Tikhonov Regularization For Solving Ill-posed Problems

When we apply Tikhonov regularization to solve linear ill-posed problems: F(x) =ywe import Tikhonov function: J_α(x) =‖y^δ- F(x)‖²+α‖x‖² According to the principle of Tikhonov regularization,we know that the minimum of the function J_α(x) is exactly the solution of the equation.And we get the minimum of J_α(x) by iterative algorithm as below: x_α^δ=(F^*F+αI)^-1F^*y^δBesides these,we have got the conclution of the convergence and convergence rates of iterative solution x_α^δto the exact solution x as below:(1).Assume x = k^*z∈k^*Y,and‖z‖≤E,chooseα(δ) =cδ/E,c＞0,then we have‖x_α^δ- x‖= O(δ^1/2;(2).Assume x = k^*kz∈k^*k(X),and‖z‖≤E,chooseα(δ) = c(δ/E)^2/3,c＞0, then we have‖x_α^δ- x‖= O(δ^2/3);(3).When we use Tikhonov regularization for linear ill-posed problems,the iterative sequence {x_α^δ} converges to the exact solution with the speed of O(δ^2/3) at most. when we apply the above methods to nonlinear ill-posed problems,because of the ill-posedness of nonlinear problems,the solution of equation usually doesn't depend on the data condition continuously or isn't unique or doesn't exist.In order to overcome the first problems,we make several assumptions as below:(â…°).F is continuous;(â…±).F is weakly-closed,which means for any sequences {x_n}(?) D(F),if x_n converges to x in X and F(x_n) converges to y in Y,then we have x∈D(F),and F(x)=y.In order to overcome the unique problem of the solution,we format Tikhonov function in this way: J_α(x) =‖y^δ- F(x)‖²+α‖x-x^*‖² x⁺ choose x^*-minimum norm solution,which means x⁺ = min_x∈D(F){‖x-x^*‖: F(x)=y}.In the next discussion,we assume that the x^*-minimum norm solution always exists,and the existence of the solution and the weakly-closedness of F can guarantee that.At present the study of Tikhonov regularization for nonlinear ill-posed problems are all based on several assumptions on initial conditions and boundary conditions. This paper summarizes the former study,and analyze,study and pack up initial conditions and boundary conditions.After comparison to linear ill-posed problems,we come to the conclusion of unification of linear and nonlinear problems,and we pack up the generic condition assumptions of initial conditions and boundary conditions: smoothness assumptions and nonlinear assumptions(Assumption 2.0.1-2.0.5).Based on these assumptions,especially nonlinear conditions,we come to the conclusion of the convergence of Tikhonov regularization. Theorem 0.0.1.Assume x_α^δis the solution of nonlinear ill-posed problems F(x)=y, there existν∈Y,satisfying:x⁺ -x^* = F'(x⁺)^*ν,and existsω∈Y,p≥1,satisfying x⁺- x^* = F'(x⁺)^*(F'(x⁺)F'(x⁺)^*)(p-1)/2ωholds at p∈.Choose radius r,satisfying B_r(x⁺)(?) D(F),Frechet derivative F'(·) is Lispschitz continuous in the ball B_r(x⁺),which means there exists a constant L≥0,satisfying:‖F'(x) - F'(x₀)‖≤L‖x - x₀‖,(?)x,x₀∈B_r(x⁺) holds in the condition r =δ/a^1/2+ 2‖x⁺ - x^*‖and L‖ν‖≤γ＜1,then we have: and if the regularization parameterαis chosen asα= O(δ^2/(p+1),then‖x⁺- x_α^δ‖=We can conclude that the smoothness and convergence of the solution has a lot of correlation with the initial conditions and boundary conditions from the above theorem,and the Tikhonov regularization for nonlinear problems can get similar conclusions as linear problems:(1).if p = 1,thenα= O(δ),‖x⁺ - x_α^δ‖= O(δ^1/2);(2).if 1≤p≤2,thenα= O(δ^2/(p+1)),‖x⁺-x_α^δ‖= O(δ^p/(p+1).The above theorem is usually called the priori principle of regularization parameter. It is oriented to analyze the convergence of Tikhonov regularization and the corresponding stability and convergence rates.For the reason of numerical computation or other application,this paper bring forward another strategy of regularization parameter on the basis of the above theorem,which is the posteriori principle. Theorem 0.0.2.Assume the above theorem holds,and whereα_j∈D_M(α) and C_z≥1/(1-L‖ν‖)^1/2,then the error estimate‖x⁺- x_i+‖≤cδ⁽p/(p+1) holds,c is a constant independent ofδ.This paper is organized as below:we introduce the ill-posed problems and Tikhonov regularization in chapter 1;we make several important assumptions for solving nonlinear ill-posed problems;we result the convergence and convergence rates under the assumptions of chapter 2 in chapter 3;we prove the results in chapter 4;chapter 5 is the key chapter,and in this chapter we discuss the application in numerical computation of Tikhonov regularization,including the posteriori rule based on balancing principle,optimal and quasi-optimal choice of the posteriori rule, and numerical algorithm of the adaptive selection of the parameter.