Summary.
For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented.
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Received August 29, 1994 / Revised version received September 19, 1995
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Plato, R. On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations . Numer. Math. 75, 99–120 (1996). https://doi.org/10.1007/s002110050232
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DOI: https://doi.org/10.1007/s002110050232