Study On Newton Method Based On Heuristic Stopping Criterion

Recently,inverse problems have attracted more and more attention.However,most of the inverse problems are ill-posed,regularization methods are needed to solve this kind of problems.The commonly used regularization methods include Tikhonov regularization method and iterative regularization method,etc.Since the Tikhonov regularization method requires a large amount of computation when solving inverse problems,Newtontype algorithm for solving nonlinear inverse problems can greatly reduce the number of steps required for iteration,and has advantages in solving large-scale inverse problems.Therefore,this paper proposes a fixed point Gauss-Newton regularization algorithm based on heuristic stop criterion for solving nonlinear inverse problems in Banach space.The algorithm only needs the derivative value of the fixed point in each iteration,which greatly reduces the computation amount and saves the computation time.In addition,the stop criterion does not require any information of noise level,so the inverse problem with unknown noise level can be treated.Firstly,on the premise that the tapering condition holds,based on the given hypothesis and the variational source condition,the posterior error estimate of the solution is obtained by estimating the heuristic function.The error estimate can be controlled by the real noise level and the actual noise level,which have a certain influence on the convergence of the solution.The convergence order of a posteriori error function can be obtained when the actual noise level has some relationship with the real noise level.Secondly,in the absence of any source conditions,based on the basic hypothesis and the modified variational regularization hypothesis satisfied by the noise data,the general convergence result of the solution is obtained.At the end of the paper,numerical simulation is given.In this paper,TV norm is selected as a regular term to reconstruct the piecewise smooth solutions of one-dimensional and two-dimensional elliptic boundary value problems respectively.Compared with the Gauss-Newton iterative method under the general heuristic criterion,the Gauss-Newton regularization algorithm proposed in this paper converges slowly under the heuristic stop criterion,but the calculation is simple.Compared with the reconstruction results of different noise data,the smaller the noise interference,the more steps needed to stop the iteration,the smaller the relative error,and the better the reconstruction effect.