Overview

Iterative methods for linear systems of equations and linear least squares problems are based on iteration schemes that access the coefficient matrix A only via matrix-vector multiplications with A and A ^T, and they produce a sequence of iteration vectors x ^{( k)}, k = 1, 2, , that converge to the desired solution. Iterative methods are preferable to direct methods when the coefficient matrix is so large that it is too time-consuming or too memory-demanding to work with an explicit decomposition of A.

Obviously, we can apply iterative methods to the symmetric positive definite Tikhonov system ( A ^T A + ? ² L ^TL) x = A ^Tb or the equivalent least squares problem, and in this way compute a regularized solution [47], [273]. However. this approach requires a new iteration process for each new value of ?, and for general matrices it is difficult to construct efficient preconditioners.

Instead, we are interested in iterative regularization methods in which each iteration vector x ^{( k)} can be considered as a regularized solution, with the iteration number k playing the role of the regularization parameter. Hence, we need iteration schemes with the intrinsic property that they, when applied to discrete ill-posed problems, initially pick up those singular value decomposition (SVD) components ( u ^T _i b/ ? _i) v _i corresponding to the largest singular values...