Eintrag in der Universitätsbibliographie der TU Chemnitz
Volltext zugänglich unter
URN: urn:nbn:de:bsz:ch1-qucosa2-963856
Bergermann, Kai
Stoll, Martin ; Güttel, Stefan ; Tudisco, Francesco (Gutachter)
Matrix functions in multiplex network analysis
Kurzfassung in englisch
The fast and accurate approximation of matrix functions and their action on vectors is a core problem in numerical linear algebra. Prominently, the matrix exponential fueled research on matrix functions due to applications in differential equations. Krylov subspace methods are among the most efficient numerical methods for the approximation of the action of a matrix function on a vector. They produce (near-) optimal polynomial or rational approximations for a wide class of matrix functions by interpolating the scalar function at approximate eigenvalues of the matrix. Their computational efficiency hinges on fast matrix-vector products in the polynomial and the choice of suitable poles as well as fast linear system solves in the rational case. Complex systems from various scientific disciplines can be modeled by graphs or networks recording pairwise interactions between a finite set of entities. These complex networks possess natural linear algebraic representations and many fundamental problems in network science can be formulated as classical numerical linear algebra problems such as eigenvalue problems, linear systems, or matrix function expressions. In recent years, generalized network models such as multiplex networks, which record different types of relationships or interactions between the same entities in different layers have received considerable attention due to their increased flexibility in modeling complex real-world phenomena. Multiplex networks can be represented by structured matrices and the generalization of well-studied single-layer network methods to the multiplex case is an ongoing endeavor in the network science community. In this thesis, we consider several methods for the analysis of structural and dynamical network properties that rely on matrix function expressions and generalize them from single-layer to multiplex networks. We discuss centrality measures, the solution of stiff systems of non-linear differential equations with exponential Runge--Kutta integrators, as well as un- and semi-supervised community detection---putting a special focus on the efficient numerical treatment of the problems. Besides leveraging standard Krylov techniques, we present novel combinations of computational methods and advance state-of-the-art methodology. For instance, we prove an a-posteriori error estimate for rational Krylov approximations of the action of the matrix exponential on vectors, which enables a novel adaptive rational Krylov procedure with approximately constant iteration numbers and a near-linear scaling of the runtime with respect to the problem size. We present highly scalable linear-complexity methods for all considered problems, which allows insights into large-scale multiplex networks from social, transport, and imaging applications.
| Universität: | Technische Universität Chemnitz | |
| Institut: | Professur Wissenschaftliches Rechnen | |
| Fakultät: | Fakultät für Mathematik | |
| Dokumentart: | Dissertation | |
| Betreuer: | Stoll, Martin | |
| DOI: | doi:10.60687/2025-0055 | |
| SWD-Schlagwörter: | Matrixfunktion , Numerische lineare Algebra , Netzwerkanalyse , Netzwerk | |
| Freie Schlagwörter (Englisch): | matrix functions , numerical linear algebra , complex networks , multiplex networks | |
| DDC-Sachgruppe: | Numerische Analysis | |
| Sprache: | englisch | |
| Tag der mündlichen Prüfung | 06.03.2025 |
