Fast collocation methods for solving ill-posed integral equations of the first kind

We consider ill-posed Fredholm integral equations of the first kind. A fast piecewise polynomial collocation method is introduced for solving the second kind of integral equation obtained by using the Tikhonov regularization from the original ill-posed equation. The method is developed based on a matrix compression strategy resulting from using multiscale piecewise polynomial basis functions and their corresponding multiscale collocation functionals. A priori and a posteriori regularization parameter choice strategies are proposed. Convergence rates of the regularized solutions are established. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.