On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations |
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Authors: | Robert Plato |
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Affiliation: | Fachbereich Mathematik, Technische Universit?t Berlin, Stra?e des 17. Juni 135, D-10623 Berlin, Germany, DE
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Abstract: | 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.
Received August 29, 1994 / Revised version received September 19, 1995 |
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Keywords: | Mathematics Subject Classification (1991):65J20 65R30 |
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