Quadrature based collocation methods for integral equations of the first kind

Problem of solving integral equations of the first kind, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int_a^b k(s, t) x(t)\, dt = y(s),\, s\in [a, b]$\end{document} arises in many of the inverse problems that appears in applications. The above problem is a prototype of an ill-posed problem. Thus, for obtaining stable approximate solutions based on noisy data, one has to rely on regularization methods. In practice, the noisy data may be based on a finite number of observations of y, say y(τ1), y(τ2), ..., y(τn) for some τ1, ..., τn in [a, b]. In this paper, we consider approximations based on a collocation method when the nodes τ1, ..., τn are associated with a convergent quadrature rule. We shall also consider further regularization of the procedure and show that the derived error estimates are of the same order as in the case of Tikhonov regularization when there is no approximation of the integral operator is involved.