Fast multiscale Galerkin methods for solving ill-posed integral equations via a coupled system under general source conditions

Fast multiscale regularized Galerkin methods for solving Fredholm integral equations of the first kind are investigated. We apply a multiscale Galerkin method with a matrix compression strategy to discretize the integral equation of the first kind and obtain a sparse system. To dealing with the ill-posedness and keeping sparsity of the discrete system, we use an equivalent coupled system form of the Tikhonov regularization. Then the multilevel augmentation method is applied to solve the coupled system. A priori and a posteriori regularization parameter choice strategies are proposed. Convergence rates of the regularized solutions are established under general source conditions. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed methods.