On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat "hat". In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M -> M' between two globally minimal real analytic hypersurfaces in C^n (n > 1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish that the reflection function R_h' associated to such a C^infty-smooth CR diffeomorphism between two globally minimal hypersurfaces in C^n always extends holomorphically to a neighborhood of the graph of \bar h in M \times \overline M', without any nondegeneracy condition on M'. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every C^infty-smooth CR mapping h: M to M' between two real analytic hypersurfaces containing no complex curves is real analytic at every point of M, without any rank condition on h

An efficient algorithm for the ℓp\ell_{p} norm based metric nearness problem

Given a dissimilarity matrix, the metric nearness problem is to find the nearest matrix of distances that satisfy the triangle inequalities. This problem has wide applications, such as sensor networks, image processing, and so on. But it is of great challenge even to obtain a moderately accurate solution due to the $O(n^{3})$ metric constraints and the nonsmooth objective function which is usually a weighted $\ell_{p}$ norm based distance. In this paper, we propose a delayed constraint generation method with each subproblem solved by the semismooth Newton based proximal augmented Lagrangian method (PALM) for the metric nearness problem. Due to the high memory requirement for the storage of the matrix related to the metric constraints, we take advantage of the special structure of the matrix and do not need to store the corresponding constraint matrix. A pleasing aspect of our algorithm is that we can solve these problems involving up to $10^{8}$ variables and $10^{13}$ constraints. Numerical experiments demonstrate the efficiency of our algorithm. In theory, firstly, under a mild condition, we establish a primal-dual error bound condition which is very essential for the analysis of local convergence rate of PALM. Secondly, we prove the equivalence between the dual nondegeneracy condition and nonsingularity of the generalized Jacobian for the inner subproblem of PALM. Thirdly, when $q(\cdot)=\|\cdot\|_{1}$ or $\|\cdot\|_{\infty}$, without the strict complementarity condition, we also prove the equivalence between the the dual nondegeneracy condition and the uniqueness of the primal solution

Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems

A semianalytical method is derived for finding the existence and stability of single-impact periodicorbits born in a boundary equilibrium bifurcation in a generaln-dimensional impacting hybridsystem. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3Dexamples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how themethod can easily be embedded within numerical continuation, and some remarks are made aboutnecessary and sufficient conditions in arbitrary dimensional system

Nonlinear resonance and excitability in interconnected systems

Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications. The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods. The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity

COMPASS

Die vorliegende Arbeit präsentiert COMPASS, einen global konvergenten Lösungsalgorithmus für gemischte Komplementaritätsprobleme. Die zu Grunde liegende math ematische Theorie basiert auf dem PATH Solver, dem standard Lösungsalgorithmus für diese Art von Problemen. COMPASS ist unter der “GNU General Public License” Lizenz veröffentlicht und ist daher Freie Software. Das Fundament von COMPASS ist eine stabilisierte Newton Methode: Das gemischte Komplementaritätsproblem wird in der Form der Normalgleichung (normal equation) reformuliert. Eine allgemeine Approximation erster Ordnung dieser Normalgleichung kann als lineares gemischtes Komplementaritätsproblem dargestellt und mit Hilfe einer Pivot Technik gelöst werden. Diese Lösung entspricht dem Newton Punkt im standard Newton Verfahren, und wird daher hier auch so bezeichnet. In der Pivot Technik wird neben der Lösung auch ein stückweise linearer Pfad generiert, der den letzten Iterationspunkt und den Newton Punkt verbindet. Ob dieser Punkt als nächster Iterationspunkt akzeptiert wird hängt von einem nicht-monotonen Stabilisierungsverfahren ab, das eine Watchdog Technik beinhaltet. Außerdem existiert eine glatte “merit” Funktion, basierend auf einer modifizierten Fischer-Burmeister Funktion, die den Erfolg des Fortschritt misst, und bei der Lösung des Komplementaritätsproblems eine Nullstelle besitzt. Gemäß der Verbesserung des Wertes dieser merit Funktion sind gewisse nicht monotone Abstiegskriterien definiert. Diese werden jedoch nicht in jedem Schritt getestet, um die Anzahl der Funktions- und Gradientenauswertungen zu minimieren. Wenn die Lösung aus der Pivot Technik, also der Newton Punkt, diesen Abstiegskriterien genügt, wird er als neuer Iterationspunkt verwendet. Falls nicht, geht der Algorithmus zurück zum letzten “Checkpoint”, dem letzten Punkt, der dem Test mit dem Abstiegskriterium erfolgreich unterzogen wurde. Der Pfad zwischen diesem Checkpoint, und dem Newton Punkt nach diesem Checkpoint (der Newton Punkt wird nach jedem Checkpoint gespeichert) wird dann nach einem die Abstiegskriterien erfüllenden Punkt durchsucht. Sollte kein passender Punkt gefunden werden, geht der Algorithmus zurück zum “Bestpoint”, dem Punkt mit dem bisher niedrigsten Wert der merit Funktion, und macht einen projizierten Gradientenschritt. Ein globaler Konvergenzbeweis dieser Theorie ist in der Arbeit enthalten. Der Algorithmus wurde im Zuge dieser Arbeit in MATLAB/Octave implementiert, und steht auf http://www.mat.univie.ac.at/~neum/software/compass/COMPASS.html zum Download und zur freien Benutzung zur Verfügung. Eine Simulation wurde anhand von zufällig generierten Problemen durchgeführt und dokumentiert das erfolgreiche Lösen von Problemen des Algorithmus zumindest bis zu einer Größenordnung von 200 Variablen. Eine kurze geschichtliche Einführung über den Zusammenhang zwischen gemischten Komplementaritätsproblemen und ökonomischen Modellen ist in der Arbeit enthalten, sowie eine Anleitung anhand eines Beispiels, wie solche Modelle in der Form von Komplementaritätsproblemen forumliert werden können.This thesis presents COMPASS, a globally convergent algorithm for solving the Mixed Complementarity Problem (MCP). The mathematical theory behind it is based on the PATH solver, the standard solver for complementarity problems. The fundament of COMPASS is a stabilized Newton method; the MCP is reformulated as the problem of finding a zero of a non-smooth vector valued function, the normal equation. A pivot technique is used to solve a general first order approximation of the normal equation at the current point in order to obtain a piecewise linear path connecting consecutive iterates. A general descent framework uses a smooth merit function establishing non-monotone descent criteria; a non-monotone stabilization scheme employs a watchdog technique and pathsearches reducing the number of function and gradient evaluations. An implementation in MATLAB/Octave was developed and is an integral part of this thesis. Simulation results on random problems, as well as a short course on economic models as an example of a field of application are included

Nonlinear stability and ergodicity of ensemble based Kalman filters

The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are data assimilation methods used to combine high dimensional, nonlinear dynamical models with observed data. Despite their widespread usage in climate science and oil reservoir simulation, very little is known about the long-time behavior of these methods and why they are effective when applied with modest ensemble sizes in large dimensional turbulent dynamical systems. By following the basic principles of energy dissipation and controllability of filters, this paper establishes a simple, systematic and rigorous framework for the nonlinear analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the dynamical properties of boundedness and geometric ergodicity. The time uniform boundedness guarantees that the filter estimate will not diverge to machine infinity in finite time, which is a potential threat for EnKF and ESQF known as the catastrophic filter divergence. Geometric ergodicity ensures in addition that the filter has a unique invariant measure and that initialization errors will dissipate exponentially in time. We establish these results by introducing a natural notion of observable energy dissipation. The time uniform bound is achieved through a simple Lyapunov function argument, this result applies to systems with complete observations and strong kinetic energy dissipation, but also to concrete examples with incomplete observations. With the Lyapunov function argument established, the geometric ergodicity is obtained by verifying the controllability of the filter processes; in particular, such analysis for ESQF relies on a careful multivariate perturbation analysis of the covariance eigen-structure.Comment: 38 page