Multiscale collocation methods for ill-posed integral equations with a modified posteriori parameter selection

摘要

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.