Deep-QLP and Spline based Quadrature Rules for Fredholm Integral Equations of the First Kind

In this paper, we propose a numerical method to construct an approximate solution to Fred-holm integral equations of the first kind. These equations are common in image processing, spectroscopy, and other engineering sciences but are often ill-posed. Moreover, the right hand side of such equations could be error-contaminated. To effectively tackle the mentioned challenges, we propose suitable quadrature rules to discretize the continuous problem. The resulting discrete system is solved in a regularized least-squares framework by adopting the Deep-QLP factorization: a novel algorithm that automatically computes an approximate truncated singular value decomposition based on a user-defined tolerance. Additionally, two regularization approaches are introduced. In the first one, given a perturbed right-hand side, a Hankel matrix is constructed and then, the so-called de-Hankelization is performed by adopting the same Deep-QLP algorithm. In the second approach, we rely on locally refined discretization spaces to increase the accuracy of the computed approximate solution, by keeping low the number of the used degrees of freedom. Numerical experiments are conducted on some famous Fredholm integral equations available in the literature.