Regularization Methods For The Cauchy Problems Of The Elliptic Equations

In this thesis, we consider three kinds of Cauchy problems for elliptic equa-tions:the Cauchy problem for homogeneous elliptic equation, the Cauchy prob-lem for semi-linear elliptic equation, the Cauchy problem for elliptic equation with variable coefficients.On the the Cauchy problem of homogeneous elliptic equation, we mainly con-sider the case that the reconstructed solution is a discontinuous function. Firstly, we transform the original Cauchy problem into a Fredholm integral equation of the first kind, this integral equation is ill-posed, we use three regularization methods (L2. H1, MTV-regularizations) to deal with it. During the procedure of the regu-larization, we give the variational form in using three methods to seek the regular-ization solution, then using the necessary condition of the first, order, we derive the Euler equation which the regularization solution of each method satisfies. In the numerical discretization, we discrete the Euler equation of each method to seek its regularization solution. For the modified total variation rogularization(MTV-regularization), Euler equation is a nonlinear equation, we adopt a fixed point iter-ative method to solve this nonlinear equation and obtain the MTV-regularization solution. In the process of numerical simulation, the regularization parameters for three methods are chosen by Morozov’s discrepancy principle, and we compare the merit and shortcoming of each method in constructing the discontinuous solution. Numerical results show that, compared with L2.H1-regularization. the modified total variation regularization (MTV-regularization) technique is more robust for reconstructing the discontinuous solution. However, the MTV-regularization will lead to a nonlinear integro-differential equation of elliptic type and need much more computational time. In addition, we alone analysis the influence of some parameters on MTV-regularization solution by a numerical example.Up to now. we do not find the related references in which the Cauchy problem of semi-linear elliptic equation has been considered. In Chapter3, we study this kind of problem for the first time. Firstly, we use the Fourier truncated regularization method and modified quasi-boundary value method to deal with an abstract Cauchy problem of semi-linear elliptic equation, respectively, and we give and prove the convergence estimates for two regularization methods under an a-priori assumption for the exact solution. Numerical results show that the Fourier truncated regularization method and modified quasi-boundary value method are stable and feasible in solving the Cauchy problem of semi-linear elliptic equation. In addition, in this part we also consider a Cauchy problem for semi-linear Poisson equation. Note that, although this problem is a special case of the abstract Cauchy problem for semi-linear elliptic equation, but here we use a kind of similar method with modified quasi-boundary value met hod to deal with it. This method can be seen as a generalization for modified quasi-boundary value method.Chapter4investigates a Cauchy problem for the elliptic equation with vari-able coefficients. Firstly, we adopt two iterative methods to solve this problem, note that, even if we add a stronger a-priori bound assumption for the exact solu-tion, the first kind of iterative method also can not obtain the convergence rate of the regularization solution on the boundary. In order to overcome this problem, we give the second kind of iterative method, this method is a modification for the first method, and it is an iterative method based on the spectrum truncated technique. In doing the convergence estimates for two methods, we respectively adopt the a-priori and a-posteriori rules to select the regularization parameter, and obtain the convergence estimates of optimal order for two algorithms. In addition, we use modified quasi-boundary value method and simplified Tikhonov regularization method to deal with this problem. And under the a-priori bound assumption for the exact solution, we give and prove the convergence estimates of two methods. On the aspect of numerical computation, we give some examples to verity the regularized efficiency of each method.