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Accelerated Landweber iterations for the solution of ill-posed equations. (English) Zbl 0745.65038

Author’s summary: The potentials of so-called linear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill-conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.

If the problem’s right hand side data are contaminated by noise, semiiterative methods may be used as regularization methods. Assuming optimal rate of convergence of iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.

To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results on fast decreasing polynomials are applied to answer an open question of H. Brakhage [Notes Rep. Math. Sci. Eng. 4, 165-175 (1987; Zbl 0642.65042)].

Numerical examples are given including a comparison to the method of conjugate gradients.


MSC:
65J10Equations with linear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47A50Equations and inequalities involving linear operators, with vector unknowns
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