Search

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper. [ABSTRACT FROM AUTHOR]

We propose a multilevel iteration method for the numerical solution of nonlinear ill-posed problems in the Hilbert space by using the Tikhonov regularization method. This leads to fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. An adaptive choice of an a posteriori rule is suggested to choose the stopping index of iteration, and the rates of convergence are also derived. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, multilevel Jacobi and Gauss-Seidel type iteration methods with compression technique are developed for solving ill-posed integral equations by making use of the multiscale structure of the matrix representation of the integral operator. The methods are based on the combination of Tikhonov regularization and multiscale Galerkin methods, and lead to fast solutions of discrete regularization methods for the equations. Choice for an a posteriori regularization parameter is proposed. An optimal convergence order for the method with the choices of parameters is established. Numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper we develop a fast multiscale Galerkin method solving ill-posed integral equations with not exactly given input data via Tikhonov regularization. The method leads to fast solutions of discrete Tikhonov regularization. The convergence rates of the Tikhonov regularization are achieved by using a modified discrepancy principle. Finally, numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]