Multiscale collocation methods for ill-posed integral equations via a coupled system

Multiscale collocation methods are developed for solving a system of integral equations which is a reformulation of the Tikhonov-regularized second-kind equation of an ill-posed integral equation of the first kind. Direct numerical solutions of the Tikhonov regularization equation require one to generate a matrix representation of the composition of the conjugate operator with its original integral operator. Generating such a matrix is computationally costly. To overcome this challenging computational issue, rather than directly solving the Tikhonov-regularized equation, we propose to solve an equivalent coupled system of integral equations. We apply a multiscale collocation method with a matrix compression strategy to discretize the system of integral equations and then use the multilevel augmentation method to solve the resulting discrete system. A priori and a posteriori parameter choice strategies are developed for these methods. The convergence order of the resulting regularized solutions is estimated. Numerical experiments are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.