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Iteration methods for convexly constrained ill-posed problems in Hilbert space. (English) Zbl 0769.65026

The author deals with minimizing a quadratic objective functional \(f \to \| Af - g\|^ 2\) over a closed convex constraint set \(C\), where \(A\) is a bounded linear operator. When the minimum is not unique, the author’s suggestion is to look for the solution of minimal norm. In case the problem is ill-posed, i.e. the solution does not depend continuously on the data, then the problem can be solved by means of the Tikhonov- Phillips iterative regularization method.
The regularities of three iterative methods, which are the projected Landweber iteration, the method of smooth solutions, and the damped projected Landweber iteration, are the main issue of this paper. Finally, the author applies these methods to specific problems and gives numerical results.

MSC:

65J10 Numerical solutions to equations with linear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47A50 Equations and inequalities involving linear operators, with vector unknowns
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