Due to the increase in the advances in wireless communication, there has been an increase in the use of multi-agents systems to complete any given task. In various applications, multi-agent systems are required to solve an underlying optimization problem to obtain the best possible solution within a feasible region. Solving such multi-agent optimization problems in a distributed framework preferable over centralized frameworks as the former ensures scalability, robustness, and security. Further distributed optimization problem becomes challenging when the decision variables of the individual agents are coupled. In this thesis, a distributed optimization problem with coupled constraints is considered, where a network of agents aims to cooperatively minimize the sum of their local objective functions, subject to individual constraints. This problem setup is relevant to many practical applications like formation flying, sensor fusion, smart grids, etc. For practical scenarios, where agents can solve their local optimal solution efficiently and require fewer assumptions on objective functions, the Alternating Direction Method of Multipliers(ADMM)-based approaches are preferred over gradient-based approaches. For such a constraint coupled problem, several distributed ADMM algorithms are present that guarantee convergence to optimality but they do not discuss the complete analysis for the rate of convergence. Thus, the primary goal of this work is to improve upon the convergence rate of the existing state-of-the-art Tracking-ADMM (TADMM) algorithm to solve the above-distributed optimization problem. Moreover, the current analysis in literature does not discuss the convergence in the case of a time-varying communication network. The first part of the thesis focuses on improving the convergence rate of the Tracking-ADMM algorithm to solve the above-distributed optimization problem more efficiently. To this end, an upper bound on the convergence rate of the TADMM algorithm is derived in terms of the weight matrix of the network. To achieve faster convergence, the optimal weight matrix is computed using a semi-definite programming (SDP) formulation. The improved convergence rate of this Fast-TADMM (F-TADMM) is demonstrated with a simple yet illustrative, coupled constraint optimization problem. Then, the applicability of F-TADMM is demonstrated to the problem of distributed optimal control for trajectory generation of aircraft in formation flight. In the second part of the thesis, the convergence analysis for TADMM is extended while considering a time-varying communication network. The modified algorithm is named as Time-Varying Tracking (TV-TADMM). The formal guarantees on asymptotic convergence are provided with the help of control system analysis of a dynamical system that uses Lyapunov-like theory. The convergence of this TV-TADMM is demonstrated on a simple yet illustrative, coupled constraint optimization problem with switching topology and is compared with the fixed topology setting.

Indiana University-Purdue University Indianapolis (IUPUI) / The interplay between biomechanics and blood perfusion in the optic nerve head (ONH) has a critical role in ocular pathologies, especially glaucomatous optic neuropathy. Elucidating the complex interactions of ONH perfusion and tissue structure in health and disease using current imaging methodologies is difficult, and mathematical modeling provides an approach to address these limitations. The biophysical phenomena governing the ONH physiology occur at different scales in time and space and porous media theory provides an ideal framework to model them. We critically review fundamentals of porous media theory, paying particular attention to the assumptions leading to a continuum biphasic model for the phenomenological description of fluid flow through biological tissues exhibiting viscoelastic behavior. The resulting system of equations is solved via a numerical method based on a novel hybridizable discontinuous Galerkin finite element discretization that allows accurate approximations of stresses and discharge velocities, in addition to solid displacement and fluid pressure. The model is used to theoretically investigate the influence of tissue viscoelasticity on the blood perfusion of the lamina cribrosa in the ONH. Our results suggest that changes in viscoelastic properties of the lamina may compromise tissue perfusion in response to sudden variations of intraocular pressure, possibly leading to optic disc hemorrhages.

Eine Vielzahl von Anwendungen in der numerischen Simulation der Strömungsdynamik und der Festkörpermechanik begründen die Entwicklung von zuverlässigen und effizienten Algorithmen für nicht-standard Methoden der Finite-Elemente-Methode (FEM). Um Freiheitsgrade zu sparen, wird in jedem Durchlauf des adaptiven Algorithmus lediglich ein Teil der Gebiete verfeinert. Einige Gebiete bleiben daher möglicherweise verhältnismäßig grob. Die Analyse der Konvergenz und vor allem die der Optimalität benötigt daher über die a priori Fehleranalyse hinausgehende Argumente. Etablierte adaptive Algorithmen beruhen auf collective marking, d.h. die zu verfeinernden Gebiete werden auf Basis eines Gesamtfehlerschätzers markiert. Bei adaptiven Algorithmen mit separate marking wird der Gesamtfehlerschätzer in einen Volumenterm und in einen Fehlerschätzerterm aufgespalten. Da der Volumenterm unabhängig von der diskreten Lösung ist, kann einer schlechten Datenapproximation durch eine lokal tiefe Verfeinerung begegnet werden. Bei hinreichender Datenapproximation wird das Gitter dagegen bezüglich des neuen Fehlerschätzerterms wie üblich level-orientiert verfeinert. Die numerischen Experimente dieser Arbeit liefern deutliche Indizien der quasi-optimalen Konvergenz für den in dieser Arbeit untersuchten adaptiven Algorithmus, der auf separate marking beruht. Der Parameter, der die Verbesserung der Datenapproximation sicherstellt, ist frei wählbar. Dadurch ist es erstmals möglich, eine ausreichende und gleichzeitig optimale Approximation der Daten innerhalb weniger Durchläufe zu erzwingen. Diese Arbeit ermöglicht es, Standardargumente auch für die Konvergenzanalyse von Algorithmen mit separate marking zu verwenden. Dadurch gelingt es Quasi-Optimalität des vorgestellten Algorithmus gemäß einer generellen Vorgehensweise für die drei Beispiele, dem Poisson Modellproblem, dem reinen Verschiebungsproblem der linearen Elastizität und dem Stokes Problem, zu zeigen. / Various applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). To reduce the number of degrees of freedom, in adaptive algorithms only a selection of finite element domains is marked for refinement on each level. Since some element domains may stay relatively coarse, even the analysis of convergence and more importantly the analysis of optimality require new arguments beyond an a priori error analysis. In adaptive algorithms, based on collective marking, a (total) error estimator is used as refinement indicator. For separate marking strategies, the (total) error estimator is split into a volume term and an error estimator term, which estimates the error. Since the volume term is independent of the discrete solution, if there is a poor data approximation the improvement may be realised by a possibly high degree of local mesh refinement. Otherwise, a standard level-oriented mesh refinement based on an error estimator term is performed. This observation results in a natural adaptive algorithm based on separate marking, which is analysed in this thesis. The results of the numerical experiments displayed in this thesis provide strong evidence for the quasi-optimality of the presented adaptive algorithm based on separate marking and for all three model problems. Furthermore its flexibility (in particular the free steering parameter for data approximation) allows a sufficient and optimal data approximation in just a few number of levels of the adaptive scheme. This thesis adapts standard arguments for optimal convergence to adaptive algorithms based on separate marking with a possibly high degree of local mesh refinement, and proves quasi-optimality following a general methodology for three model problems, i.e., the Poisson model problem, the pure displacement problem in linear elasticity and the Stokes equations.

Die vorliegende Arbeit "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courant- und nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen. Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz. Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden. Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden. / The thesis "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes.

Garantierte untere Eigenwertschranken (GLB) für elliptische Eigenwertprobleme partieller Differentialgleichungen sind in der Theorie sowie in praktischen Anwendungen relevant. Auf Grund des Rayleigh-Ritz- (oder) min-max-Prinzips berechnen alle konformen Finite-Elemente-Methoden (FEM) garantierte obere Schranken. Ein Postprocessing nichtkonformer Methoden von Carstensen und Gedicke (Math. Comp., 83.290, 2014) sowie Carstensen und Gallistl (Numer. Math., 126.1, 2014) berechnet GLB. In diesen Schranken ist die maximale Netzweite ein globaler Parameter, das kann bei adaptiver Netzverfeinerung zu deutlichen Unterschätzungen führen. In einigen numerischen Beispielen versagt dieses Postprocessing für lokal verfeinerte Netze komplett. Diese Dissertation präsentiert, inspiriert von einer neuen skeletal-Methode von Carstensen, Zhai und Zhang (SIAM J. Numer. Anal., 58.1, 2020), einerseits eine modifizierte hybrid-high-order Methode (m=1) und andererseits ein allgemeines Framework für extra-stabilisierte nichtkonforme Crouzeix-Raviart (m=1) bzw. Morley (m=2) FEM. Diese neuen Methoden berechnen direkte GLB für den m-Laplace-Operator, bei denen eine leicht überprüfbare Bedingung an die maximale Netzweite garantiert, dass der k-te diskrete Eigenwert eine untere Schranke für den k-ten Dirichlet-Eigenwert ist. Diese GLB-Eigenschaft und a priori Konvergenzraten werden für jede Raumdimension etabliert. Der neu entwickelte Ansatz erlaubt adaptive Netzverfeinerung, die für optimale Konvergenzraten auch bei nichtglatten Eigenfunktionen erforderlich ist. Die Überlegenheit der neuen adaptiven FEM wird durch eine Vielzahl repräsentativer numerischer Beispiele illustriert. Für die extra-stabilisierte GLB wird bewiesen, dass sie mit optimalen Raten gegen einen einfachen Eigenwert konvergiert, indem die Axiome der Adaptivität von Carstensen, Feischl, Page und Praetorius (Comput. Math. Appl., 67.6, 2014) sowie Carstensen und Rabus (SIAM J. Numer. Anal., 55.6, 2017) verallgemeinert werden. / Guaranteed lower eigenvalue bounds (GLB) for elliptic eigenvalue problems of partial differential equation are of high relevance in theory and praxis. Due to the Rayleigh-Ritz (or) min-max principle all conforming finite element methods (FEM) provide guaranteed upper eigenvalue bounds. A post-processing for nonconforming FEM of Carstensen and Gedicke (Math. Comp., 83.290, 2014) as well as Carstensen and Gallistl (Numer. Math., 126.1,2014) computes GLB. However, the maximal mesh-size enters as a global parameter in the eigenvalue bound and may cause significant underestimation for adaptive mesh-refinement. There are numerical examples, where this post-processing on locally refined meshes fails completely. Inspired by a recent skeletal method from Carstensen, Zhai, and Zhang (SIAM J. Numer. Anal., 58.1, 2020) this thesis presents on the one hand a modified hybrid high-order method (m=1) and on the other hand a general framework for an extra-stabilized nonconforming Crouzeix-Raviart (m=1) or Morley (m=2) FEM. These novel methods compute direct GLB for the m-Laplace operator in that a specific smallness assumption on the maximal mesh-size guarantees that the computed k-th discrete eigenvalue is a lower bound for the k-th Dirichlet eigenvalue. This GLB property as well as a priori convergence rates are established in any space dimension. The novel ansatz allows for adaptive mesh-refinement necessary to recover optimal convergence rates for non-smooth eigenfunctions. Striking numerical evidence indicates the superiority of the new adaptive eigensolvers. For the extra-stabilized nonconforming methods (a generalization of) known abstract arguments entitled as the axioms of adaptivity from Carstensen, Feischl, Page, and Praetorius (Comput. Math. Appl., 67.6, 2014) as well as Carstensen and Rabus (SIAM J. Numer. Anal., 55.6, 2017) allow to prove the convergence of the GLB towards a simple eigenvalue with optimal rates.