Arcangeli’s method for Fredholm equations of the first kind

Abstract:

It is well known that a linear operator equation of the first kind, with an operator having nonclosed range, is ill-posed, that is, the solution depends discontinuously on the data. Tikhonov’s method for approximating the solution depends on the choice of a positive parameter which effects a trade-off between fidelity and regularity in the approximate solution. If the parameter is chosen according to Morozov’s discrepancy principle, then the approximations converge to the true solution as the error level in the data goes to zero. If the operator is selfadjoint and positive and semidefinite, then "simplified" approximations can be formed. We show that Morozov’s criterion for the simplified approximations does not result in a convergent method, however, Arcangeli’s criterion does lead to convergence. We then prove the uniform convergence of Arcangeli’s method for Fredholm integral equations of the first kind with continuous kernel.