtains convergence results under appropriate conditions.Because of the complexity of practical problems,we can not accurately obtain the priori information of the solution,so this paper adopts the posterior parameter selection principle,among which the Morozov discrepancy principle is the most famous.In this paper,the generalized discrepancy principle is used as the stopping criterion of iterative scheme,combined with Hilbert space theory and Cauchy inequality,the convergence results of the method are obtained under the appropriate assumptions of the regularization parameters.Considering that the real noise level is not available or accurate,we can't get the number of iterative termination steps that depend only on the noise level and the right end of the data.Therefore,the heuristic stopping criterion is applied to the iterative method constructed in this paper to obtain the parameter that only depends on the perturbed data.However,in the worst case,it is impossible to obtain the convergence result without the noise level information.Therefore,when it is proved that the iterative scheme converges under the Hanke-Raus stopping criterion,some auxiliary conditions,such as weak nonlinear conditions,are introduced in this paper.It is proved that the iterative scheme is convergent under these conditions.At the end of this paper,aiming at the inversion of elliptic differential equation parameters,the numerical results are given,and the simulation results show that the iterative scheme uses fewer iterations under the Hanke-Raus stopping criterion,and the approximate solution is more stable when the noise level is unknown.