Variational Methods For Several Kinds Of Inverse Problems And Singular Perturbation Problems

The inverse problems and singular perturbation problems appear widely in the science and physical engineering,inverse problems are almost ill-posed,the exact solution of singular perturbation problems can produce dramatic changes in the solution area caused by the influence of the singular perturbation parameters,which are the reasons that the usual numerical methods can not get the numerically stable approximate solution of these problems.So more and more scholars pay attention to the stable numerical algorithms.In this thesis,we study on the variational methods of inverse problems and singular perturbation problems,the advantage of the variational methods based on the variational principle is that it provides a powerful theoretical framework for solving practical problems in which the ill-posed problems can be expressed as well posed problems.The main research contents are as follows:The second section gives the inversion methods of the unknown source and coefficient inverse problems of partial differential equations,the direct method is proposed to recover the inverse problems.Based on this idea,an equivalent nonlinear equation is obtained by the given additional conditions,then the approximate solution of the equivalent equation is used to approximate the exact solution.Due to the error accumulation of the solution progress,the variation iteration method is employed to giving the numerical solution of the equivalent equation,the characteristic of this method is that it can get very good numerical results with fewer iteration steps.Some examples are given to explain the implementation process and disturbance experiments are carried out by disturbing the additional conditions.The third section studies on singular perturbation problems,three modified variation iteration methods are presented for solving the nonlinear singular perturbation of second order initial and boundary value problems.On the one hand,in view of the deficiency of the traditional variation iteration method,a piecewise variation iteration method is developed for solving the long interval Lane-Emden equation.In the long interval,the long interval is divided into some subintervals in which we apply the variation iteration method to solve the equation,this can overcome the deficiency that the traditional variational iteration method is only convergent in a small region.In order to assure the stability of this method,the convergence and error estimation are analyzed.On the other hand,for the perturbation problems,variation iteration method is combined with the perturbation technique,the perturbation technique of small parameters can eliminate the influence of a small perturbation parameter to the numerical process.Meanwhile the variation iteration method can obtain the convergent solution rapidly in many cases.The variation iterationperturbation method combines the advantages of these two methods so that it can solve the nonlinear second order perturbation problems effectively.In addition,for the boundary value problems,the iteration function from the variation iteration method can not be deduced to satisfy the boundary value conditions directly,the existing method is to join the undetermined constants in the initial iteration function,but when the nonlinear item is very complex,the undetermined constants are difficult to determine.Therefore,the shooting method and variation iteration method are used to solve this problem well,which leads the shooting-variation iteration method of high convergence speed and precision.In the fourth section,the practical application of the variational principle in inverse problems is the image denoising.A variational model based on the coupling energy functional of convex and nonconvex functions is proposed,under this model,the corresponding partial differential equation is deduced to image denoising by the variational principle.The partial differential equation is a forward–backward diffusion equation which is very suitable for removing noises of the piecewise constant image.Owing to the introduction of a nonconvex function,it is needed to consider the existence and uniqueness of solution,so the Young measure theory is introduced to prove the existence of weak solution.Under certain conditions,the uniqueness of solution is proved.At the same time,the properties of the Young measure solution explain the rationality of the model.In the numerical experiments,the discrete formate of PM method and the AOS discrete formate of the partial differential equation to numerical calculations are given.Moreover,the peak signal to noise ratio and structural similarity are proposed to evaluate the effect of denoising.The numerical results show that the presented model not only can eliminate the speckle effect of PM model,avoid the step effect of TV model,but also can protect borders.Compared with the two classical models,the new model is superior to other models in the visual effect and the image denoising evaluation indexes.