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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.

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.:List of Figures xi List of Tables xiii List of Algorithms xv List of Acronyms xvii List of Symbols xix Publications xxi 1 Introduction 1 1.1 Notation 2 1.2 Structure of this Thesis 3 2 Basics 5 2.1 Linear Algebra and Eigenvalues 6 2.1.1 The Eigenvalue Problem 7 2.1.2 Dense Matrix Algorithms 9 2.2 Integral Operators and Integral Equations 14 2.2.1 Definitions 14 2.2.2 Example - BEM 16 2.3 Introduction to Hierarchical Arithmetic 17 2.3.1 Main Idea 17 2.3.2 Definitions 19 2.3.3 Hierarchical Arithmetic 24 2.3.4 Simple Hierarchical Matrices (Hl-Matrices) 30 2.4 Examples 33 2.4.1 FEM Example 33 2.4.2 BEM Example 36 2.4.3 Randomly Generated Examples 37 2.4.4 Application Based Examples 38 2.4.5 One-Dimensional Integral Equation 38 2.5 Related Matrix Formats 39 2.5.1 H2-Matrices 40 2.5.2 Diagonal plus Semiseparable Matrices 40 2.5.3 Hierarchically Semiseparable Matrices 42 2.6 Review of Existing Eigenvalue Algorithms 44 2.6.1 Projection Method 44 2.6.2 Divide-and-Conquer for Hl(1)-Matrices 45 2.6.3 Transforming Hierarchical into Semiseparable Matrices 46 2.7 Compute Cluster Otto 47 3 QR Decomposition of Hierarchical Matrices 49 3.1 Introduction 49 3.2 Review of Known QR Decompositions for H-Matrices 50 3.2.1 Lintner’s H-QR Decomposition 50 3.2.2 Bebendorf’s H-QR Decomposition 52 3.3 A new Method for Computing the H-QR Decomposition 54 3.3.1 Leaf Block-Column 54 3.3.2 Non-Leaf Block Column 56 3.3.3 Complexity 57 3.3.4 Orthogonality 60 3.3.5 Comparison to QR Decompositions for Sparse Matrices 61 3.4 Numerical Results 62 3.4.1 Lintner’s H-QR decomposition 62 3.4.2 Bebendorf’s H-QR decomposition 66 3.4.3 The new H-QR decomposition 66 3.5 Conclusions 67 4 QR-like Algorithms for Hierarchical Matrices 69 4.1 Introduction 70 4.1.1 LR Cholesky Algorithm 70 4.1.2 QR Algorithm 70 4.1.3 Complexity 71 4.2 LR Cholesky Algorithm for Hierarchical Matrices 72 4.2.1 Algorithm 72 4.2.2 Shift Strategy 72 4.2.3 Deflation 73 4.2.4 Numerical Results 73 4.3 LR Cholesky Algorithm for Diagonal plus Semiseparable Matrices 75 4.3.1 Theorem 75 4.3.2 Application to Tridiagonal and Band Matrices 79 4.3.3 Application to Matrices with Rank Structure 79 4.3.4 Application to H-Matrices 80 4.3.5 Application to Hl-Matrices 82 4.3.6 Application to H2-Matrices 83 4.4 Numerical Examples 84 4.5 The Unsymmetric Case 84 4.6 Conclusions 88 5 Slicing the Spectrum of Hierarchical Matrices 89 5.1 Introduction 89 5.2 Slicing the Spectrum by LDLT Factorization 91 5.2.1 The Function nu(M − µI) 91 5.2.2 LDLT Factorization of Hl-Matrices 92 5.2.3 Start-Interval [a, b] 96 5.2.4 Complexity 96 5.3 Numerical Results 97 5.4 Possible Extensions 100 5.4.1 LDLT Slicing Algorithm for HSS Matrices 103 5.4.2 LDLT Slicing Algorithm for H-Matrices 103 5.4.3 Parallelization 105 5.4.4 Eigenvectors 107 5.5 Conclusions 107 6 Computing Eigenvalues by Vector Iterations 109 6.1 Power Iteration 109 6.1.1 Power Iteration for Hierarchical Matrices 110 6.1.2 Inverse Iteration 111 6.2 Preconditioned Inverse Iteration for Hierarchical Matrices 111 6.2.1 Preconditioned Inverse Iteration 113 6.2.2 The Approximate Inverse of an H-Matrix 115 6.2.3 The Approximate Cholesky Decomposition of an H-Matrix 116 6.2.4 PINVIT for H-Matrices 117 6.2.5 The Interior of the Spectrum 120 6.2.6 Numerical Results 123 6.2.7 Conclusions 130 7 Comparison of the Algorithms and Numerical Results 133 7.1 Theoretical Comparison 133 7.2 Numerical Comparison 135 8 Conclusions 141 Theses 143 Bibliography 145 Index 153

Die vorliegende Arbeit zum Thema der numerischen Analysis von Eigenwertproblemen befasst sich mit fünf wesentlichen Aspekten der numerischen Analysis von Eigenwertproblemen. Der erste Teil präsentiert einen Algorithmus von asymptotisch quasi-optimaler Rechenlaufzeit, der die adaptive Finite Elemente Methode mit einem iterativen algebraischen Eigenwertlöser kombiniert. Der zweite Teil präsentiert explizite beidseitige Schranken für die Eigenwerte des Laplace Operators auf beliebig groben Gittern basierend auf einer Approximation der zugehörigen Eigenfunktion in dem nicht konformen Finite Elemente Raum von Crouzeix und Raviart und einem Postprocessing. Die Effizienz der garantierten Schranke des Eigenwertfehlers hängt von der globalen Gitterweite ab. Der dritte Teil betrachtet eine adaptive Finite Elemente Methode basierend auf Verfeinerungen von Knoten-Patchen. Dieser Algorithmus zeigt eine asymptotische Fehlerreduktion der adaptiven Sequenz von einfachen Eigenwerten und Eigenfunktionen des Laplace Operators. Die hier erstmals bewiesene Eigenschaft der Saturation des Eigenwertfehlers zeigt Zuverlässigkeit und Effizienz für eine Klasse von hierarchischen a posteriori Fehlerschätzern. Der vierte Teil betrachtet a posteriori Fehlerschätzer für Konvektion-Diffusion Eigenwertprobleme, wie sie von Heuveline und Rannacher (2001) im Kontext der dual-gewichteten residualen Methode (DWR) diskutiert wurden. Zwei neue dual-gewichtete a posteriori Fehlerschätzer werden vorgestellt. Der letzte Teil beschäftigt sich mit drei adaptiven Algorithmen für Eigenwertprobleme von nicht selbst-adjungierten Operatoren partieller Differentialgleichungen. Alle drei Algorithmen basieren auf einer Homotopie-Methode die vom einfacheren selbst-adjungierten Problem startet. Neben der Gitterverfeinerung wird der Prozess der Homotopie sowie die Anzahl der Iterationen des algebraischen Löser adaptiv gesteuert und die verschiedenen Anteile am gesamten Fehler ausbalanciert. / This thesis "on the numerical analysis of eigenvalue problems" consists of five major aspects of the numerical analysis of adaptive finite element methods for eigenvalue problems. The first part presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for a symmetric eigenvalue problem of asymptotic quasi-optimal computational complexity. The second part introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. The third part presents an adaptive finite element method (AFEM) based on nodal-patch refinement that leads to an asymptotic error reduction property for the adaptive sequence of simple eigenvalues and eigenfunctions of the Laplace operator. The proven saturation property yields reliability and efficiency for a class of hierarchical a posteriori error estimators. The fourth part considers a posteriori error estimators for convection-diffusion eigenvalue problems as discussed by Heuveline and Rannacher (2001) in the context of the dual-weighted residual method (DWR). Two new dual-weighted a posteriori error estimators are presented. The last part presents three adaptive algorithms for eigenvalue problems associated with non-selfadjoint partial differential operators. The basis for the developed algorithms is a homotopy method which departs from a well-understood selfadjoint problem. Apart from the adaptive grid refinement, the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources.

Gegenstand dieser Arbeit ist die numerische Approximation von Eigenwerten elliptischer Differentialoperatoren vermittels der adaptiven finite-Elemente-Methode (AFEM). Durch lokale Netzverfeinerung können derartige Verfahren den Rechenaufwand im Vergleich zu uniformer Verfeinerung deutlich reduzieren und sind daher von großer praktischer Bedeutung. Diese Arbeit behandelt adaptive Algorithmen für Finite-Elemente-Methoden (FEMs) für drei selbstadjungierte Modellprobleme: den Laplaceoperator, das Stokes-System und den biharmonischen Operator. In praktischen Anwendungen führen Störungen der Koeffizienten oder der Geometrie auf Eigenwert-Haufen (Cluster). Dies macht simultanes Markieren im adaptiven Algorithmus notwendig. In dieser Arbeit werden optimale Konvergenzraten für einen praktischen adaptiven Algorithmus für Eigenwert-Cluster des Laplaceoperators (konforme und nichtkonforme P1-FEM), des Stokes-Systems (nichtkonforme P1-FEM) und des biharmonischen Operators (Morley-FEM) bewiesen. Fehlerabschätzungen in der L2-Norm und Bestapproximations-Resultate für diese Nichtstandard-Methoden erfordern neue Techniken, die in dieser Arbeit entwickelt werden. Dadurch wird der Beweis optimaler Konvergenzraten ermöglicht. Die Optimalität bezüglich einer nichtlinearen Approximationsklasse betrachtet die Approximation des invarianten Unterraums, der von den Eigenfunktionen im Cluster aufgespannt wird. Der Fehler der Eigenwerte kann dazu in Bezug gesetzt werden: Die hierfür notwendigen Eigenwert-Fehlerabschätzungen für nichtkonforme Finite-Elemente-Methoden werden in dieser Arbeit gezeigt. Die numerischen Tests für die betrachteten Modellprobleme legen nahe, dass der vorgeschlagene Algorithmus, der bezüglich aller Eigenfunktionen im Cluster markiert, einem Markieren, das auf den Vielfachheiten der Eigenwerte beruht, überlegen ist. So kann der neue Algorithmus selbst im Fall, dass alle Eigenwerte im Cluster einfach sind, den vorasymptotischen Bereich signifikant verringern. / The numerical approximation of the eigenvalues of elliptic differential operators with the adaptive finite element method (AFEM) is of high practical interest because the local mesh-refinement leads to reduced computational costs compared to uniform refinement. This thesis studies adaptive algorithms for finite element methods (FEMs) for three model problems, namely the eigenvalues of the Laplacian, the Stokes system and the biharmonic operator. In practice, little perturbations in coefficients or in the geometry immediately lead to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. This thesis proves optimality of a practical adaptive algorithm for eigenvalue clusters for the conforming and nonconforming P1 FEM for the eigenvalues of the Laplacian, the nonconforming P1 FEM for the eigenvalues of the Stokes system and the Morley FEM for the eigenvalues of the biharmonic operator. New techniques from the medius analysis enable the proof of L2 error estimates and best-approximation properties for these nonstandard finite element methods and thereby lead to the proof of optimality. The optimality in terms of the concept of nonlinear approximation classes is concerned with the approximation of invariant subspaces spanned by eigenfunctions of an eigenvalue cluster. In order to obtain eigenvalue error estimates, this thesis presents new estimates for nonconforming finite elements which relate the error of the eigenvalue approximation to the error of the approximation of the invariant subspace. Numerical experiments for the aforementioned model problems suggest that the proposed practical algorithm that uses marking with respect to all eigenfunctions within the cluster is superior to marking that is based on the multiplicity of the eigenvalues: Even if all exact eigenvalues in the cluster are simple, the simultaneous approximation can reduce the pre-asymptotic range significantly.

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.

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