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Multiscale collocation methods are developed for solving ill-posed Fredholm integral equations of the first kind in Banach spaces, if the associated resolvent integral operator fulfils a condition with respect to a interval. We apply a multiscale collocation method with a matrix compression strategy to discretize the integral equation of the second kind obtained by using the Lavrentiev regularization from the original ill-posed integral equation and then use the multilevel augmentation method to solve the resulting discrete equation. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We develop a multilevel iteration method with compression technique for solving ill-posed integral equations. The method leads to fast solutions of discrete regularization methods for the equations. Choice for a posteriori regularization parameter is proposed. An optimal convergence order for the method with the choices of parameters is established. The theoretical analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper we develop a fast multiscale Galerkin method to solve the ill-posed integral equation via iterated Tikhonov regularization. This method leads to fast solutions of discrete iterated Tikhonov regularization. The convergence rates of iterated Tikhonov regularization are achieved by using a modified discrepancy principle. Finally, numerical experiments are given to illustrate the efficiency of the method. [Copyright &y& Elsevier]

We apply fast multiscale methods for solving ill-posed integral equations via the Tikhonov regularization. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]