TY - THES
T1 - Iterative regularization methods for the solution of the split feasibility problem in Banach spaces
A1 - Schöpfer,Frank
Y1 - 2007/11/15
N2 - We develop iterative methods for the solution of the split feasibility problem (SFP) in Banach spaces and analyze stability and regularizing properties. The SFP consists in finding a common point in the intersection of finitely many closed convex sets, whereby some of the sets arise by imposing constraints in the range of a linear operator. In principle the SFP can be solved by cyclically projecting onto the individual sets. In applications such projection algorithms are efficient if the projections onto the individual sets are relatively simple to calculate. If the sets arise by imposing constraints in the range of a linear operator then it is in general too difficult or too costly to project onto these sets in each iterative step. In finite-dimensional euclidean spaces Byrne suggested the CQ algorithm for the solution of the SFP, which avoids projecting directly onto such sets by using gradients of suitable functionals. To solve the SFP in Banach spaces we generalize this algorithm via duality mappings, metric projections and Bregman projections. We provide the necessary theoretical framework and extend it by some further contributions. We prove convergence of the resulting methods, show how approximate data may be used, and analyze their regularizing properties by applying a discrepancy principle. Especially we are also concerned with the computation of projections onto affine subspaces that are given via the nullspace or the range of a linear operator. To this end we can use the same iterative scheme as for the SFP and we also propose generalized sequential subspace and conjugate gradient methods.
KW - Iteration
KW - Banach-Raum
KW - Konvexe Menge
CY - Saarbrücken
PB - Saarländische Universitäts- und Landesbibliothek
AD - Postfach 151141, 66041 Saarbrücken
UR - http://scidok.sulb.uni-saarland.de/volltexte/2007/1332
ER -