Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs

Relational datasets are often modeled as an unsigned, undirected graph due the nice properties of the resulting graph Laplacian, but information is lost if certain attributes of the graph are not represented. This thesis presents two generalizations of Algebraic Multigrid (AMG) solvers with graph Laplacian systems for different graph types: applying Gremban’s expansion to extend unsigned graph Laplacian solvers to signed graph Laplacian systems and generalizing techniques in Lean Algebraic Multigrid (LAMG) to a new multigrid solver for unsigned, directed graph Laplacian systems.Signed graphs extend the traditional notion of connections and disconnections to in- clude both favorable and adverse relationships, such as friend-enemy social networks or social networks with “likes” and “dislikes.” Gremban’s expansion is used to transform the signed graph Laplacian into an unsigned graph Laplacian with twice the number of unknowns. By using Gremban’s expansion, we extend current unsigned graph Laplacian solvers’ to signed graph Laplacians. This thesis analyzes the numerical stability and applicability of Grem- ban’s expansion and proves that the error of the solution of the original linear system can be tightly bounded by the error of the expanded system.In directed graphs, some subset of relationships are not reciprocal, such as hyperlink graphs, biological neural networks, and electrical power grids. A new algebraic multigrid algorithm, Nonsymmetric Lean Algebraic Multigrid (NS-LAMG), is proposed, which uses ideas from Lean Algebraic Multigrid, nonsymmetric Smoothed Aggregation, and multigrid solvers for Markov chain stationary distribution systems. Low-degree elimination, intro- duced in Lean Algebraic Multigrid for undirected graphs, is redefined for directed graphs.A semi-adaptive multigrid solver, inspired by low-degree elimination, is instrumented in the setup phase, which can be adapted for Markov chain stationary distributions systems. Nu- merical results shows that NS-LAMG out performs GMRES(k) for real-world, directed graph Laplacian linear systems. Both generalizations enable more choices in modeling decisions for graph Laplacian systems.Due the successfulness of NS-LAMG and other various nonsymmetric AMG (NS-AMG) solvers, a further study of theoretical convergence properties are discussed in this thesis. In particular, a necessary condition known as “weak approximation property”, and a sufficient one, referred to as “strong approximation property” as well as the “super strong approx- imation property” are generalized to nonsymmetric matrices and the various relationships between the approximation properties are proved for the nonsymmetric case. In NS-AMG, if P ̸= R the two-grid error propagation operator for the coarse-grid correction is an oblique projection with respect to any reasonable norm, which can cause the error to increase. A main focal point of this paper is a discussion on the conditions in which the error propagation operator is bounded, as the stability of the error propagation operator and the approxima- tion properties play an important role in proving convergence of the two-grid method for NS-AMG, which was studied in [37]

Bridging the Theory-Practice Gap of Laplacian Linear Solvers

Solving Laplacian linear systems is an important task in a variety of practical and theoretical applications. Laplacians of structured graphs, such as two and three dimensional meshes, have long been important in finite element analysis and image processing. More recently, solving linear systems on the Laplacians of large graphs without mesh-like structure has emerged as an important computational task in network analysis. A number of theoretical solvers with good asymptotic complexity have been proposed over the past couple decades, but these ideas have not made their way into practical solvers. Nor is it clear that a class of challenging problems exist which would benefit from asymptotically fast solvers. Yet it seems that one of the following should be true: either existing solvers have tighter Big-O bounds than currently believed, or there are some problems where recent asymptotically fast (but theoretical) algorithms should be useful.This work considers the latter possibility; we aim to bridge the gap between theoretical and practical Laplacian algorithms by experimenting with Laplacian solvers and by searching for difficult test problems. We examine the performance of existing algorithms for solving Laplacian linear systems and identify the strengths and weaknesses of different methods on different test problems. We perform an extensive evaluation of the KOSZ solver, one of the recently proposed Õ(m) Laplacian algorithms. We test various extensions of KOSZ which we propose to try and improve its performance in practice. We introduce heavy path graphs, a novel class of graphs for experimenting with Laplacian solvers.To challenge existing solver implementations, we propose the use of genetic algorithms to create difficult test graphs for existing solvers. At the same time, these algorithms could be used to find graphs with good performance for recently proposed solvers. Searching for graphs which satisfy both objectives could be instrumental towards bridging the theory-practice gap of Laplacian solvers. We demonstrate the successful evolution of graphs which are difficult for conjugate gradient with diagonal scaling, while relatively simple for KOSZ. Such graph evolution techniques could be useful for finding graphs with a variety of combinatorial properties

Algebraic Multigrid for Meshfree Methods

This thesis deals with the development of a new Algebraic Multigrid method (AMG) for the solution of linear systems arising from Generalized Finite Difference Methods (GFDM). In particular, we consider the Finite Pointset Method, which is based on GFDM. Being a meshfree method, FPM does not rely on a mesh and can therefore deal with moving geometries and free surfaces is a natural way and it does not require the generation of a mesh before the actual simulation. In industrial use cases the size of the linear systems often becomes large, which means that classical linear solvers often become the bottleneck in terms of simulation run time, because their convergence rate depends on the discretization size. Multigrid methods have proven to be very efficient linear solvers in the domain of mesh-based methods. Their convergence is independent of the discretization size, yielding a run time that only scales linearly with the problem size. AMG methods are a natural candidate for the solution of the linear systems arising in the FPM, as this thesis will show. They need to be tuned to the specific characteristics of GFDM, though. The AMG methods that are developed in this thesis achieve a speed-up of up to 33x compared to the classical linear solvers and therefore allow much more accurate simulations in the future.Diese Dissertation beschäftigt sich mit der Entwicklung einer neuen Algebraischen Mehrgittermethode für die Lösung linearer Gleichungssysteme aus Generalisierten Finite Differenzen Methoden. Im Speziellen betrachten wir die sogenannte Finite Pointset Method, eine gitterfreie Lagrange Methode, welche auf Generalisierten Finite Differenzen Methoden basiert. Die Finite Pointset Method wurde insbesondere für Simulationen von Vorgängen mit freien Oberflächen und bewegten Geometrien entwickelt, bei denen der gitterfreie Charakter der Methode besonders große Vorteile liefert: An den freien Oberflächen und nahe der Geometrie muss zu keinem Zeitpunkt – auch nicht zu Beginn der Simulation – ein Gitter erstellt oder angepasst werden. Dies ist ein großer Vorteil gegenüber klassischen gitterbasierten Methoden. Wie in gitterbasierten Methoden entstehen auch in der Finite Pointset Method und anderen Generalisierten Finite Differenzen Methoden große, dünn besetze lineare Gleichungssysteme. Das Lösen dieser Gleichungssysteme wird bei fein aufgelösten Simulationen, wie sie in der Industrie oft nötig sind, schnell zum zeitlichen Flaschenhals der Gesamtsimulation. Ohne eine geeignete Methode zur Lösung dieser Gleichungssysteme dauern Simulationen oft sehr lange oder sind praktisch nicht durchführbar. Auch kann es vorkommen, dass klassische Lösungsverfahren divergieren und die Simulation damit unmöglich wird. Im Kontext von gitterbasierten Methoden sind Mehrgittermethoden ein etabliertes Werkzeug, um die entstehenden linearen Gleichungssysteme effizient und robust zu lösen. Besonders hervorzuheben ist dabei die lineare Skalierbarkeit dieser Methoden in der Größe der Matrix. Damit eignen sie sich besonders für fein aufgelöste Simulationen. Algebraische Mehrgittermethoden sind natürliche Kandidaten für die Lösung der Gleichungssysteme aus Generalisierten Finite Differenzen Methoden, wie diese Dissertation zeigen wird. Außerdem entwickeln wir eine neue Algebraische Mehrgittermethode, die auf den Einsatz in der Finite Pointset Method zugeschnitten ist und die Besonderheiten dieser Methode beachtet. Dazu zählen die Eigenschaften der einzelnen Matrizen, die wir ebenfalls analysieren werden, und auch die Veränderung der Matrizen über mehrere Zeitschritte hinweg, die im Vergleich mit gitterbasierten Verfahren eine größere Schwierigkeit darstellt. Wir evaluieren unsere neue Methode anhand von akademischen und realen Beispielen, sowohl mit nur einem Prozess als auch mit mehreren (MPI-)Prozessen. Die hier neu entwickelte Algebraische Mehrgittermethode ist um ein Vielfaches schneller als klassische Verfahren zur Lösung linearer Gleichungssysteme und erlaubt damit neue, genauere Simulationen mit gitterfreien Methoden

Scalable Algorithms for the Analysis of Massive Networks

Die Netzwerkanalyse zielt darauf ab, nicht-triviale Erkenntnisse aus vernetzten Daten zu gewinnen. Beispiele für diese Erkenntnisse sind die Wichtigkeit einer Entität im Verhältnis zu anderen nach bestimmten Kriterien oder das Finden des am besten geeigneten Partners für jeden Teilnehmer eines Netzwerks - bekannt als Maximum Weighted Matching (MWM). Da der Begriff der Wichtigkeit an die zu betrachtende Anwendung gebunden ist, wurden zahlreiche Zentralitätsmaße eingeführt. Diese Maße stammen hierbei aus Jahrzehnten, in denen die Rechenleistung sehr begrenzt war und die Netzwerke im Vergleich zu heute viel kleiner waren. Heute sind massive Netzwerke mit Millionen von Kanten allgegenwärtig und eine triviale Berechnung von Zentralitätsmaßen ist oft zu zeitaufwändig. Darüber hinaus ist die Suche nach der Gruppe von k Knoten mit hoher Zentralität eine noch kostspieligere Aufgabe. Skalierbare Algorithmen zur Identifizierung hochzentraler (Gruppen von) Knoten in großen Graphen sind von großer Bedeutung für eine umfassende Netzwerkanalyse. Heutigen Netzwerke verändern sich zusätzlich im zeitlichen Verlauf und die effiziente Aktualisierung der Ergebnisse nach einer Änderung ist eine Herausforderung. Effiziente dynamische Algorithmen sind daher ein weiterer wesentlicher Bestandteil moderner Analyse-Pipelines. Hauptziel dieser Arbeit ist es, skalierbare algorithmische Lösungen für die zwei oben genannten Probleme zu finden. Die meisten unserer Algorithmen benötigen Sekunden bis einige Minuten, um diese Aufgaben in realen Netzwerken mit bis zu Hunderten Millionen von Kanten zu lösen, was eine deutliche Verbesserung gegenüber dem Stand der Technik darstellt. Außerdem erweitern wir einen modernen Algorithmus für MWM auf dynamische Graphen. Experimente zeigen, dass unser dynamischer MWM-Algorithmus Aktualisierungen in Graphen mit Milliarden von Kanten in Millisekunden bewältigt.Network analysis aims to unveil non-trivial insights from networked data by studying relationship patterns between the entities of a network. Among these insights, a popular one is to quantify the importance of an entity with respect to the others according to some criteria. Another one is to find the most suitable matching partner for each participant of a network knowing the pairwise preferences of the participants to be matched with each other - known as Maximum Weighted Matching (MWM). Since the notion of importance is tied to the application under consideration, numerous centrality measures have been introduced. Many of these measures, however, were conceived in a time when computing power was very limited and networks were much smaller compared to today's, and thus scalability to large datasets was not considered. Today, massive networks with millions of edges are ubiquitous, and a complete exact computation for traditional centrality measures are often too time-consuming. This issue is amplified if our objective is to find the group of k vertices that is the most central as a group. Scalable algorithms to identify highly central (groups of) vertices on massive graphs are thus of pivotal importance for large-scale network analysis. In addition to their size, today's networks often evolve over time, which poses the challenge of efficiently updating results after a change occurs. Hence, efficient dynamic algorithms are essential for modern network analysis pipelines. In this work, we propose scalable algorithms for identifying important vertices in a network, and for efficiently updating them in evolving networks. In real-world graphs with hundreds of millions of edges, most of our algorithms require seconds to a few minutes to perform these tasks. Further, we extend a state-of-the-art algorithm for MWM to dynamic graphs. Experiments show that our dynamic MWM algorithm handles updates in graphs with billion edges in milliseconds

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Pacific Symposium on Biocomputing 2023

The Pacific Symposium on Biocomputing (PSB) 2023 is an international, multidisciplinary conference for the presentation and discussion of current research in the theory and application of computational methods in problems of biological significance. Presentations are rigorously peer reviewed and are published in an archival proceedings volume. PSB 2023 will be held on January 3-7, 2023 in Kohala Coast, Hawaii. Tutorials and workshops will be offered prior to the start of the conference.PSB 2023 will bring together top researchers from the US, the Asian Pacific nations, and around the world to exchange research results and address open issues in all aspects of computational biology. It is a forum for the presentation of work in databases, algorithms, interfaces, visualization, modeling, and other computational methods, as applied to biological problems, with emphasis on applications in data-rich areas of molecular biology.The PSB has been designed to be responsive to the need for critical mass in sub-disciplines within biocomputing. For that reason, it is the only meeting whose sessions are defined dynamically each year in response to specific proposals. PSB sessions are organized by leaders of research in biocomputing's 'hot topics.' In this way, the meeting provides an early forum for serious examination of emerging methods and approaches in this rapidly changing field