On the monotone error rule for parameter choice in iterative and continuous regularization methods

We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y(delta) satisfying parallel to y(delta) - yparallel to less than or equal to delta. Regularized approximations x(r)(delta) to the minimum-norm solution x(dagger) of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r(ME) (n = n(ME)) as the largest r - value with the guaranteed monotonical decrease of the error parallel tox(r)(delta) - x(dagger)parallel to for r is an element of [0, r(ME)] (parallel tox(n)(delta) - x(dagger)parallel to < ∥ x(n-1)(delta) - x(dagger)parallel to for n = 1, 2,..., n(ME)). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.