## A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization StrategiesConvergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynamics

 dc.contributor.advisor Krause, Rolf dc.contributor.author Groß, Christian dc.date.accessioned 2020-04-14T02:47:56Z dc.date.available 2020-04-14T02:47:56Z dc.date.issued 08.10.2009 dc.identifier.uri http://hdl.handle.net/20.500.11811/4129 dc.description.abstract Nonlinear right preconditioned globalization strategies for the solution of nonlinear programming problems of the following kind $u \in \mathcal B \subset \mathbb R^n: J(u) = \min!$ where $\mathcal B$ is a convex set of admissible solutions, $n\in \mathbb N$, and $J: \mathbb R^n \to \mathbb R$, sufficiently smooth, are presented. Preconditioned globalization strategies are traditional Linesearch or Trust-Region strategies in combination with a nonlinear update operator which results from a nonlinear solution process for smaller, but related, nonlinear programming problems. We will formulate conditions on this abstract operator, in order to ensure global convergence, i.e., convergence to first-order critical points, of the resulting method. In addition, we introduce particular implementations of this abstract operator, i.e., nonlinear multiplicatively preconditioned Trust-Region (MPTS) and Linesearch strategies (MPLS), as well as nonlinear additively preconditioned Trust-Region (APTS) and Linesearch (APLS) strategies. As it turns out, these additive strategies are novel parallel, locally adaptive and robust solution methods for nonlinear programming problems. Moreover, the MPTS strategy generalizes the RMTR concepts in [GK08] in order to allow also for the application of alternating nonlinear domain decomposition methods. On the other hand, the MPLS method simplifies and generalizes the concepts in [WG08] giving rise to a novel solution strategy for pointwise constrained nonlinear programming problems. The respective nonlinear solution strategies are analyzed and global convergence is shown. In addition, global convergence is also shown for combined nonlinear additively and multiplicatively preconditioned Trust-Region and Linesearch strategies. Moreover, we show the efficiency and reliability of these methods in the context of problems arising from the field of nonlinear elasticity in 3d. Particular emphasis has been placed on the formulation and analysis of the resulting minimization problems. Here, we show that these problems satisfy the assumptions stated to show convergence of the respective preconditioned globalization strategies. Moreover, various elasto-static and elasto-dynamic examples are presented in order to compare the convergence rates and runtimes of the different strategies. dc.language.iso eng dc.rights In Copyright dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ dc.subject Nichtlineare Programmierung dc.subject Gebietszerlegungsverfahren dc.subject Mehrgitterverfahren dc.subject Finite Elemente dc.subject Elastizitätstheorie dc.subject Nonlinear Programming dc.subject Domain Decomposition methods dc.subject Multigrid methods dc.subject Finite Elements dc.subject Elasticity theory dc.subject.ddc 510 Mathematik dc.title A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies dc.title.alternative Convergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynamics dc.type Dissertation oder Habilitation dc.publisher.name Universitäts- und Landesbibliothek Bonn dc.publisher.location Bonn dc.rights.accessRights openAccess dc.identifier.urn https://nbn-resolving.org/urn:nbn:de:hbz:5N-18682 ulbbn.pubtype Erstveröffentlichung ulbbnediss.affiliation.name Rheinische Friedrich-Wilhelms-Universität Bonn ulbbnediss.affiliation.location Bonn ulbbnediss.thesis.level Dissertation ulbbnediss.dissID 1868 ulbbnediss.date.accepted 2009-09-11 ulbbnediss.fakultaet Mathematisch-Naturwissenschaftliche Fakultät dc.contributor.coReferee Harbrecht, Helmut
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