This thesis is concerned with the finite element analysis and the a posteriori error estimation for eigenvalue problems for general operator pencils on two-dimensional manifolds. A specific application of the presented theory is the computation of corner singularities. Engineers use the knowledge of the so-called singularity exponents to predict the onset and the propagation of cracks. All results of this thesis are explained for two model problems, the Laplace and the linear elasticity problem, and verified by numerous numerical results.

This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach. By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem. Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established. Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed. Taking the symmetry of the graph into account, a particular optimal edge weighting exists. Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.

In large and complex systems, failures can have dramatic consequences, such as black-outs, pandemics or the loss of entire classes of an ecosystem. Nevertheless, it is a centuries-old intuition that by using networks to capture the core of the complexity of such systems, one might understand in which part of a system a phenomenon originates. I investigate this intuition using spectral methods to decouple the dynamics of complex systems near stationary states into independent dynamical modes. In this description, phenomena are tied to a specific part of a system through localized eigenvectors which have large amplitudes only on a few nodes of the system's network. Studying the occurrence of localized eigenvectors, I find that such localization occurs exactly for a few small network structures, and approximately for the dynamical modes associated with the most prominent failures in complex systems. My findings confirm that understanding the functioning of complex systems generally requires to treat them as complex entities, rather than collections of interwoven small parts. Exceptions to this are only few structures carrying exact localization, whose functioning is tied to the meso-scale, between the size of individual elements and the size of the global network. However, while understanding the functioning of a complex system is hampered by the necessary global analysis, the prominent failures, due to their localization, allow an understanding on a manageable local scale. Intriguingly, food webs might exploit this localization of failures to stabilize by causing the break-off of small problematic parts, whereas typical attempts to optimize technological systems for stability lead to delocalization and large-scale failures. Thus, this thesis provides insights into the interplay of complexity and localization, which is paramount to ascertain the functioning of the ever-growing networks on which we humans depend.:1 Introduction 2 Concepts and Tools 2.1 Networks 2.2 Food webs 2.3 Dynamics on networks 2.4 Steady state operating modes 2.5 Bifurcations affecting operating modes 2.6 Dynamical modes 2.7 Generalized models for food webs 3 Perturbation Impact 3.1 Impact of perturbations on food webs 3.2 Examples 3.3 Impact formulation with dynamical modes 3.4 Influence and sensitivity of species 3.5 Localized dynamical modes 3.6 Iterative parameter estimation 3.7 Most important parameters and species 3.8 Discussion 4 Exact Localization 4.1 Graph symmetries 4.2 Localized dynamics on symmetries 4.3 Exactly localized dynamics 4.4 Symmetry reduction in networks 4.5 Application to food webs 4.6 Localization on asymmetric structures 4.7 Nearly-exact localization 4.8 Other systems 4.9 Discussion 5 Approximate Localization 5.1 Spread of a dynamical mode 5.2 Examples for localized instabilities 5.3 Localization of extreme eigenvalues 5.4 Dependence on the system size 5.5 Localization in the model of R. May 5.6 Finding motifs that carry localization 5.7 (Self-)stabilization of food webs 5.8 Repairing localized instabilities 5.9 Discussion 6 Conclusions Acknowledgments Appendix A Parametrization of the Gatun Lake food web B The Master Stability Function approach C Approximate localization on larger structures Bibliography

In this work we calculate the binding energies and root-mean-square radii for A−body nuclear bound state systems, where A ≥ 3. To study three−body systems, we employ the three−dimensional differential Faddeev equations with nucleon-nucleon semi-realistic potentials. The equations are solved numerically. For this purpose, the equations are transformed into an eigenvalue equation via the orthogonal collocation procedure using triquintic Hermite splines. The resulting eigenvalue equation is solved using the Restarted Arnoldi Algorithm. Ground state binding energies of the 3H nucleus are determined. For A > 3, the Potential Harmonic Expansion Method is employed. Using this method, the Schr¨odinger equation is transformed into coupled Faddeev-like equations. The Faddeevlike amplitudes are expanded on the potential harmonic basis. To transform the resulting coupled differential equations into an eigenvalue equation, we employ again the orthogonal collocation procedure followed by the Gauss-Jacobi quadrature. The corresponding eigenvalue equation is solved using the Renormalized Numerov Method to obtain ground state binding energies and root-mean-square radii of closed shell nuclei 4He, 8Be, 12C, 16O and 40Ca. / Physics / M. Sc. (Physics)

In this thesis we present results about some eigenvalue statistics of the beta-Hermite ensembles, both in the classical cases corresponding to beta = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matrices) respectively. We also look at the less explored general beta-Hermite ensembles (consisting of real tridiagonal symmetric matrices). Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statistics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distributions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results. For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices. Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the beta-Hermite ensembles. / I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från beta-Hermite ensemblerna, först i de klassiska fallen då beta = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella symmetriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella beta-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser). Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika normeringar av det största egenvärdet. De resultat vi presenterar för dessa statistikor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illustrerar även dessa resultat med hjälp av simuleringar. För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess nivåtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemblen presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser. Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egenvärdena av beta-Hermite ensemblerna.

We consider the problem of maximizing the second smallest eigenvalue of the weighted Laplacian of a (simple) graph over all nonnegative edge weightings with bounded total weight. We generalize this problem by introducing node significances and edge lengths. We give a formulation of this generalized problem as a semidefinite program. The dual program can be equivalently written as embedding problem. This is fifinding an embedding of the n nodes of the graph in n-space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by the respective edge length, and the embedded nodes are spread as much as possible. (The sum of the squared norms is maximized.) We proof the following necessary condition for optimal embeddings. For any separator of the graph at least one of the components fulfills the following property: Each straight-line segment between the origin and an embedded node of the component intersects the convex hull of the embedded nodes of the separator. There exists always an optimal embedding of the graph whose dimension is bounded by the tree-width of the graph plus one. We defifine the rotational dimension of a graph. This is the minimal dimension k such that for all choices of the node significances and edge lengths an optimal embedding of the graph can be found in k-space. The rotational dimension of a graph is a minor monotone graph parameter. We characterize the graphs with rotational dimension up to two.:1 Introduction 1.1 Notations and Preliminaries 1.2 The Algebraic Connectivity 1.3 Two applications 1.4 Outline 2 The Embedding Problem 2.1 Semidefinite formulation 2.2 The dual as geometric embedding problem 2.3 Physical interpretation and examples 2.4 Formulation without fifixed barycenter 3 Geometrical Operations 3.1 Congruent transformations 3.2 Folding a flat halfspace 3.3 Folding and Collapsing 4 Structural properties of optimal embeddings 4.1 Separator-Shadow 4.2 Separators containing the origin 4.3 The tree-width bound 4.4 Application to trees 5 The Rotational Dimension of a graph 5.1 Defifinition and basic properties 5.2 Characterization of graphs with small rotational dimension 5.3 The Colin de Verdi ere graph parameter List of Figures Bibliography Theses

本論文は,様々なネットワーク最適化問題の数学的構造について様々な観点から調べたものである.主たる結果はネットワーク最適化問題の代表例である最大流問題に,関するいくつかの結果と,Min-Plus代数に値をもつ行列の固有値と固有ベクトルに関する特徴づけに関する結果からなっている. / 博士(理学) / Doctor of Philosophy in Science / 同志社大学 / Doshisha University

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[pt] Neste trabalho apresentam-se três tipos de técnicas de análise do colapso estrutural através do método dos elementos finitos: análise linearizada da carga crítica, análise incremental da carga crítica e análise não linear completa. Na análise linearizada da carga crítica formulou-se um problema de autovalor empregando matrizes de rigidez baseadas na configuração indeformada da estrutura e materiais com comportamento linear elástico. No caso da análise incremental da carga crítica, o problema de autovalor foi formulado empregando matrizes de rigidez incrementais para levar em consideração os grandes deslocamentos e propriedades não lineares do material. Finalmente, na análise não linear completa a configuração deformada da estrutura e propriedades não lineares do material são atualizadas durante todo o processo incremental-iterativo até atingir a carga crítica. Desenvolveu-se uma implementação computacional para estudar as três técnicas de análise em estruturas planas como vigas, colunas, pórticos e arcos, empregando elementos isoparamétricos bidimensionais para estado plano de tensões. A configuração deformada da estrutura, devido aos grandes deslocamentos e rotações dos elementos, foi considerada através de uma formulação Lagrangeana Total, enquanto o comportamento inelástico do material foi modelado empregando um modelo elastoplástico de Von Mises (J2) com encruamento isotrópico. Nos exemplos apresentados mostrou-se a influência da não linearidade geométrica e física na estimativa de cargas críticas e no comportamento pós-crítico, podendo ocorrer bifurcações ao longo da trajetória de equilíbrio fundamental definida no espaço carga-deslocamentos. / [en] This work presents three kinds of techniques for collapse analysis using the finite element method: linear buckling analysis, nonlinear buckling analysis and full nonlinear analysis. The linear buckling analysis requires the definition of an eigenvalue problem using a stiffness matrix formulation based on the initial configuration of the structure and under the assumption of a linear elastic material behavior. In the case of nonlinear buckling analysis, the eigenvalue problem was formulated employing an incremental stiffness matrix in order to consider the effects of large displacements and nonlinear material properties in the critical load estimation. Finally, the full nonlinear analysis takes into account the deformed configuration and the nonlinear material properties of the structure, updating both of them through all the incremental-iterative process up to reaching the critical load. A Finite Element computational program, using plane stress isoperimetric bidimensional elements, was developed to study the three analysis techniques applied to plane structures such as beams, columns, frames and arches. The deformed configuration of the structure, due to large displacements and rotations, was considered through the Total Lagrangian formulation, whereas the inelastic material behavior was modeled using the Von Mises plasticity model with isotropic hardening. The examples presented in this article show the influence of geometric and material nonlinearity in the critical load estimation and the postcritical behavior, being this the reason for the potential occurrence of bifurcation points over the fundamental equilibrium path defined in the load-displacement space.

This thesis is on the numerical computation of eigenvalues of symmetric hierarchical matrices. The numerical algorithms used for this computation are derivations of the LR Cholesky algorithm, the preconditioned inverse iteration, and a bisection method based on LDLT factorizations. The investigation of QR decompositions for H-matrices leads to a new QR decomposition. It has some properties that are superior to the existing ones, which is shown by experiments using the HQR decompositions to build a QR (eigenvalue) algorithm for H-matrices does not progress to a more efficient algorithm than the LR Cholesky algorithm. The implementation of the LR Cholesky algorithm for hierarchical matrices together with deflation and shift strategies yields an algorithm that require O(n) iterations to find all eigenvalues. Unfortunately, the local ranks of the iterates show a strong growth in the first steps. These H-fill-ins makes the computation expensive, so that O(n³) flops and O(n²) storage are required. Theorem 4.3.1 explains this behavior and shows that the LR Cholesky algorithm is efficient for the simple structured Hl-matrices. There is an exact LDLT factorization for Hl-matrices and an approximate LDLT factorization for H-matrices in linear-polylogarithmic complexity. This factorizations can be used to compute the inertia of an H-matrix. With the knowledge of the inertia for arbitrary shifts, one can compute an eigenvalue by bisectioning. The slicing the spectrum algorithm can compute all eigenvalues of an Hl-matrix in linear-polylogarithmic complexity. A single eigenvalue can be computed in O(k²n log^4 n). Since the LDLT factorization for general H-matrices is only approximative, the accuracy of the LDLT slicing algorithm is limited. The local ranks of the LDLT factorization for indefinite matrices are generally unknown, so that there is no statement on the complexity of the algorithm besides the numerical results in Table 5.7. The preconditioned inverse iteration computes the smallest eigenvalue and the corresponding eigenvector. This method is efficient, since the number of iterations is independent of the matrix dimension. If other eigenvalues than the smallest are searched, then preconditioned inverse iteration can not be simply applied to the shifted matrix, since positive definiteness is necessary. The squared and shifted matrix (M-mu I)² is positive definite. Inner eigenvalues can be computed by the combination of folded spectrum method and PINVIT. Numerical experiments show that the approximate inversion of (M-mu I)² is more expensive than the approximate inversion of M, so that the computation of the inner eigenvalues is more expensive. We compare the different eigenvalue algorithms. The preconditioned inverse iteration for hierarchical matrices is better than the LDLT slicing algorithm for the computation of the smallest eigenvalues, especially if the inverse is already available. The computation of inner eigenvalues with the folded spectrum method and preconditioned inverse iteration is more expensive. The LDLT slicing algorithm is competitive to H-PINVIT for the computation of inner eigenvalues. In the case of large, sparse matrices, specially tailored algorithms for sparse matrices, like the MATLAB function eigs, are more efficient. If one wants to compute all eigenvalues, then the LDLT slicing algorithm seems to be better than the LR Cholesky algorithm. If the matrix is small enough to be handled in dense arithmetic (and is not an Hl(1)-matrix), then dense eigensolvers, like the LAPACK function dsyev, are superior. The H-PINVIT and the LDLT slicing algorithm require only an almost linear amount of storage. They can handle larger matrices than eigenvalue algorithms for dense matrices. For Hl-matrices of local rank 1, the LDLT slicing algorithm and the LR Cholesky algorithm need almost the same time for the computation of all eigenvalues. For large matrices, both algorithms are faster than the dense LAPACK function dsyev.