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1	Tikhonov regularization with oversmoothing penalties Gerth, Daniel 21 December 2016 (has links) (PDF) In the last decade l1-regularization became a powerful and popular tool for the regularization of Inverse Problems. While in the early years sparse solution were in the focus of research, recently also the case that the coefficients of the exact solution decay sufficiently fast was under consideration. In this paper we seek to show that l1-regularization is applicable and leads to optimal convergence rates even when the exact solution does not belong to l1 but only to l2. This is a particular example of over-smoothing regularization, i.e., the penalty implies smoothness properties the exact solution does not fulfill. We will make some statements on convergence also in this general context. Tikhonov Regularisierung l1 Regularisierung Überglätten Konvergenzraten Tikhonov regularization convergence rates l1-regularization oversmoothing ddc:510 Mathematik
2	Tikhonov regularization with oversmoothing penalties Gerth, Daniel 21 December 2016 (has links) In the last decade l1-regularization became a powerful and popular tool for the regularization of Inverse Problems. While in the early years sparse solution were in the focus of research, recently also the case that the coefficients of the exact solution decay sufficiently fast was under consideration. In this paper we seek to show that l1-regularization is applicable and leads to optimal convergence rates even when the exact solution does not belong to l1 but only to l2. This is a particular example of over-smoothing regularization, i.e., the penalty implies smoothness properties the exact solution does not fulfill. We will make some statements on convergence also in this general context. info:eu-repo/classification/ddc/510 ddc:510 Mathematik
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	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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