Equivalence of regularization and truncated iteration for general ill-posed problems (Q1911416)

The linear algebraic equations deduced from discretizing a Fredholm integral equation of the first kind is usually very ill-posed. One standard way to obtain stable solutions is to modify the problem with the Tikhonov regularization. Another is to apply an iterative method to the normal equations. \textit{H. E. Fleming} [ibid. 130, 133-150 (1990; Zbl 0714.65045)] has established an equivalence between the two types of methods if the coefficient matrix has full rank. It is proven that every direct regularization method of a very general type is equivalent to a truncated iterative method and vice versa. This paper extends these results to incomplete-rank matrices. By using a formula for general iterative methods, a simpler and unified proof is given for both the overdetermined and the underdetermined case.