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1	Parameter choice in Banach space regularization under variational inequalities Hofmann, Bernd, Mathé, Peter 17 April 2012 (has links) (PDF) The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established. Nichtlineare inkorrekte Probleme inkorrekt gestellte Probleme Banachraum-Regularisierung Konvergenzraten Nonlinear ill-posed problems Banach space regularization convergence rates ddc:510 Regularisierung Tichonov-Regularisierung
2	Parameter choice in Banach space regularization under variational inequalities Hofmann, Bernd, Mathé, Peter January 2012 (has links) The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established. info:eu-repo/classification/ddc/510 ddc:510
3	The impact of a curious type of smoothness conditions on convergence rates in l1-regularization Bot, Radu Ioan, Hofmann, Bernd 31 January 2013 (has links) (PDF) Tikhonov-type regularization of linear and nonlinear ill-posed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the l1-norm is used as penalty term, the l1-regularization was studied by numerous authors although the non-reflexivity of the Banach space l1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the l1-regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components. l1-Regularisierung Konvergenzraten Quellbedingungen nonlinear ill-posed problems Tikhonov-type regularization l1-regularization sparsity constraints convergence rates solution decay variational inequalities source conditions discrepancy principle ddc:510 ddc:519 Regularisierung Konvergenz
4	Preconditioned Newton methods for ill-posed problems / Vorkonditionierte Newton-Verfahren für schlecht gestellte Probleme Langer, Stefan 21 June 2007 (has links) No description available. 510 Mathematik Mathematics and Computer Science 31.76 31.80 31.45 33.06 EEHA 520: Ill-posed problems EGFF 220: Ill-posedness electromagnetic theory: General}
5	The impact of a curious type of smoothness conditions on convergence rates in l1-regularization Bot, Radu Ioan, Hofmann, Bernd January 2013 (has links) Tikhonov-type regularization of linear and nonlinear ill-posed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the l1-norm is used as penalty term, the l1-regularization was studied by numerous authors although the non-reflexivity of the Banach space l1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the l1-regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components. info:eu-repo/classification/ddc/510 ddc:510 info:eu-repo/classification/ddc/519 ddc:519 Regularisierung; Konvergenz

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