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1	Eigenvalue Problem for the 1-Laplace Operator / Das Eigenwertproblem für den 1-Laplace-Operator Milbers, Zoja 27 March 2009 (has links) (PDF) We consider the eigenvalue problem associated to the 1-Laplace operator. We define higher eigensolutions by means of weak slope and establish existence of a sequence of eigensolutions by using nonsmooth critical point theory. In addition, we deduce a new necessary condition for the first eigenvalue of the 1-Laplace operator by means of inner variations. / Wir betrachten das zum 1-Laplace-Operator gehörige Eigenwertproblem. Wir definieren höhere Eigenlösungen mittels weak slope und weisen die Existenz einer Folge von Eigenlösungen nach, indem wir die nichtglatte Theorie kritischer Punkte anwenden. Zusätzlich leiten wir eine neue notwendige Bedingung für den ersten Eigenwert des 1-Laplace-Operators mittels innerer Variationen her. 1-Laplace nonsmooth critical point theory weak slope inner variations 1-Laplace nichtglatte Theorie kritischer Punkte weak slope innere Variationen ddc:510 rvk:SK 620
2	The Eigenvalue Problem of the 1-Laplace Operator Littig, Samuel 19 February 2015 (has links) (PDF) As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered. 1-Laplace Operator Nichtglatte Theorie kritischer Punkte Geschlecht Störungsresultate Bifurkation Vektorwertiger Totalvariationsfluss Symmetrisierter 1-Laplace Operator BD 1-Laplace operator nonsmooth critical point theory genus perturbation bifurcation vectorial total variation flow symmetrized 1-Laplace operator BD ddc:510 rvk:SK 620
3	Rational Krylov Methods for Operator Functions Güttel, Stefan 26 March 2010 (has links) (PDF) We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included. rational Krylov operator functions decompositions approximations Rayleigh-Ritz method PAIN method hybrid methods error estimators inexact solves parallel rational Arnoldi parameter optimization potential theory superlinear convergence rationale Krylow-Verfahren Operatorfunktionen Krylow-Zerlegungen Krylow-Approximationen Rayleigh-Ritz Methode PAIN Methode hybride Krylow-Verfahren Fehlerschätzer inexakte Gleichungslöser Parallelisierung optimale Parameter Potenzialtheorie ddc:510 rvk:SK 620 rvk:SK 915

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