A discretization scheme for stably solving ill-posed operator equations Tx=y between Hilbert spaces, where the operator's range is not closed so no continuous inverse exists. Instead of the usual bounded finite-rank projections on the domain or codomain, it applies finite-rank 'projection-like' operators that need not be bounded and whose ranges need not lie inside the codomain, then solves the resulting finite matrix system for a minimum-norm approximation. Students learn how error bounds for the discrete solution are inherited from Tikhonov regularization through the operator-gap quantity ||T*T - Tn*Tn||, and how the setting unifies classical projection methods with quadrature-based collocation for first-kind integral equations.
This arxiv paper proposes a discrete regularization approach for solving ill-posed operator equations, with specific application to quadrature-based collocation methods for Fredholm integral equation…