Research On Proximal Iterative Algorithms For Solving Nonlinear Inverse Problems

In this paper,investigate the proximal iterative algorithm for solving nonlinear inverse problems.The inverse problem is usually ill-posed,especially when the solution is unstable depending on the perturbation of the data,that is,a small perturbation of the data will lead to a large deviation of the solution.Because of the ill-posedness of the inverse problem,as we all know,Tikhonov regularization method,also known as l2 norm regularization,is usually used to solve stable approximate solutions of ill-posed inverse problems.However,the l2 norm penalty function may cause the solution to be excessively smooth,especially when the solution of the inverse problem is sparse.Therefore,a better alternative is needed.Another method to avoid the above disadvantages is the so-called sparse regularization,which also called l1 norm regularization.But,this will lead to the non-differentiability of the target function.Therefore,this paper is mainly carried out from the following aspects:(1)Firstly,the proximal operator is introduced and combined with the gradient algorithm to construct the proximal gradient algorithm to solve the sparse constraint target functional.Secondly,an accelerated version of this iterative method is proposed by adopting the BB step rule and including multiple iterations in the update process.Finally,in the context of proximal regularization Landweber iteration method,the convergence of the proximal method is discussed.(2)The inverse problem is applied to the full waveform inversion.First,we apply the variable projection method to the data correction in the frequency domain full waveform inversion.More specifically,the source weight of each frequency is calculated by the minimum norm solution between measured and simulated data,so that the inversion process is no longer dependent on the source wavelet.Secondly,numerical examples show that both the proposed proximal gradient algorithm and the accelerated proximal gradient algorithm can achieve good reconstruction results without knowing the source,the accelerated reconstruction algorithm is more accurate in locating the anomalous body and estimating the velocity more accurately.(3)The inverse problem is applied to the resistivity inversion.First,we give a complete electrode model.Secondly,the simulation of resistivity imaging is carried out.The experimental results show that the proximal regularized Landweber iterative method(PRLI)and the regularized Landweber iterative method(RLI)can be reconstructed reasonably,but the PRLI method is more accurate and has higher resolution than the traditional RLI method.