Talks

Workshops

Seminars

  • Elena Isaak (Bielefeld) : Numerical analysis of the balanced Milstein method
    2013/12/13, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Balanced Milstein methods (BMM) have been proposed for solving numerically stochastic ordinary differential equations with large noise coefficients. In this talk we discuss consistency, bistability and convergence of the BMM. The main ingredient of the analysis is a stochastic version of Spijker's norm. We show that the order of consistency for the BMM in this norm is one, and we prove bistability which leads to two-sided estimates of the strong error of convergence.

  • Kathrin Glau (TU München): Kolmogorov backward equations for option pricing in Lévy models
    2013/12/09, 16:00, V5-148, Seminar Numerical Analysis
    Abstract: One major task mathematical finance sets itself is modeling, pricing and calibration of financial instruments. (Semi)martingale theory is used for modeling and derivative prices are written as conditional expectations. Typically, the latter are not available in closed form and, thus, computational methods become necessary. Essentially three approaches to compute the expectations are being used: Monte Carlo simulation, Fourier based valuation methods and the representation of prices as solutions of partial integro-differential equations (PIDEs). In this context we focus on Galerkin methods for solving PIDEs arising in Lévy models. We classify Lévy processes according to the solution spaces of the associated parabolic PIDEs and point out the role of the symbol. Furthermore, we derive Feynman-Kac representations of variational solutions. We discuss applications to option pricing and give an outlook on a Finite Element solver based on the symbol.

  • L´ubomír Banas (Bielefeld): Phase field models for multiphase flow: modelling, numerics and applications.
    2013/12/02, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Understanding and accurate prediction of multiphase multicomponent flows is of essential interest for a large number of scientific and engineering applications. Despite intensive past and present research efforts, it is still not clear how to accurately and efficiently simulate multiphase fluid flow for the full range of physical parameters and regimes such as, e.g., densities, viscosities, capillary relations, number of fluid phases, interface geometry, dynamic or static contact angles, etc. We review of a promising strategy for the modelling of incompressible multiphase flow based on the phase-field approach. We discuss advantages of the approach from the modelling and computational point of view. We also present a framework for multiscale flow simulations and discuss applications to multiphase flow in porous media.

  • Marian Slodicka (Gent): Inverse source problems in parabolic equations
    2013/11/29, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Inverse coefficient and source problems for partial differential equations represent a well-known and established area of mathematical research in the last decades. They appear in various applied technologies (geophysics, optic, tomography, remote sensing, radar-location, etc.). Inverse source problems for evolutionary (parabolic, hyperbolic, Navier-Stokes) settings have been intensively studied by many authors. We study a problem of source identification from given data for the parabolic heat equation in several dimensions. The temperature \(u\), heat source \(F\) and the initial temperature distribution \(u_0(x)\) then satisfy \[\partial_tu + A u = F in\ \ \Omega\times (0,T),\] \[u(x,0) = u_0(x) for\ \ x\in \Omega\] along with appropriate boundary conditions. Here, \(A\) is a strongly elliptic, linear differential operator of second-order. The right-hand side \(F\) is assumed to be separable in both variables \(x\) and \(t\), i.e. \[F(x,t)=g(x)h(t).\] Two kinds of inverse problems will be addressed:
    -- reconstructing the source \(g(x)\) (when \(h(t)\) is given) from the additional information \(u(x,T) = \psi_T(x)\quad\mbox{for}\ \ x\in \Omega\)
    -- identification of the unknown function \(h(t)\) from additional data, assuming that \(g(x)\) is known. The unknown function \(h(t)\) is then recovered from a single point measurement \(u(y,t),\ t\in\ I\) at a given point \(y\in\overline\Omega\).

  • Ludwig Gauckler (TU Berlin): Mathematical and numerical analysis of Hamiltonian partial differential equations on long time intervals
    2013/11/25, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Qualitative properties of Hamiltonian partial differential equations on long time intervals are to be discussed in the talk, and the preservation of these properties by a numerical discretization will be studied. In the first part of the talk we will discuss for some numerical methods the long-time near-conservation of the energy, an important conserved quantity of these equations. In the second part of the talk we will study the long-time stability of plane wave solutions to the nonlinear Schrödinger equation, first for the exact solution and then for the numerical discretization of the equation by the popular split-step Fourier method.

  • Sebastian Paul : Ein endliches Kriterium zur Transversalität homokliner Orbits
    2013/11/22, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Ziel des Vortrags ist ein numerisches Verfahren, mit dem wir homokline Orbits eines diskreten, autonomen dynamischen Systems auf Transversalität prüfen können. Unter Verwendung des im ersten Vortrag bewiesenen Satzes reicht es dazu aus, eine exponentielle Dichotomie der Variationsgleichung auf einem hinreichend großen, endlichen Intervall \([-T,T], T \in \mathbb{N}\) nachzuweisen. Im Anschluss daran werden wir die Ergebnisse anhand der Hénon-Abbildung illustrieren.

  • Zdzislaw Brzezniak (York): Stochastic Euler equations in unbounded 2-Ddomains
    2013/11/18, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: I will speak about the existence of a martingale solution to stochastic Euler Equations (with general multiplicative noise) in the Sobolev \(H^{1,q}\cap H^{1,2}\) spaces (based on a 2001 paper with Peszat) and about the existence and uniquence of solutions with bounded vorticity for a specila multiplicative noise (based on a recent unfished work with Flandoli and Maurielli). Somce comments about the existence to stochastic NSEs in unboded domains (based on a work with E. Motyl) wil also be made.

  • Simon Pelster : Spektren kontinuierlicher Dynamischer Systeme
    2013/11/15, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Der Vortrag stellt das Ljapunow-Spektrum und das berechnete Ljapunow-Spektrum für diskrete dynamische Systeme der Form \(u_{t+1}=A_t u_t\) vor. Es wird bewiesen, dass die Ljapunow-Exponenten für Systeme mit \(\sup_{t\in \mathbb{Z}} \|A_t\|=M < \infty\) endlich sind. Außerdem wird ein Kriterium, das die Äquivalenz der beiden Spektren im diskreten Fall liefert, präsentiert.

  • Andrea Lunari (Bielefeld): Optimal Control for a Phase-Field Model of Multiphase Flow
    2013/11/11, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: The optimal control of a multiphase flow is a very interesting and challenging subject from the analytical point of view and leads to concrete industrial applications (e.g. binary alloy, polymer fluids). In this talk I discuss the optimal control of a two-phase fluid flow described by a coupled Cahn-Hilliard-Stokes system, showing the mathematical settings for the problem and its features from an analytical point of view. Then I present a space-time dicretization and an algorithm for the solution of the associated discrete optimal control problem. Finally, I show some first numerical experiments.

  • Adam Andersson (Göteborg): A new approach to weak convergence of SPDEs
    2013/11/08, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Weak convergence of numerical approximations for non-linear SPDEs has previously been proved by a use of the Itô formula and the Kolmogorov equation. In this talk I will present a new method to prove weak convergence for semilinear equations with additive noise, not relying neither on the Itô formula nor on the Kolmogorov equation. We linearize the weak error and obtain a remainder term with high order of convergence. For the linearized term we use the fact that the gradient of the test function, evaluated at the solution of the SPDE, is a Malliavin smooth random variable. This allows us to estimate this term by taking the supremum over a bounded subset of random variables from the Malliavin space. After some analysis a use of the Gronwall Lemma is possible. The order of weak convergence, is as expected, twice that of strong convergence. The novelty this result, except for being a new method of proof, is that it allows for test functions with polynomial growth, meaning that we have proved convergence of any moment. This is joint work with Raphael Kruse (ETH) and Stig Larsson (Chalmers).

  • Gerhard Unger (TU Graz): Boundary element methods for eigenvalue problems in acoustics
    2013/10/28, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk we present an overview about boundary element methods for eigenvalue problems in acoustics. Our approach is based on a reduction of eigenvalue problems to the boundary of the considered domain. Therefore boundary element methods are in particular suitable for problems which are posed in unbounded domains. Boundary integral formulations of eigenvalue problems lead to nonlinear eigenvalue problems even if the original eigenvalue problem is a linear one. The reason for that is that the eigenvalue parameter occurs nonlinearly in the fundamental solution which is related to the underlying partial differential equation. The used boundary integral formulations for the eigenvalue problems can be considered as eigenvalue problems for holomorphic Fredholm operator-valued functions for which a comprehensive theory is well-established. Within this theory convergence results for the Galerkin discretization of boundary integral formulations of eigenvalue problems are derived. For the numerical solution of the discretized eigenvalue problems the contour integral method is applied. We present several numerical examples of acoustic and vibro-acoustic eigenvalue problems which demonstrate the feasibility of our approach.

  • Koray Arslan : Energieerhaltende Integratoren für Poisson Systeme
    2013/07/22, 16:00, V5-148, Seminar Numerical Analysis
    Abstract: Mit den sogenannten Poisson Systemen betrachten wir in diesem Vortrag nicht-kanonische Hamiltonsche Systeme und zeigen, dass jede Erhaltungsgröße dieses Systems - insbesondere die Hamiltonsche und die Casimir Funktion - entlang der exakten Lösung erhalten bleibt. Wir stellen anschließend eine neue Klasse von numerischen Verfahren vor, die von E. Hairer & D. Cohen 2011 eigens für Poisson Systeme entwickelt wurde. Wir zeigen, dass diese Verfahren die Hamiltonsche Funktion und quadratische Casimir Funktionen erhalten und dass sie invariant unter linearen Transformationen sind.

  • Marc Winter: Taylor-Approximation invarianter Faserbündel
    2013/07/19, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Wir betrachten eine nicht-autonome Differenzengleichung der Form \(x(n+1)=A(n)x(n)+F(x(n),n)\), \(n \in \mathbb{Z}\). Wir definieren die lokalen und globalen invarianten Faserbündel der Differenzengleichung als Analogon zu den Mannigfaltigkeiten autonomer Differenzengleichungen. Dann untersuchen wir, unter welchen Voraussetzungen diese Faserbündel existieren. Anschließend werden wir diese Faserbündel mit Hilfe der Taylorentwicklung approximieren.

  • Jessica Jandulski : Das Sharkovskii-Theorem für periodische Differenzengleichungen - Teil 2
    2013/07/15, 16:00, V5-148, AG Dynamische Systeme
    Abstract: Während des ersten Vortrags lernten wir das folgende Sharkovskii-Theorem für periodische Differenzengleichungen kennen: Sei \(f_{n\mod p}(x_n)=x_{n+1}\), \(n\in\mathbb{N}\), \(f_n\in C(I,I)\) für alle \(n\in\mathbb{N}\) eine \(p\)-periodische Differenzengleichung, die einen minimalen \(r\)-Zyklus mit \(r\in A_{p,l}\) hat. Dann hat die Differenzengleichung für alle \(A_{p,q}\) mit \(A_{p,l}>A_{p,q}\) in der \(p\)-Sharkovskii-Ordnung einen minimalen \(k\)-Zyklus mit \(k\in A_{p,q}\). Ziel des zweiten Vortrags wird es sein, das nicht-autonome System unter Verwendung des Konzepts der Schiefprodukt-Flüsse in ein autonomes System zu überführen und dann das Theorem mithilfe des klassischen Sharkovskii-Theorems zu beweisen.

  • Martin Rasmussen (Imperial College London) : Bifurcations of random dynamical systems
    2013/07/08, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Despite its importance for applications, relatively little progress has been made towards the development of a bifurcation theory for random dynamical systems. In this talk, I will demonstrate that adding noise to a deterministic mapping with a pitchfork bifurcation does not destroy the bifurcation, but leads to two different types of bifurcations. The first bifurcation is characterized by a breakdown of uniform attraction, while the second bifurcation can be described topologically. Both bifurcations do not correspond to a change of sign of the Lyapunov exponents, but I will explain that these bifurcations can be characterized by qualitative changes in the dichotomy spectrum and collisions of attractor-repeller pairs. This is joint work with M. Callaway, T.S. Doan, J.S.W Lamb (Imperial College) and C.S. Rodrigues (MPI Leipzig)

  • Raphael Kruse (ETH Zürich): Konsistenz und Stabilität von Galerkin finite Elemente Methoden für Reaktions-Diffusions-Gleichungen
    2013/07/01, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Wir betrachten Galerkin finite Elemente Methoden für (deterministische) semilineare Reaktions-Diffusions-Gleichungen und betten diese in den abstrakten Rahmen der diskreten Approximationstheorie ein. Basierend auf einer passend für Halbgruppen verallgemeinerten Spijker Norm beweisen wir die Bistabilität des Verfahrens und leiten zweiseitige Fehlerabschätzungen her. Anschließend vervollständigen wir den Konvergenzbeweis mit einer Analyse des Konsistenzfehlers.

  • Paul Voigt (Bielefeld): Das Dirichletproblem für nicht-lokale nicht-symmetrische Operatoren
    2013/06/28, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Im Vortrag wird das elliptische Dirichletproblem für lineare nicht-lokale und nicht notwendigerweise symmetrische Operatoren studiert. Aufgrund der Nicht-lokalität des Operators werden die Randdaten - im Gegensatz zum klassischen Dirichletproblem - auf dem Komplement einer gegebenen Menge vorgegeben. Das Problem wird in einem klassischen Hilbertraum-Setting formuliert. Existenz und Eindeutigkeit von Lösungen wird mit Standardtechniken, wie z.B. der Fredholm-Alternative bewiesen.

  • Alina Girod: Diskretisierung homokliner Orbits im nicht-autonomen Fall
    2013/06/24, 16:00, V5-148, AG Dynamische Systeme
    Abstract: Wir betrachten ein kontinuierliches nicht-autonomes Dynamisches System, welches einen transversalen homoklinen Orbit besitzt. Dieses diskretisieren wir zunächst exakt. Diskretisieren wir mit einem Einschritt-Verfahren, welches nah am exakt diskretisierten System ist, so erhalten wir wieder einen transversalen Orbit der nah an dem exakt diskretisierten Orbit liegt. Anschließend werden wir die Theorie an einem Beispiel illustrieren.

  • Robin Flohr : Konvergenz des Strang-Operatorsplittings - Teil 2
    2013/06/14, 14:15, V5-148, AG Dynamische Systeme

  • Robin Flohr : Konvergenz des Strang-Operatorsplittings
    2013/06/14, 14:15, V5-148, AG Dynamische Systeme
    Abstract: In dem Vortrag wird das Prinzip des Operatorsplittings von Gleichungen der Form \(u_t = Au + uu_x\) vorgestellt. \(A\) ist dabei ein linearer Differentialoperator, so dass die Gleichung wohlgestellt ist. Wichtige Beispiele für \(A\) sind dabei \(Au=u_{xx}\) oder \(A=u_{xxx}\), so dass die Theorie sowohl für die Burgers-Gleichung als auch für die KdV-Gleichung anwendbar ist. Außerdem werden Regularitätsaussagen der getrennten Probleme sowie die Konvergenz erster Ordnung des Strang-Splittings bewiesen.

  • Jessica Jandulski : Das Sharkovskii-Theorem für periodische Differenzengleichungen
    2013/06/03, 16:00, V5-148, AG Dynamische Systeme
    Abstract: Das Sharkovskii-Theorem für autonome Differenzengleichungen behandelt eine stetige, reellwertige Abbildung \(f\), die einen periodischen Orbit der minimalen Periode \(n\) besitzt und liefert die Antwort auf die Frage: Für welche \(m\) existieren dann auch \(m\)-periodische Orbits? Unser Ziel wird es sein, das Sharkovskii-Theorem auf \(p\)-periodische Differenzengleichungen \(x_{n+1}=f_{n\mod p}(x_n)\) zu erweitern. Dafür lernen wir zunächst einen Spezialfall kennen, den wir mithilfe des klassischen Sharkovskii-Theorems beweisen. Anschließend betrachten wir das allgemeinere Sharkovskii-Theorem für \(p\)-periodische Differenzengleichungen und beweisen auch dieses Theorem unter Verwendung des Konzepts der Schiefprodukt-Flüsse mithilfe des klassischen Sharkovskii-Theorems.

  • Sebastian Paul : Dichotomien im Übergang von endlichen zu unendlichen Intervallen
    2013/05/31, 14:15, V5-148, AG Dynamische Systeme
    Abstract: In diesem Vortrag gehen wir zunächst von einer exponentiellen Dichotomie auf endlichen Intervallen aus. Unter welchen Voraussetzungen lässt sich daraus eine exponentielle Dichotomie auf \(\mathbb{Z}\) zusammensetzen? Wir liefern Bedingungen, unter denen dies möglich ist und beweisen diese Aussage mit Hilfe des Roughness-Theorems.

  • Rudolf Dürksen : Oberhalbstetigkeit von Pullback-Attraktoren
    2013/05/27, 16:00, V5-148, AG Dynamische Systeme
    Abstract: Wir betrachten die nichtautonome Differentialgleichung \(\dot{x}=f(x,t),\ x(s)=x_s\), wobei \(f\in C(\mathbb{R}^p\times\mathbb{R},\mathbb{R}^p)\). Im nichtautonomen Fall sind die Lösungen sowohl von der Endzeit \(t\), als auch von der Startzeit \(s\) explizit abhängig, wohingegen im autonomen Fall nur die bereits verstrichene Zeit \(t-s\) relevant ist. In diesem Zusammenhang werden wir den Begriff der Pullback-Konvergenz einführen und einen für nichtautonome Systeme geeigneten Attraktor, den sogenannten Pullback-Attraktor, definieren. Ziel des Vortrags ist es, zu sehen, dass auch der Pullback-Attraktor unter gewissen Voraussetzungen oberhalbstetig ist.

  • Wolf-Jürgen Beyn (Bielefeld): Mathematische Modellbildung, Analyse und Simulation zellulärer Prozesse
    2013/05/24, 16:15, V5-148,
    Abstract: Im Vortrag werden einige Grundprinzipien mathematischer Modellbildung diskutiert, die zum Verständnis des Zusammenwirkens zellulärer Einzelprozesse beitragen können. Zunächst wird auf die verschiedenen Modelltypen eingegangen, die sich durch Begriffspaare wie diskret-kontinuierlich, stationär-zeitabhängig, räumlich homogen-inhomogen, linear-nichtlinear und zufällig-deterministisch unterscheiden lassen. Speziell werden dann einfache Modellgleichungen für enzymatisch ablaufende Stoffwechselprozesse analysiert und Möglichkeiten aufgezeigt, um diese Modelle auf Transportprozesse und genetische regulierte Netzwerke zu erweitern. Dabei entstehen oft große Systeme nichtlinearer Differentialgleichungen, die eine Vielzahl von Parametern enthalten und deren Lösungsverhalten nicht einfach zu überblicken ist. Einerseits werden solche Systeme auf dem Computer simuliert und mit experimentellen Daten verglichen, andererseits versucht man die Modelle so zu reduzieren (Hauptkomponentenanalyse, Pseudostationarität), dass Einsichten in biochemisch relevantes Verhalten gewonnen werden können.

  • Simon Pelster : Spektren kontinuierlicher Dynamischer Systeme
    2013/05/24, 14:00, V5-148, AG Dynamische Systeme
    Abstract: Der Vortrag stellt das Ljapunow-Spektrum und das berechnete Ljapunow-Spektrum kontinuierlicher dynamischer Systeme vor. Unter der Voraussetzung integraler Getrenntheit sind die Spektren für Systeme mit oberer Dreiecksgestalt identisch. Im zweiten Teil wird die Verbindung vom berechneten Ljapunow-Spektrum zum Sacker-Sell-Spektrum für kontinuierliche Systeme hergestellt.

  • Koray Arslan : Symplektizität in numerischen Verfahren und Beispiele symplektischer Integratoren
    2013/05/17, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Nachdem in der ersten Vortragsreihe der Begriff der Symplektizität eingeführt und im Satz von Poincaré gezeigt wurde, dass der exakte Fluss eines Hamiltonschen Systems symplektisch ist, werden wir nun numerische Verfahren konstruieren, von denen wir auch fordern, dass deren diskreter Fluss symplektisch ist.
    Dazu definieren wir erst, was ein symplektisches Verfahren ist, und untersuchen anschließend verschiedene numerische Verfahren auf ihre Symplektizität.
    Dabei richtet sich unser Hauptaugenmerk auf Runge-Kutta-Verfahren, die - wie aus der Numerik II bekannt ist - eine wichtige Klasse von Einschrittverfahren für Anfangswertaufgaben bilden.

  • Koray Arslan: Symplektizität in numerischen Verfahren und Beispiele symplektischer Integratoren
    2013/05/13, 16:00, V5-148, AG Dynamische Systeme
    Abstract: Nachdem in der ersten Vortragsreihe der Begriff der Symplektizität eingeführt und im Satz von Poincaré gezeigt wurde, dass der exakte Fluss eines Hamiltonschen Systems symplektisch ist, werden wir nun numerische Verfahren konstruieren, von denen wir auch fordern, dass deren diskreter Fluss symplektisch ist.
    Dazu definieren wir erst, was ein symplektisches Verfahren ist, und untersuchen anschließend verschiedene numerische Verfahren auf ihre Symplektizität.
    Dabei richtet sich unser Hauptaugenmerk auf Runge-Kutta-Verfahren, die - wie aus der Numerik II bekannt ist - eine wichtige Klasse von Einschrittverfahren für Anfangswertaufgaben bilden.

  • Andre Schenke: Exponentielle Dichotomien für nichtinvertierbare Systeme - Teil 2
    2013/05/10, 14:15, V5-148, AG Dynamische Systeme

  • David Kiesewalter: Die Randelementmethode für das Eigenwertproblem zum Laplaceoprator
    2013/05/06, 16:00, V5-148, AG Dynamische Systeme

  • Abderrahman Boukricha (Tunis): Variational Formulation of Nonlocal and Quasilinear Elliptic Problems and Numerical Analysis of the Weighted p-Laplacian
    2013/04/26, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: The principle of variational formulation or variational approach for the resolution of well posed problems is to replace the equation by an equivalent formulation which can be solved by Hilbert space methods (in the linear case) or by Minty-Browder methods (in the nonlinear case).
    In this talk, we recall the variational approach of Gregoire Allaire in the classical case for stationary and evolution problems. We prove variational formulations for nonlocal problems (Riesz potentials) and for quasilinear elliptic problems. We then present a numerical analysis for the weighted p-Laplacian.

  • Alexander Mielke (WIAS Berlin): Gradient structures and uniform global decay for reaction-diffusion systems
    2013/04/25, 17:15, V2-210/216,

  • Andre Schenke: Exponentielle Dichotomien für nichtinvertierbare Systeme
    2013/04/19, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Dieser Vortrag behandelt die Verallgemeinerung des Dichotomiebegriffes für nichtinvertierbare dynamische Systeme. Die Definition wird motiviert und es werden elementare Eigenschaften und Beispiele behandelt sowie Unterschiede zum "gewöhnlichen" Dichotomiebegriff für invertierbare Systeme aufgezeigt. Die Projektoren etwa sind im Falle J=Z nicht eindeutig und die Charakterisierung von Bild und Kern der Projektoren mittels Quasibeschränktheit gelingt nur unvollständig. Eine Lösung dieses Problems bietet der Begriff der Regularität, der im zweiten Teil des Vortrags besprochen wird.

  • Abderrahman Boukricha (Tunis) Koautoren: Imed Ghanmi, Rochdi Jebari: Numerical Approximation of solutions of Nonlinear Partial Differential Equations by Homotopy Perturbation Method (HPM) and Adomian Decomposition Method (ADM)
    2013/04/12, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this work, the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM) are presented.
    These methods provide numerical approximations for solutions of non-linear partial differential equations. The solutions of these non-linear problems are approached by series with easily computable partial sums (using Maple for HPM and Mathematica for ADM).
    The numerical approximation in cases, where the exact solutions (resp. other approximations) are known, turns out to be very good (resp. yields better approximations). Ther results show that HPM and ADM are very effective and simple.

  • Dietrich Neumann (Bielefeld) : Diskrete Approximation nichtlinearer Eigenwertprobleme mit Multiplizitäten
    2013/01/28, 16:15, V5-148, AG Dynamische Systeme

  • Wolf-Jürgen Beyn (Bielefeld) : Continuation and Collapse of Homoclinic Tangles
    2013/01/25, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: By a classical theorem (Birkhoff, Smale, Shilnikov),transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In our work we analyze the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The main bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a set of bifurcation equations that is indexed by a symbolic sequence.
    For the Henon family we investigate in detail the bifurcation structure of multi-humped homoclinic orbits originating from several tangencies. The emerging homoclinic network is explained by combining our bifurcation result with graph-theoretical arguments.
    This is joint work with Thorsten Hüls.

  • Sebastian Schmitz (Lugano): Optimal Realiability in Design for Fatigue Life
    2013/01/22, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Fatigue describes the damage or failure of material under cyclic loading. Activation and deactivation operations of technical units are important examples in engineering where fatigue and especially low-cycle fatigue (LCF) play an essential role. A significant scatter in fatigue life for many materials results in the necessity of advanced probabilistic models for fatigue. Moreover, structural shape optimization is of increasing interest in engineering, where with respect to fatigue the cost functionals are motivated by their predictability for the integrity of the component after a certain number of load cycles. But mathematical properties such as the existence of the shape derivatives are desirable, too. Deterministic design philosophies that derive a predicted component life from the average life of the most loaded point on the component plus a safety factor accounting for the scatter band do not have this favorable property, as taking maxima is not a differentiable operation. Here, we present a new local probabilistic model for LCF. This model constitutes a new link between reliability statistics, shape optimization and structural analysis which considers the perspective of fatigue but also fits into the mathematical setting of shape optimization. The cost functionals derived in this way are too singular to be \(H^1\) lower semi-continuous. We therefore have to modify the existence proof of optimal shapes for the case of sufficiently smooth shapes using elliptic regularity, uniform Schauder estimates and compactness of certain subsets in \(C^k(\Omega^\textrm{ext},\mathbb{R})\) via the Arcela-Ascoli theorem, where \(\Omega^\textrm{ext}\) is some shape containing all admissible shapes. Moreover, we extend our existence results to high-cycle fatigue (HCF) and deterministic models of fatigue.

  • Lennart Esdar (Bielefeld): Existenz und Stabilität wandernder Wellen für eine nichtlineare Wellengleichung
    2013/01/08, 16:15, V5-148, Seminar Numerical Analysis

  • Jochen Röndigs (Bielefeld): Reaction Diffusion Systems on Infinite Lattices
    2012/11/30, 14:15, V5-148, AG Dynamische Systeme
    Abstract: This talk is about the evolution of reaction diffusion systems on an infinite dimensional space and their finite dimensional approximations. The main result is the upper semicontinuity of an approximate attractor on a finite lattice with respect to the attractor of the original system on the infinite lattice. We take a short look at characteristic properties of the equations and then derive a priori estimates which establish the existence of a global attractor, both on the finite and the infinite lattice.

  • Rudolf Dürksen: Unterhalbstetigkeit von Attraktoren
    2012/11/20, 16:15, V5-148, AG Dynamische Systeme
    Abstract: Wir betrachten eine autonome Differentialgleichung der Form \(\frac{du}{dt}=f(u)\), \(u(0)=U\in\mathbb R^p\), die mit einem Einschrittverfahren \(U_{n+1}=S_{\Delta t}^1U_n\), \(U_0=U\) gelöst wird. Für den diskretisierten Attraktor \(\mathcal{A}_{\Delta t}\) haben wir gesehen, dass die Oberhalbstetigkeit \(\text{dist}\left(\mathcal{A}_{\Delta t},\mathcal{A}\right)\rightarrow 0\) gegeben ist. Wir werden zeigen, dass unter zusätzlichen Voraussetzungen auch die Unterhalbstetigkeit \(\text{dist}\left(\mathcal{A},\mathcal{A}_{\Delta t}\right)\rightarrow 0\) gegeben ist, sodass insgesamt \(\text{dist}_H\left(\mathcal{A},\mathcal{A}_{\Delta t}\right)\rightarrow 0\) für \(\Delta t \to 0\) gilt.

  • Andre Schenke: Hyperzyklische Operatoren und der Satz von Grivaux
    2012/11/16, 14:15, V5-148, AG Dynamische Systeme
    Abstract: Der Satz von Grivaux gibt eine Charakterisierung derjenigen Mengen eines separablen unendlich-dimensionalen Banachraumes an, die Orbits eines hyperzyklischen Operators sind. Behandelt werden die grundlegenden Ideen für die Formulierung des Satzes von Grivaux, der Beweis im Banachraumfall sowie eine kurze Beweisskizze für den Fall von Fréchet-Räumen mit einer stetigen Norm.

  • Alina Girod: Diskretisierung heterokliner Orbits
    2012/10/16, 16:15, V5-148, AG Dynamische Systeme
    Abstract: Erste Betrachtung ist ein kontinuierliches Dynamisches System, wobei der Begriff nicht-entarteter Verbindungsorbit eingeführt wird. Diese und 1-tangentiale Orbits betrachten wir dann im diskretisierten Fall. Zwischenziel ist es zu zeigen, dass diskretisierte Verbindungsorbits des kontinuierlichen Systems 1-tangentiale nicht-entartete heterokline Orbits des \( \varepsilon \)-Flusses sind. Allgemeines Ziel ist es zu zeigen, dass eine Diskretisierung mit einem Einschritt-Verfahren unter gewissen Annahmen mindestens zwei 1-tangentiale nicht-entartete heterokline Orbits besitzt.

  • Thorsten Hüls: Homoclinic trajectories of non-autonomous maps
    2012/10/09, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: For time-dependent dynamical systems of the form \(x_{n+1} = f_n(x_n), n \in \mathbb{Z}\) homoclinic trajectories are the non-autonomous analog of homoclinic orbits from the autonomous world. More precisely, two trajectories \((x_n)_{n\in\mathbb{Z}}\), \((y_n)_{n\in\mathbb{Z}}\) are called homoclinic to each other, if \(\lim_{n\to \pm \infty} \|x_n - y_n\| = 0\). Two boundary value problems are introduced, the solution of which yield finite approximations of these trajectories. Under certain hyperbolicity assumptions, we prove existence, uniqueness and error estimates. Extending these ideas, we also propose adequate notions for heteroclinic orbits in non-autonomous systems. The resulting algorithms and error estimates are illustrated by an example.

  • Stefanie Hittmeyer (Auckland): Interacting global manifolds in a planar map model of wild chaos
    2012/07/20, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We study a non-invertible planar map that has been suggested by Bamon, Kiwi and Rivera-Letelier as a model for a new type of chaotic dynamics in continuous-time dynamical systems of dimension at least five; one also speaks of wild Lorenz-like chaos. This map opens up the origin (the critical point) to an open disk and wraps the plane twice around it; inside this disk there are no preimages. The bounding critical circle and its images, together with the critical point and its preimages form the so-called critical set. This set interacts with a saddle fixed point and its stable and unstable sets.
    Advanced numerical techniques enable us to study how the stable and unstable sets change as a parameter is varied along a path towards the wild chaotic regime. We find sequences of bifurcations, which are of two types. First, there are bifurcations that also occur in invertible maps, such as homoclinic tangencies. Second, we find bifurcations specific to non-invertible maps: interactions of the stable and unstable sets with the critical set, which also cause changes (such as self-intersections) of the topology of these global invariant sets. Overall, a consistent sequence of both types of bifurcations emerges, which we present as a first attempt towards explaining the geometric nature of wild chaos.

  • Martin Arnold: Ein Lie-Gruppen-DAE-Integrator
    2012/07/09, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Lage und Orientierung des Starrkörpers lassen sich ohne Singularitäten als Elemente geeigneter Lie-Gruppen beschreiben. Schon 1989 haben Géradin und Cardona diese Darstellung zu einer Finite- Elemente-Beschreibung flexibler Mehrkörpersysteme verallgemeinert. Bezieht man die für mechanische Mehrkörpersysteme typischen (holonomen) Zwangsbedingungen ein, so ergeben sich differentiell-algebraische Systeme (engl.: differential-algebraic equations, DAEs) vom Index 3 auf einer (Matrix-)Lie-Gruppe. Bei der direkten Zeitintegration dieses Index-3-Systems mit dem aus der Strukturdynamik bekannten Generalized-/(alpha/)-Verfahren) beobachtet man - ebenso wie bei Anwendung dieses Verfahrens in linearen Räumen - eine Ordnungsreduktion zu Beginn der Integration. Eine detaillierte, auf die zugrunde liegende Lie-Gruppen-Struktur zugeschnittene Konvergenzanalyse zeigt, wie die Ordnungsreduktion durch modifizierte Startwertedes Zeitintegrationsverfahrens vermieden werden kann. Die Ergebnisse der theoretischen Untersuchungen werden durch numerische Tests für einfache Benchmarkprobleme verifiziert.

  • Andrea Walther (Paderborn): On an Inexact Trust-region Approach for Inequality Constrained Optimization
    2012/06/29, 12:45, V5-148, Seminar Numerical Analysis
    Abstract: This talk presents a trust-region SQP algorithm for the solution of minimization problems with nonlinear inequality constraints. The approach works only with an approximation of the constraint Jacobian. Hence, it is well suited for optimization problems of moderate size but with dense constraint Jacobian. The accuracy requirements for the presented first-order global convergence result can be verified easily during the optimization process. Numerical results for some test problems are shown.

  • Sonja Cox (Universität Innsbruck) : Pathwise estimates for the implicit Euler scheme for SDEs in Banach spaces
    2012/06/08, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In the first part of my talk I will explain what is meant by pathwise estimates for an approximation scheme of a stochastic differential equation (SDE), and why such estimates are of importance. In recent work by Jan van Neerven and myself, we obtained pathwise estimates for the implicit Euler scheme for SDEs in Banach spaces. In the second half of the talk I will sketch how we obtained these results and indicate what challenges arise when working in Banach spaces.

  • Wilhelm Stannat (TU Berlin, Bernstein Netzwerk) : Stochastic stability of travelling waves in the Nagumo equation
    2012/06/01, 14:15, V5-148, Seminar Numerical Analysis
    Abstract:

  • Michael Scheutzow (TU Berlin) (Projekte A3, B3, B4) : Uniqueness of invariant measures via asymptotic coupling with applications to stochastic delay equations
    2012/05/25, 14:00!!, V5-148,
    Abstract: We provide sufficient conditions for the uniqueness of an invariant measure for a Markov process in terms of the existence of a generalized asymptotic coupling. This method is particularly useful in cases in which the transition probabilities for different initial conditions are mutually singular (and hence classical methods for proving uniqueness fail). We apply the result to stochastic delay equations.
    This is joint work with Martin Hairer and Jonathan Mattingly.

  • Heiko Prange : Hinreichende Bedingungen für LR-Chaos in zeitdiskreten dynamischen Systemen
    2012/05/21, 16:15, V5-148, AG Dynamische Systeme
    Abstract: Ähnlich wie die chaotische Dynamik des Shift-Operators auf der Menge der Symbole 0 und 1 lässt sich der Begriff des Links-Rechts Chaos auf Teilmengen L und R eines zeitdiskreten dynamischen Systems definieren. Die Kern-Idee zum Aufspüren von LR-Chaos in zeitdiskreten dynamischen Systemen ist es, periodische Punkte zu vorgegebenen Links-Rechts Sequenzen zu finden. Diese periodischen Punkte kann man als Fixpunkte von iterierten Funktionen betrachten. So führt der Miranda-Fixpunktsatz zu Gebieten mit LR-chaotischer Dynamik.

  • Christian Pötzsche (Klagenfurt) : Feinstruktur des Dichotomie-Spektrum
    2012/05/18, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Das Dichotomie-Spektrum (auch dynamisches oder Sacker-Sell Spektrum) ist ein wesentlicher Begriff innerhalb der Theorie nichtautonomer dynamischer Systeme, da es zentrale Information zur Stabilität, Hyperbolizität und Robustheit von Lösungen enthält. Aktuelle Anwendungen in der Verzweigungstheorie erfordern jedoch eine detailliertere Einsicht in dessen Feinstruktur. Auf dieser Basis untersuchen wir eine hilfreiche Verbindung zwischen Dichotomie-Spektrum und Operator-Theorie, welche das Langzeitverhalten nichtautonomer Gleichungen mit dem Punkt-, Surjektivitäts- oder Fredholm-Spektrum gewichteter Shift-Operatoren in Beziehung setzt. Dieser Zusammenhang führt auf entsprechende Teilmengen des Dichotomie-Spektrums, welche nicht nur Verzweigungen bereits auf linearer Ebene klassifizieren, sondern auch Beweise über das asymptotische Verhalten von Differential- und Differenzengleichungen vereinfacht.

  • Rudolf Dürksen (Bielefeld) : Oberhalbstetigkeit von Attraktoren
    2012/05/07, 16:15, V5-148, AG Dynamische Systeme
    Abstract: Sei eine autonome Differentialgleichung der Form \(\frac{du}{dt}=f(u), u(0)=U\in\mathbb R^p\), gegeben. Bekanntlich kann diese Differentialgleichung mit einem Einschrittverfahren \(U_{n+1}=S_{\Delta t}^1U_n, U_0=U\) gelöst werden. Wir werden das Verhalten des diskretisierten Attraktors \(\mathcal{A}_{\Delta t}\) untersuchen und dabei feststellen, dass in unserem Fall die Oberhalbstetigkeit gegeben ist, d. h. dass \(dist\left(\mathcal{A}_{\Delta t},\mathcal{A}\right)\rightarrow 0\) gilt.

  • Jan Giesselmann (Stuttgart) : Some ideas for the numerical discrtization of the Navier-Stokes-Korteweg model
    2012/04/30, 16:15, V5-148, Seminar Numerical Analysis
    Abstract:

  • Sergey Tikhomirov : Shadowing lemma for partially hyperbolic systems
    2012/01/30, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We say that diffeomorphism \(f\) of a manifold \(M\) is partially hyperbolic if tangent bundle of \(M\) admits an invariant splitting \(E^s + E^c + E^u\), such that \(E^s\) and \(E^u\) are uniformly hyperbolic and \(E^c\) is not. If \(E^c \) is empty diffeomorphism is uniformly hyperbolic. Shadowing lemma says that in hyperbolic systems any pseudotrajectory can be shadowed by an exact trajectory. We introduce notion of central pseudotrajectory and prove that in partially hyperbolic systems any pseudotrajectory can be shadowed by a central pseudotrajectory.

  • Annika Lang (ETH Zürich): Simulation of stochastic processes
    2012/01/23, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Hilbert-space-valued stochastic processes such as Q-Wiener processes are in general the driving noise of stochastic partial differential equations (SPDEs for short). To approximate the solution of an SPDE, it is in general necessary to approximate the driving noise. In this talk, different approximation methods for stochastic processes and the corresponding Ito integrals are introduced and their simulations are presented. As an application to the approximation of solutions of SPDEs it is shown how to equilibrate the discretization errors of the space and time approximation of the SPDE and the error of the driving noise approximation.

  • Lennart Esdar : Solitärwellen in Hamiltonschen Systemen
    2012/01/20, 14:15, V5-148, AG Dynamische Systeme
    Abstract:

  • Christian Lubich (Tübingen): Modulated Fourier expansions for highly oscillatory problems
    2011/12/09, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Modulated Fourier expansions are an analytic technique for understanding the behaviour of weakly nonlinear oscillatory problems over very long times. The technique applies to highly oscillatory ODEs, to particle systems such as the Fermi-Pasta-Ulam lattice, to Hamiltonian PDEs such as nonlinear Schrödinger and wave equations, as well as to their numerical discretizations. The approach first came up about a decade ago in the numerical analysis of highly oscillatory ODEs, where it explained remarkable long-time energy conservation properties of numerical integrators, and has since been used to analyse long-time properties of various types of problems as mentioned above, both for the continuous equations and their numerical discretizations. In addition to their role as an analytical tool originating from numerics, modulated Fourier expansions have also been found useful as a numerical approximation method for highly oscillatory problems. Most of the talk is based on joint work with Ernst Hairer, some parts also with David Cohen, Ludwig Gauckler and Daniel Weiss.

  • Matthias Ehrhardt : Absorbing Boundary Conditions for Hyperbolic Systems
    2011/12/05, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: This talk deals with absorbing boundary conditions (ABCs) for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) ABCs do not t the standard form of boundary conditions for hyperbolic systems.
    In the second part we present the approach of deriving so-called discrete absorbing boundary conditions, i.e. ABCs constructed on a purely discrete level. These discrete ABCs are better adapted to the interior scheme: they lead to less unphysical reflections and the resulting overall scheme has better stability properties. Finally, we sketch briefly how ABCs can be derived for nonlinear hyperbolic systems.

  • Thorsten Rieß (Universität Konstanz): N-heteroclinic orbits near non-reversible homoclinic snaking
    2011/11/28, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Non-reversible homoclinic snaking of a codimension-1 homoclinic orbit to an equilibrium is a phenomenon that is known to occur near certain heteroclinic equilibrium-to-periodic (EtoP) cycles. We show numerically that there exist other connecting orbits in the neighbourhood of the homoclinic snaking for a specific family of three-dimensional vector fields. In particular, we use a numerical method based on Lin's method to compute codimension-1 PtoE connecting orbits that take additional excursions along the EtoP cycle before connecting up, so-called N-heteroclinic PtoE orbits. It turns out that the N-heteroclinic PtoE orbits exist on isolas in parameter space.

  • Sergei Pilyugin (St. Petersburg): Lipschitz shadowing and structural stability: the case of flows
    2011/10/11, 16:30, V5-148, Seminar Numerical Analysis
    Abstract: Recently, it was shown that for diffeomorphisms, Lipschitz shadowing is equivalent to structural stability (S.Yu. Pilyugin and S.B. Tikhomirov, Nonlinearity, vol. 23, 2509-2515, 2010). In this talk, we discuss a similar result for flows and explain the appearing difficulties (the main difficulty is created by the absence of Mane's theorem characterizing structural stability in terms of strong transversality). This is a joint research with K.Palmer and S.Tikhomirov.

  • Richard Norton (Oxford): Finite Element Approximation of an H1 Gradient Flow
    2011/07/11, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: I consider the discretization error in space and time of an H1 gradient flow for an energy integral where the energy density is given by the sum of a double-well potential term and a bending energy term. This problem is equivalent to a nonlinear heat equation with nonlocal nonlinearity. The approach for the error analysis is to adapt standard error analysis theory developed for nonlinear heat equations to bound the discretization error in terms of the mesh size and time step as well as energy parameters. In particular, I carefully track how the size of the bending energy affects the error bounds.

  • Piotr Gwiazda: On scalar hyperbolic conservation laws with a discontinuous flux
    2011/07/04, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We study the Cauchy problem for scalar hyperbolic conservation laws with fluxes that can have jump discontinuities with respect to the unknown and only measurable with respect to the space variable. We introduce a new concept of entropy weak and measure-valued solution that is consistent with the standard one for continuous fluxes. We then answer the question as to what kind of properties the fluxes should posses in order to establish the existence and/or uniqueness of various notions of solutions. In any space dimension we establish the existence of measure-valued entropy solution for fluxes having jump discontinuities, under additional assumptions we prove existence of weak solutions.

  • Ewelina Zatorska: On a new approach applied to the time--discretization of the compressible Navier--Stokes equations
    2011/06/20, 16:15, V5-148, Seminar Numerical Analysis

  • Matthias Groncki: Markovketten und der Metropolis-Algorithmus
    2011/06/10, 14:15, V5-148, AG Dynamische Systeme

  • Gert Lube (Göttingen): A Projection-based Variational Multiscale Method for Turbulent Incompressible Flows
    2011/06/06, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider the Navier-Stokes Fourier model for time-dependent, non-isothermal, incompressible flows. This nonlinear evolution problem has a generalized solution. Moreover, in case of a regularization with the well-known Smagorinsky turbulence model, it is uniquely solvable. Unfortunately, the latter model is too diffusive in application to turbulent flows. As a remedy, a variational multiscale (VMS) method for the large-eddy simulation is considered.
    Following a general proposal in [1], our VMS-approach relies on local projection of the velocity deformation tensor and the temperature gradient together with a grad-div stabilization of the divergence-free constraint. Semi-discrete stability and a priori error estimates are derived in the case of inf-sup stable approximation of velocity and pressure, see [2, 3]. In particular, rather general nonlinear and piecewise constant coefficients of the subgrid models for the unresolved scales of velocity and pressure are allowed, including the classical Smagorinsky model. We give a critical discussion of the results in view of the employed Gronwall argument. Moreover, we discuss aspects of the time discretization and of its analysis.
    Finally, we present and discuss numerical simulations for basic benchmark problems like decaying homogeneous isotropic turbulence, channel flow and natural convection in a differentially heatedcavity.

    References
    [1] W. Layton, A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comput. 133 (2002), 147-157.
    [2] L. Röhe, and G. Lube, Analysis of a variational multiscale method for Large-Eddy simulation and its application to homogeneous isotropic turbulence, Comput. Meths. Appl. Mech. Engrg. 199 (2010), 2331-2342.
    [3] J. Löwe, and G. Lube, A projection-based variational multiscale method for Large Eddy simulation with application to non-isothermal free convection problems, NAM Preprint, Georg-August-Universität zu Göttingen, 2010. Accepted for Math. Model. Meths. Appl. Sc. (5/2011).

  • Denny Otten (Bielefeld): Exponential decay of two-dimensional rotating waves
    2011/06/03, 14:15, V5-148, AG Dynamische Systeme

  • Marcin Malogrosz: A Model of Morphogen Transport Well-Posedness and Asymptotical Behaviour
    2011/05/30, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Morphogen transport (MT) is a process occurring in the tissue of life organisms, affecting cell differentiation. There is a vast literature concerning modeling of MT but as for now there is still no consensus on what is the exact mechanism of the movement of morphogen particles. Various types of diffusion, bucket brigade, reactions with other particles are among those being considered.
    I will present my recent results concerning well-posedness and asymp- totical behavior of the solutions of the model proposed in [1] (semilinear parabolic PDE coupled with ODE), where MT is being modeled by passive diffusion and binding-unbinding reactions with receptors.
    My results are nontrivial extension of those obtained in [2], where the same model is being considered in 1D setting. Using theory of analytic semigroups I improve dimension of the domain (from 1 to arbitrary) and topology of the convergence of solution to unique equilibrium (from \(L_2 \times L_2\) to \(C^{1,\alpha} \times C^{0,\alpha}\)).

    References
    [1] Lander, A. D., Nie, Q., Wan, Y. M. Do Morphogen Gradients Arise by Diffusion? Dev. Cell, Vol. 2, pp. 785-796.
    [2] Krzyzanowski, P., Laurençcot, P., Wrzosek, D. Well-posedness and con- vergence to the steady state for a model of morphogen transport, SIAM J.MATH. ANAL. Vol. 40, No. 5, pp. 1725-1749.

  • Raphael Kruse (Bielefeld): FEniCS: Finite Elements in Computer Science eine Software zur Loesung elliptischer und parabolischer Aufgaben
    2011/05/13, 14:15, V5-148, AG Dynamische Systeme

  • Maria Lopez Fernandez (Uni Zürich): Contour integral methods for parabolic equations
    2011/05/09, 16:15, V5-148, Seminar Numerical Analysis

  • Andrzej Warzynski: 30 Years of Residual Distribution Schemes for Hyperbolic Conservation Laws
    2011/05/02, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: The most popular schemes for hyperbolic conservation laws are based on 1-dimensional concepts (i. e. Riemann solver) which are then heuristically extended and applied to flow problems in more than 1 space dimension. Recently, intensive research has been being carried out focused on the development of `multidimensional upwind' schemes. This new class of numerical algorithms is far better able to incorporate genuinely multidimensional phenomena described by conservation laws and thus to predict the fluid flow more accurately. In this talk I shall first briefly justify the need for genuinely multidimensional approach and then describe a particular class of multidimensional upwind schemes, namely those developed within the Residual Distribution (RD) framework. This will include design principles for the steady state RD schetime-dependent problems. Finally, I shall focus on the recently proposed explicit Runge-Kutta RD [1] and discontinuous RD schemes [2], and discuss the possibility of combining these two approaches. This is an ongoing research conducted in collaboration with M. E. Hubbard and M. Ricchiutto. [1] R. Abgrall, M. Ricchiuto 'Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case.' J. Comput. Phys. 229(16), 5653--5691, 2010. [2] M. E. Hubbard 'Discontinuous fluctuation distribution.' J. Comput. Phys. 227(24), 10125--10147, 2008.

  • Janning Barembruch (Bielefeld): Diskretisierung der Hopf-Bifurkation
    2011/04/15, 14:15, V5-148, AG Dynamische Systeme

  • Guy Vallet (Pau): On some Barenblatt's problems
    2011/04/11, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk we will be interested in the problem of Barenblatt's type: \(f(\partial_{t}u)-\Delta_pu - \epsilon \Delta \partial_{t}u =g\quad \textrm { in }\ Q.\) In a first part, we will consider the hilbertian case \(p=2\), then when \(p> \frac{2d}{d+2}\) and when \(p\) is a function of \(x\). We finish with some numerical simulations of the hilbertian case.

  • Janning Barembruch (Bielefeld): Diskretisierung der Hopf-Bifurkation
    2011/04/08, 14:15, V5-148, AG Dynamische Systeme

  • Raphael Kruse : Finite-Elemente Methoden für PDEs
    2011/02/04, 14:15, V5-148, AG Dynamische Systeme

  • Heiko Berninger (FU Berlin): On Domain Decomposition Methods for Nonlinear Transmission Problems
    2011/01/31, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Consider a nonoverlapping decomposition of a domain \(\Omega\) into subdomains \(\Omega_1\) and \(\Omega_2\). It is well known that instead of solving \(-\triangle u = f\) on \(\Omega\), one can solve this equation on \(\Omega_1\) and on \(\Omega_2\) if one additionally obeys the transmission conditions \(u_1|_\Gamma = u_2|_\Gamma\) and \(\frac{\partial u_1}{\partial n} = \frac{\partial u_2}{\partial n}\) across the interface \(\Gamma = \Omega_1 \cap \Omega_2\). Furthermore, domain decomposition methods like the Dirichlet-Neumann or the Robin method can be used to solve the transmission problem. But what if the continuity condition \(u1|_\Gamma = u2|_\Gamma\) is replaced by the more general one \(F_1(u_1|_\Gamma) = F_2(u_2|_\Gamma)\) with - possibly nonlinear - operators \(F_1, F_2\) acting on the trace space \(H^{\frac{1}{2}}_{00} (\Gamma)\)? We discuss variations of this situation, sketch proofs for well-posedness and convergence of corresponding nonlinear domain decomposition methods, and address open questions as well as limitations inherent in the proofs. These situations occur in nature when we intend to simulate saturated - unsaturated groundwater flow in a way that we find quite nice. This is because we have a solver-friendly discretization for the Richards equation in homogeneous soil and monotone multigrid as a powerful solver in this case. We present numerical examples which also address optimization of the domain decomposition methods. Joint work with: R. Kornhuber, O. Sander (FU Berlin), M. Discacciati (EPFL Lausanne)

  • Raphael Kruse : Finite-Elemente Methoden für PDEs
    2011/01/28, 14:15, V5-148, AG Dynamische Systeme

  • José Augusto Ferreira (Coimbra):
    2011/01/24, 17:15, V5-148, Seminar Numerical Analysis
    Abstract: Integro-differential equations of Volterra type arise, naturally, in many applications such as for instance heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. The aim of this talk is to presente supraconvergent finite difference methods for such integro-differential equations. As these finite difference methods can be seen as piecewise linear finite elements method combined with special quadrature formulas, our results are superconvergence results in the finite element language.

  • Bernd Simeon (Kaiserslautern): Transiente Sattelpunktprobleme und Anwendungen in der Mechanik
    2011/01/24, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: n. a.

  • Jens Rottmann-Matthes : Optimale Integralabschätzungen für die Laplacetransformation
    2011/01/21, 14:15, V5-148, AG Dynamische Systeme

  • Yi Zhou : Die Takens-Bogdanov Singularitaet
    2011/01/14, 14:15, V5-148, AG Dynamische Systeme

  • Sven Kreimer-Huenke : Spektralmethoden fuer Evolutionsgleichungen
    2010/12/17, 14:15, V5-148, AG Dynamische Systeme

  • Filip Rindler (Oxford): Rigidity for some differential inclusions involving the gradient and the symmetrized gradient
    2010/12/14, 18:00, V5-148, Seminar Evolutionsgleichungen
    Abstract: We look at (smooth) functions \(u:\mathbb{R}^2\to\mathbb{R}^2\) whose gradient can be written in the form \(\nabla u(x)=P g(x)\) for a fixed \(\mathbb{R}^{2\times 2}\)-matrix \(P\) and a smooth scalar function \(g:\mathbb{R}^2\to \mathbb{R}\). Similarly, replacing the gradient by the symmetrized gradient, we consider (smooth) solutions \(u:\mathbb{R}^2\to\mathbb{R}^2\) of \(\left( \nabla u(x)+ (\nabla u(x))^T\right)/2 = Pg(x),\) with \(P\) a fixed symmetric \(\mathbb{R}^{2\times 2}\)-matrix and \(g\) as before. I will establish conditions on the existence of such solutions, depending on the value of \(P\), and I will prove some (probably unexpected) 'rigidity' properties of any function \(u\) satisfying one of the above properties. The situation in two dimensions already contains all the essential features and exposes fascinating connections to Harmonic Function Theory and Complex Analysis. Whereas the presented results are new (particularly in the case of the symme trized gradient) and are a core ingredient in the results described in my se minar talk, the discussion is completely elementary and involves only first-year multi-dimensional Differential Calculus. I will also point out a few references for further study of such 'rigidity arguments'.

  • Filip Rindler (Oxford): Minimization problems in the space BD of functions of bounded deformation
    2010/12/13, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: The space BD of functions of bounded deformation consists of all L^1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient (Δ u + Δ u^T)/2) is representable as a finite Radon measure. Such functions play an essential role in modern theories of (linear) elasto-plasticity in a variational framework. In this talk, I will present the first general theorem on solvability of minimization problems for integral functionals with linear growth on the whole space BD. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative, corresponding to fractal phenomena in nature. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, which is based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to prove the crucial lower semicontinuity result without an Alberti-type theorem in BD, which is not available at present. A similar strategy also allows to considerably simplify the proof of the classical lower semicontinuity theorem in the space BV of functions of bounded variation.

  • Eduard Feireisl (Prag): Asymptotic behavior of compressible viscous fluids
    2010/12/09, 17:15, V3-201, Mathematical Colloquium
    Abstract: We discuss the long-time behavior of solutions to energetically closed fluid systems. By this we mean that the system possesses an energy functional, the value of which is conserved in time, and an entropy, the total amount of which is nondecreasing. Various topics are addressed: Equilibrium solutions, thermodynamic stability, the existence of attractors, boundedness of globaltrajectories

  • Eduard Feireisl (Prag): Singular limits of compressible fluids driven by large external forces
    2010/12/07, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider a compressible fluid excited by large external forces. Using the abstract result of Kato we show that the acoustic component of the velocity vanishes in the incompressible regime although there is a strong interaction between the force and acoustic waves. Applications are given to rotating fluid systems.

  • Lina Fruendt : Symbolische Dynamik und transversale homokline Punkte
    2010/12/03, 14:15, V5-148, AG Dynamische Systeme

  • Martin Stynes (Cork): A new finite element method for singularly perturbed reaction-diffusion problems
    2010/11/30, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Consider the singularly perturbed linear reaction-diffusion problem -ε^2 Δ u + bu = f in Ω ⊂ R^d, u=0 on δΩ, where 0 < ε << 1, b > 0 and d≥ 2. It is argued that for this type of problem, the standard energy norm is too weak a norm to meas ure adequately the errors in solutions computed by finite element methods. A stronger norm is introduced and a mixed finite element constructed whose solution is quasi-optimal in this new norm, and a duality argument is used to show that this solution attains a higher order of convergence in the L^2 norm. Error bounds derived from these analyses are presented for the cases d=2,3. For a problem posed on the unit square in R^2, an error bound that is uniform in ε is derived when the new method is implemented on a Shishkin mesh. Numerical results are presented to show the superiority of the new method over current finite element methods for singularly perturbed reaction-diffusion problems.

  • Lina Fruendt : Symbolische Dynamik und transversale homokline Punkte
    2010/11/26, 14:15, V5-148, AG Dynamische Systeme

  • Aneta Wróblewska: Generalized Stokes system in Orlicz spaces
    2010/11/22, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: The talk concerns the generalized Stokes system ∂_t u - div S(t, x,Du) + ∇ p = f in (0, T) × Ω div u = 0 in (0, T) × Ω u(0, x) = u_0 in Ω u(t, x) = 0 on (0, T) × ∂ Ω, with the nonlinear viscous term having growth conditions prescribed by an N-function. We will consider the case of monotone functions. Our main interest is directed to relaxing the assumptions on the N-function. To prove existence of weak solutions to our equations we will show the Korn-Sobolev inequality for anisotropic Orlicz spaces and the fact that closures of smooth compactly supported functions w.r.t. modular and weak star topology of symmetric gradient coincides.

  • Jens Rottmann-Matthes : Laplace-Transformation und Stabilitaet der konvektiven Waermeleitungsgleichung
    2010/11/19, 14:15, V5-148, AG Dynamische Systeme

  • François Murat (Paris) : How to solve second order elliptic equations with right-hand side in L^1
    2010/11/18, 17:15, V3-201, Mathematical Colloquium
    Abstract: In this lecture, I will consider the problem: find u such that -div(A(x)Du) = f in Ω, u = 0 on Ω when the matrix A is coercive with measurable bounded coefficients and when f belongs to L^1(Ω). The main difficulty of the problem is to define a convenient notion of solution. Such a definition (the 'solution by transposition') was introduced by G. Stampacchia in 1973. However, this definition is essentially restricted to the linear case. In this lecture, I will present the notion of 'renormalized solution', which can be extended in a natural way to the case of a second order monotone operator in divergence form posed on W^{1,p}_0 (Ω) Definition: u is a renormalized solution of the problem if u : Ω → ℝ is measurable and a.e. finite T_n(u) ∈ H^1_0 (Ω) for every n > 0, 1/n ∫ |DT_n(u)|^2 → 0 as n → + ∞, -div(h(u)A(x)Du)+h'(u)A(x)DuDu = h(u)f in D'(Ω) for every h ∈ C^1_c (ℝ) This definition allows one to prove that the problem has a renormalized solution, that this renormalized solution is unique, and that it depends continuously on f, i.e. that in this framework the problem is well posed in the sense of Hadamard.

  • Thomas Hanning : Numerische Berechnung invarianter Mannigfaltigkeiten
    2010/11/05, 16:00, V5-148, AG Dynamische Systeme

  • Wolf-Juergen Beyn (Bielefeld) : Eine Integralmethode fuer nichtlineare Eigenwertprobleme
    2010/10/29, 14:15, V5-148, AG Dynamische Systeme

  • Petra Wittbold (Essen): On a nonlinear elliptic-parabolic integro-differential equation with L^1-data
    2010/10/27, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider an initial-boundary-value problem for a nonlinear elliptic-parabolic integro-differential equation. Equations of this type have been proposed to model flow through porous media with memory effects and nonlinear heat flow in certain dielectric materials at low temperatures. Following classical results by Gripenberg, existence and uniqueness of a generalized solution to an associated abstract operator integro-differential equation can be shown for abitrary L^1-data. We study the question of regularity of this generalized solution and discuss conditions that ensure that the generalized solution is a strong, weak, entropy or renormalized solution.

  • Thorsten Rohwedder (Berlin): The electronic Schrödinger equation and an analysis for the continuous Coupled Cluster method
    2010/10/25, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Many properties of atoms, molecules and solid states are described quite accurately by solutions Ψ of the electronic Schrödinger equation H Ψ = E Ψ , an extremely high-dimensional operator eigenvalue equation for the Hamiltonian H of the system under consideration. Of utmost interest is the smallest eigenvalue of H and the corresponding eigenfunction, giving the ground state energy and the electronic wave function describing the ground state, respectively.
    In the first part of this talk, the audience is introduced to the electronic Schrödinger equation and the typical problems that arise when dealing with this equation. We will then introduce the Coupled Cluster method, a method that is standardly used in quantum chemistry for highly accurate calculations. Coupled Cluster (CC) is standardly formulated as an ansatz for the approximation of the Galerkin solution of the Schrödinger equation with in a given discretisation [1].
    We globalise this ansatz to infinite dimensional spaces, thus obtaining an equivalent reformulation of the original, continuous Schrödinger equation in terms of a root equation for a nonlinear operator A, corresponding to the finite dimensional CC function f. We show local strong monontonicity of the CC function, derive some existence and uniqueness results and prove a goal-oriented error estimator [2] for the ground state energy. We conclude with a short discussion of the algorithmic treatment of the CC root equation.
    [1] R. Schneider, Num. Math. 113, 3, 2009.
    [2] R. Becker, R. Rannacher, Acta Numerica 2000 (A. Iserlet, ed.), Cambridge University Press, 2001

  • Etienne Emmrich (Bielefeld): Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation
    2010/10/19, 18:00, V5-148, Seminar Evolutionsgleichungen
    Abstract: The initial-value problem for doubly nonlinear evolution equations of the type u'' + Au' + Bu = f is studied. The time-dependent operator A is assumed to satisfy a certain growth condition and is supposed to be the sum of a monotone, coercive and hemicontinuous principal part and a strongly continuous perturbation. The operator B is supposed to be the sum of an operator that induces an inner product and a non-monotone perturbation fulfilling a certain local Hölder-type continuity condition.
    Examples are viscous regularisations of equations arising in elasticity.
    For a full discretisation combining a modification of the Stürmer-Verlet method with an inner approximation scheme, weak convergence of piecewise polynomial prolongations of the discrete solutions are proven. These results also imply the existence of a weak solution.
    For the time discretisation, also variable time grids are allowed as long as the deviation from equidistance is not too large.
    A crucial step in the convergence proof is the equivalence with an integro-differential equation of parabolic type.
    In the case of non-monotone perturbations, a priori estimates in fractional Sobolev-Slobodetskii spaces are a prerequisite for the necessary compactness argument.
    This is joint work with Mechthild Thalhammer (Innsbruck, Austria).

  • Boris Andreianov (Besançon): A one dimensional hyperbolic particle-fluid interaction model, theory and numerical approximation
    2010/10/18, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: D'Alambert paradox states that, if the fluid viscosity is neglected, the resulting force of the fluid action on an immersed moving solid is zero. In other words, planes could not fly if the air viscosity was totally neglected ...
    In this talk we present a simple ''particle-in-Burgers'' model combining Burgers equation without viscosity and a viscous interaction prenomenon between the fluid and a point particle. The fluid equation becomes u_t + ( (u^2)/2 )_x = -λ (u-h'(t)) δ_0(x-h(t)) where x=h(t) is the particle path, and δ_0 is the Dirac-δ. The equation is coupled to an evolution equation for the particle path h(⋅); the resulting force on the particle is proportional to the jump of the normal fluxes of the fluid across the particle path, and the whole model conserves the quantity of movement.
    We study well-posedness and numerical approximation of the coupled problem by (as simple as possible) finite volume schemes. Analysis tools have much in common with the case of conservation laws with discontinuous flux.
    Joint work with F. Lagoutiere, N. Seguin and T. Takahashi.

  • Rajesh Kumar (Magdeburg): Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage equations
    2010/09/20, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk I will discuss the stability and convergence analysis of a finite volume scheme for solving aggregation-breakage equation which is a non-linear integropartial differential equation.
    First we rewrite this equation in conservative form and then we apply the well known finite volume method. By showing Lipschitz continuity of the numerical fluxes we found that the scheme is second order convergent independently of the meshes for pure breakage problem while for pure aggregation, due to non-linearity, it shows second order convergent only on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a non-uniform random grid. Finally, for the coupled problem as well, a second order of convergence is shown on uniform and non-uniform smooth meshes whereas first order convergence is obtained on random grid.
    The mathematical results of convergence analysis are also validated numerically.In this talk I will discuss the stability and convergence analysis of a finite volume scheme for solving aggregation-breakage equation which is a non-linear integropartial differential equation.
    First we rewrite this equation in conservative form and then we apply the well known finite volume method. By showing Lipschitz continuity of the numerical fluxes we found that the scheme is second order convergent independently of the meshes for pure breakage problem while for pure aggregation, due to non-linearity, it shows second order convergent only on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a non-uniform random grid. Finally, for the coupled problem as well, a second order of convergence is shown on uniform and non-uniform smooth meshes whereas first order convergence is obtained on random grid.
    The mathematical results of convergence analysis are also validated numerically.

  • Hagen Eichel (Magdeburg): Supercloseness und Superkonvergenz stabilisierter Finite-Elemente-Diskretisierungen niedriger Ordnung des Stokes-Problems
    2010/09/20, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Der Vortrag handelt von Supercloseness und Superkonvergenz stabilisierter Finite-Elemente-Methoden angewandt auf das Stokes-Problem. Es werden sowohl konsistente residuenbasierte Stabilisierungsmethoden, als auch inkonsistente Stabilisierungstypen, basierend auf lokaler Projektion, betrachtet. Weiterhin wird die Supercloseness des Linearteils der MINI-Element-Lösung gezeigt. Die Resultate über Supercloseness werden für Dreieckselemente, sowie achsenparallele Rechtecke und Quader hergeleitet, allerdings auch Erweiterungen auf allgemeine Gitter diskutiert. Durch einen geeigneten Post-Prozess können dann Superkonvergenz-Resultate erzielt werden. Anschließend werden numerische Berechnungen zur Unterstützung der theoretischen Betrachtung präsentiert.The supercloseness and superconvergence property of stabilized finite element methods apllied to the Stokes problem are studied. We consider consistent residual based stabilization methods as well as inconsistent local projection type stabilizations. Moreover, we are able to show the supercloseness of the linear part of the MINI-element solution which has been previously observed in practical computations. The results on supercloseness hold on three-directional triangular, axiparallel rectangular, and bricktype meshes, respectively, but extensions to more general meshes are also discussed. Applying an appropriate postprocess to the computed solution, we establish superconvergence results. Numerical examples illustrate the theoretical predictions.

  • Eskil Hansen (Lund): Time stepping schemes for nonlinear parabolic problems and a theorem by Brezis and Pazy
    2010/08/16, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: There is a rich theory describing the approximation of nonlinear semigroups. At its core one finds the results by Brezis and Pazy, who generalize the classical linear results of Trotter and Chernoff. Even though the theory was derived in the early seventies, it is virtually unknown within the numerical community. The aim of this talk is therefore to illustrate how this nonlinear theory can be used as a corner stone when deriving convergence for time stepping schemes applied to fully nonlinear parabolic equations. In particular, we will illustrate our framework by deriving the convergence for splitting schemes and DIRK methods under minimal regularity assumptions.

  • Mechthild Thalhammer (Innsbruck): Exponential operator splitting methods for nonlinear evolutionary problems involving critical parameters
    2010/08/02, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk, the error behaviour of exponential operator splitting methods for nonlinear evolutionary problems is investigated. In particular, an exact local error representation that is suitable in the presence of critical parameters is deduced. Essential tools in the theoretical analysis including time-dependent nonlinear Schrödinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients are an abstract formulation of differential equations on function spaces and the formal calculus of Lie-derivatives. The general mechanism is exposed on the basis of the least technical example method, the first-order Lie-Trotter splitting. The conclusion that exponential operator splitting methods are favourable for the time-integration of nonlinear Schrödinger equations in the semi-classical regime with Wentzel-Kramers-Brillouin initial condition under the time stepsize restriction h = O(p√{ε}), where 0 < ε < < 1 denotes the critical parameter and p the order of the splitting method, is confirmed by a numerical example for the time-dependent Gross-Pitaevskii equation.

  • Yuri Latushkin (Columbia, Missouri): Birman-Schwinger operators and the Evans function
    2010/07/19, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: This is a review of some recent work related to the spectral theory of the Birman-Schwinger type integral operators familiar from quantum mechanics, and the Evans function, a popular tool in stability analysis of traveling waves. Some formulas are given relating the Evans function and its derivative and the modified Fredholm determinants of the respective Birman-Schwinger operators.

  • Günter M. Ziegler (TU Berlin): A strictly colored Tverberg theorem
    2010/07/15, 16:15, V3-201, Mathematical Colloquium
    Abstract: Im Herbst 1964 saß der junge norwegische Mathe matiker Helge Tverberg in einem Hotelzimmer in Manchester, fror, und bewies ein bemerkenswertes Resultat: Wenn man 3r-2 Punkte in der Ebene hat, so kann man diese in r Gruppen von höchstens drei Punkten aufteilen, so dass die r dadurch bestimmten Dreiecke, Strecken und Punkte einen gemeinsamen Schnittpunkt haben. Ein Punkt weniger reicht nicht. Eine d-dimensionale Version des Satzes, mit (d+1)(r-1)+1 Punkten, hat Tverberg gleich mitbewiesen.
    1992 haben dann Vrecica und Zivaljevic einen "gefärbten Tverberg-Satz" präsentiert. Für den Beweis wurden elegante topologische Methoden und kombinatorische Strukturen verwendet, aber das Resultat war nicht scharf - Vrecica und Zivaljevic brauchten mehr Punkte als erwartet. Jetzt gibt es eine überraschende neue, scharfe "gefärbte" Version des ursprünglichen Satzes von Tverberg, neue Beweismethoden kommen zum Einsatz - es gibt Fortschritt! Darüber will ich berichten.
    (Gemeinsame Arbeit mit Pavle V. M. Blagojevic und Benjamin Matschke)In autumn 1964 the young norwegian mathematician Helge Tverberg sitting freezing in a hotel room in Manchester proved a remarkable result: 3r-2 points in the plain can be divided into r groups of not more then three points such that the r triangles, lines and points have a common intersection point. One point less is not enough. Also, a d-dimensional version of this theorem was proven with (d+1)(r-1)+1 points.
    In 1992, Vrecica and Zivaljevic presented a "colored Tverberg theorem". Elegant topological methods and combinatorial structures are used for the proof. Nevertheless, the result was not sharp - Vrecica and Zivaljevic needed more points than expected. Now there is progress: A new and surprising sharp colored version of the original theorem of Tverberg has arised, together with new proof methods. This is the subject of the talk.
    (Joint work with Pavle V. M. Blagojevic and Benjamin Matschke

  • Alexander Ostermann (Innsbruck): Numerical analysis of operator splitting methods
    2010/07/12, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Splitting methods form a large class of competitive time discretisations of evolution equations. The reason for their frequent use is that the splitting procedure yields time stepping schemes which dramatically reduce the required computational effort, compared to schemes based on the full vector field.
    After a brief introduction to the concept of splitting and the non-stiff convergence theory, we concentrate on problems with unbounded operators. It turns out that the non-stiff order conditions are sufficient to get optimal convergence orders for exponential splitting methods in the stiff case. We discuss analytic frameworks for proving (optimal) convergence results, and we introduce a new setting that is applicable for a wide variety of linear equations and their dimension splittings. In particular, we analyse parabolic problems with homogeneous Dirichlet or Neumann boundary conditions on bounded domains.
    We further discuss a new class of splitting methods of orders up to fourteen based on complex coefficients. These results resolve the open question whether there exist splitting schemes with convergence rates greater than two in the context of analytic semigroups. As a concrete application we consider once more parabolic equations and their dimension splittings. The sharpness of our theoretical error bounds is illustrated by numerical experiments.

  • Caren Tischendorf (Köln): Solution Approaches for Abstract Differential-Algebraic Equations
    2010/07/05, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: The simulation of complex systems describing different physical effects becomes more and more of interest in various applications, for instance, in chip design, in structural mechanics, in biomechanics and in medicine. The modeling of complex processes often lead to coupled systems that are composed of ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs).
    Such coupled systems can be regarded in the general framework of abstract differential-algebraic equations of the form
    tischendorf1
    This equation is to be understood as an operator equation with operators tischendorf2, tischendorf3 and tischendorf4 acting in real Hilbert spaces where tischendorf5 is the solution belonging to a problem adapted space.
    If the Hilbert spaces are chosen to be the finite dimensional space Rm, then we obtain a differential-algebraic equation. Choosing tischendorf6 and tischendorf7 as the natural embedding operators, we obtain an evolution equation. If, additionally, tischendorf8 is a second-degree differential operator in space, it leads to a parabolic differential equation. For elliptic differential equations, the operators tischendorf6 and tischendorf7 are identically zero.
    For most coupled systems, the operators tischendorf6 and tischendorf7 are neither identically zero nor invertible on the time interval [t0,T]. A general theory of abstract differential-algebraic equations (ADAEs) does not exist and can not be expected to be given considering alone the complexity of problems simulating partial differential equations. However, special classes of ADAEs have recently been successfully analyzed and simulated.
    In particular, we discuss solvability and perturbation results via two different approaches. The first one addresses linear ADAEs with constant coefficients using Laplace transformation. The second one handles linear ADAEs with monotone, time dependent coefficients by a Galerkin approach.

  • Christian Kuehn (Cornell):
    2010/06/14, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: We start with an introduction to fast-slow systems. The geometric viewpoint of the theory will be emphasized. Then we discuss the three-dimensional FitzHugh-Nagumo (FHN) equation and its bifurcations. The singular limit bifurcation diagram of the FHN equation will be derived. The computation and interaction of different types of invariant manifolds will be emphasized to explain the dynamics. We shall also briefly look at mixed-mode oscillations (MMOs) in the FHN equation and more general fast-slow system.

  • Dimitri Puhst (TU Berlin): Fractional derivatives and their applications
    2010/06/11, 14:15, V5-148, Seminar Numerical Analysis

  • Jens Rademacher (CWI Amsterdam): Mechanisms of semi-strong interaction in multiscale reaction diffusion systems
    2010/06/07, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: In spatial multiscale reaction diffusion systems where some diffusion lengths are much shorter than the rest, interfaces can form where only the components with the short scale localise. The interaction between such interfaces is called semi-strong as it is driven by the nonlocalised components. Cases where the interface motion is of the order of the square of the short diffusion lengths ('second order') have been studied over the past decade. By formal expansions and numerical studies we show that the interaction strength can also be of the same order as the short diffusion length ('first order').
    We illustrate these mechanisms in the Schnakenberg model and investigate interaction manifolds and their stability. Taking a model independent point of view, starting only from a dichotomy in diffusion lengths, characteristic equations of motion of interfaces for first and second order semi-strong interaction can be derived. For first order pulse interaction with a single long diffusion length and under certain natural assumptions several explicit Lyapunov-functionals such as the largest interpulse distance are found.
    This is partly joint work with J. Ehrt and M. Wolfrum (WIAS, Berlin).

  • Mario Botsch (Bielefeld): Polyhedral Finite Elements
    2010/05/31, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Finite element simulations of deformable objects are typically based on spatial discretizations using either tetrahedral or hexahedral elements. This allows for simple and efficient computations, but in turn requires complicated remeshing in case of topological changes or adaptive simulations. In this talk I will show how arbitrary polyhedral elements can be used in FEM simulations, thereby avoiding the need for remeshing (and thus simplifying) adaptive refinement, interactive cutting, and fracturing of the simulation domain.

  • Rolf Dieter Grigorieff (TU Berlin): Qualokation bei periodischen Pseudo-Differentialoperatoren
    2010/05/03, 16:15, V5-148, Seminar Numerical Analysis
    Abstract: Qualokation bezeichnet ein auf I. Sloan zurückgehendes Diskretisierungsverfahren, das zwischen dem Kollokations- und dem Petrov-Galerkin-Verfahren angesiedelt ist. Der Vortrag handelt von der Konvergenzanalyse des Verfahrens angewandt auf periodische Pseudo-Differentialoperatoren unter Verwendung von Splines mit mehrfachen Knoten als Ansatz- und Testraum. Eine Grundlage dafür sind Approximationseigenschaften der Splineräume, die ebenfalls vorgestellt werden. Besondere Aufmerksamkeit verdient die Bestimmung von Bedingungen an die Parameterwahl des Verfahrens, mit denen eine zusätzliche Konvergenzordnung einhergeht. Qualocation denotes a discretization scheme going back to I. Sloan located somewhere between the collocation method and the Petrov-Galerkin method. The talk covers the convergence analysis of the method applied to periodic pseudo-differential operators using splines with multiple vertices as an ansatz- and test space. This analysis is based on approximation properties of the also introduced spline spaces. We focus on a proper parameter choice providing an additional order of convergence.

  • Jussi Behrndt (TU Berlin): Spektraleigenschaften einer Klasse elliptischer Differentialoperatoren auf beschraenkten und unbeschraenkten Gebieten
    2010/04/22, 17:15, V3-201, Mathematical Colloquium
    Abstract: In this lecture we consider a formally symmetric second order elliptic differential expression L on a bounded or unbounded domain Ω with smooth boundary ∂Ω. Our aim is to describe the spectral properties of a family of selfadjoint realizations of L in L2(Ω) with nonlocal boundary conditions on ∂Ω. For this we apply boundary triplet and Dirichlet-to-Neumann techniques, as well as general perturbation methods from abstract operator theory.

  • Mechthild Thalhammer (Innsbruck): High-order time-splitting spectral methods for nonlinear Schrödinger equations
    2010/02/22, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk, I will address the issue of efficient numerical methods for the time integration of nonlinear Schrödinger equations. As model problems, I will consider systems of coupled Gross-Pitaevskii equations that arise in quantum physics for the description of multi-component Bose-Einstein condensates. My intention is to study the quantitative and qualitative behaviour of high-accuracy discretisations that rely on time-splitting Fourier and Hermite spectral methods. In particular, this includes a stability and convergence analysis of high-order exponential operator splitting methods for evolutionary Schrödinger equations. Numerical examples illustrate the theoretical results.

  • Aneta Wróblewska (Warsaw): Unsteady flows of nonhomogeneous non-Newtonian incompressible fluids with growth conditions in Orlicz spaces
    2010/01/25, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Our purpose is to show existence of weak solutions to unsteady flow of non-Newtonian incompressible nonhomogeneous fluids with nonstandard growth conditions of the stress tensor. We are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening. Since we are interested in flows with the rheology more general then power-law-type, we describe the growth conditions with help of general x-dependent convex function and formulate our problem in generalized Orlicz spaces.

  • Jos Gesenhues (Bremen): The Kuramoto model: Modelling, analysis and simulation
    2010/01/11, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Das Kuramoto-Modell ist ein Modell für das Verhalten einer großen Anzahl schwach gekoppelter Oszillatoren. Insbesondere können an ihm Synchronisierungsphänomene untersucht werden. In dieser Arbeit werden sowohl das klassische diskrete Kuramoto-Modell als auch ein kontinuierlicher Ansatz, der auf eine partielle, nichtlineare Integro-Differentialgleichung führt, modelliert und analysiert. Das zentrale Ergebnis der Analyse ist ein kritischer Kopplungswert, ab dem Synchronität auftritt. Dieser wird für beide Modelle in Simulationen bestätigt. The Kuramoto model is a model describing the behavior of a large number of weakly coupled oszillators. In particular, synchronization phenomena can be studied with it. In this work the classical discrete Kuramoto model is modeled and analyzed as well as a continous ansatz leading to a partial nonlinear integro-differential equation. The central result of the analysis is a critical coupling value, from which on synchronization occurs. This will be verified for both models by simulation.

  • David Šiška (London): Finite-Difference Approximations for Normalized Bellman Equations
    2010/01/11, 10:15, U5-133, Seminar Numerical Analysis
    Abstract: A class of stochastic optimal control problems involving optimal stopping is considered. Methods of Krylov (Appl. Math. Optim. 52(3):365-399, 2005) are adapted to investigate the numerical solutions of the corresponding normalized Bellman equations and to estimate the rate of convergence of finite difference approxi- mations for the optimal reward functions.

  • Christopher Hartleb (TU Ilmenau): Lower bounds for the independence number of a graph in terms of the numbers of vertices and edges
    2009/12/22, 10:15, V4-119, Seminar Numerical Analysis

  • Volker Mehrmann (TU Berlin / MATHEON): Theory and numerical methods for the stability analysis of differential algebraic systems
    2009/12/14, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of various kinds can be analyzed via the resulting underlying ODE. Some essential differences between the spectral theory for ODEs and that for DAEs are pointed out. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.

  • Arjen Doelmann (Leiden): Busse balloons and Hopf dances, or: Bifurcations of Spatially Periodic Patterns
    2009/12/07, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk we propose to study generic aspects of the Busse balloon associated to spatially periodic patterns in reaction-diffusion equations. The Busse balloon -- named after the physicist Friedrich Busse -- is defined as the region in (wave number, parameter space) for which stable periodic patterns exist; the boundary of the Busse balloon is determined by bifurcations/destabilizations. After a general introduction, in which we interpret the Turing bifurcation and the associated side band instability mechanism, as a well-studied and well-understood generic aspect of the Busse balloon, we introduce a novel destabilization mechanism for reversible spatially periodic patterns. This Hopf dance mechanism occurs for long wavelength patterns near the homoclinic tip of a Busse balloon. Here, the boundary of the Busse balloon locally has a fine-structure of two intertwining ?dancing? Hopf destabilization curves (or manifolds) that limit on the Hopf bifurcation value of a homoclinic limit pulse.

  • Thorsten Hüls: On r-periodic orbits of k-periodic maps
    2009/11/30, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk, we analyze r-periodic orbits of k-periodic difference equations, i.e.
    huels09
    and their stability. We discuss that, depending on the values of r and k, such orbits generically only occur in finite dimensional systems that depend on sufficiently many parameters, i.e. they have a large codimension in the sense of bifurcation theory. As an example, we consider the periodically forced Beverton-Holt model, for which explicit formulas for the globally attracting periodic orbit, having the minimal period k=r, can be derived. When r factors k the Beverton-Holt model with two time-variant parameters is an example that can be studied explicitly and that exhibits globally attracting r-periodic orbits. For arbitrarily chosen periods r and k, we develop an algorithm for the numerical approximation of an r-periodic orbit and of the associated parameter set, for which this orbit exists. We apply the algorithm to the generalized Beverton-Holt and another example that exhibits periodic orbits with r and k relatively prime.

  • Georgy Kitavtsev (WIAS Berlin): Reduced ODE models describing coarsening dynamics of slipping droplets
    2009/11/23, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk the topic of reduced ODE models corresponding to a family of one-dimensional lubrication equations derived by Münch et al. 06' is addressed. This family describes the dewetting process of nanoscopic thin liquid films on hydrophobized polymer substrates due to the presence of several intermolecular forces and takes account of different ranges of slip-lengths at the polymer substrate interface. Reduced ODE models derived from underlying lubrication equations allow for an efficient analytical and numerical investigation of the latest stage of the dewetting process: coarsening dynamics of the remaining droplets. We first give an asymptotical derivation of these models and use them to investigate the influence of slip-length on the coarsening dynamics. In a so called strong-slip case we find a unique critical slip-length at which the direction for migration of droplets changes. In the second part of the talk we present a new geometric approach which can be used for an alternative derivation and justification of above reduced ODE models and is based on a center-manifold reduction recently applied by Mielke and Zelik 08' to a certain class of semilinear parabolic equations. One of the main problems for a rigorous justification of this approach is investigation of the spectrum of a lubrication equation linearized at the stationary solution, which describes physically a single droplet. The corresponding eigenvalue problem turns out to be a singularly perturbed one with respect to a small parameter ε tending to zero. For this problem we show existence of an ε-dependent spectral gap between a unique exponentially small eigenvalue and the rest of the spectrum.

  • Sergei Pilyugin (St. Petersburg): Lipschitz shadowing property
    2009/11/09, 14:15, V5-148, Seminar Numerical Analysis

  • Dario Götz (TU Berlin): Existenz von schwachen Lösungen und Zeitdiskretisierung der Bewegungsgleichung verallgemeinerter nicht-Newtonscher Fluide
    2009/10/19, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Nicht-Newtonsche Fluide verhalten sich, meist aufgrund von zugrundeliegenden Mikrostrukturen, anders, als man es von gewöhnlichen Flüssigkeiten erwartet; so zum Beispiel Blut, Lava, Farbe, Tomatenketchup, Polymere, Emulsionen oder Suspensionen.
    Im Vortrag untersuchen wir die schwache Lösbarkeit der instationären Bewegungsgleichung für inkompressible Fluide mit scherratenabhängiger Viskosität, wobei wir für den Spannungstensor eine p-Struktur voraussetzen. Im Fall goetz1 kann die schwache Lösbarkeit mithilfe der Theorie monotoner Operatoren gezeigt werden. Mathematisch anspruchsvoller ist der Fall p < 2, der hier behandelt werden soll. Ziel ist es, die schwache Lösbarkeit des Problems für alle goetz2 zu zeigen. Ein Problem, das sich dabei ergibt, ist die fehlende Regularität der zeitlichen Ableitung der Lösung.
    Die Beweisidee beruht auf einer Zeitdiskretisierung durch ein implizites Euler-Verfahren. Mithilfe der sogenannten parabolischen Lipschitz-Truncation-Methode und geeigneten Regularitätsaussagen in gebrochenen Sobolew-Räumen wird das Problem der fehlenden Regularität der zeitlichen Ableitung gelöst. Damit geht eine Repräsentation des in der schwachen Formulierung verschwindenden Druckes einher, die essenziell für diese Beweisidee ist. Non-Newton fluids behave, typically due to underlying microstructures, different than we expect from ordinary fluids; e.g. blood, lava, color, tomato ketchup, polymeres, emulsions or suspensions.
    In this talk we analyze the existence of weak solutions of the non-stationary equation of motion for incompressible fluids with shear rate dependent viskosity and p-structure for the stress tensor. In the case goetz1 the existence of weak solutions follows from the theory of monotone operators. The more interesting case, that we focus on here, is p < 2. We aim to show the existence of a weak solution of the problem for all goetz2. A problem that occurs is the missing regularity of the time derivations of the solution.
    The idea of the proof is based on a time discretization by the implicit Euler method. Using the so-called Lipschitz truncation method and proper regularities of the Sobolev spaces the problem of the missing regularity can be solved. A representation of the pressure vanishing in the weak formulation comes along with this ansatz and is essential for the idea of the proof.

  • Arnulf Jentzen (Frankfurt): Taylor expansions for stochastic partial differential equations
    2009/07/24, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Taylor expansions of stochastic partial differential equations (SPDEs) of evolutionary type and their first applications to numerical analysis are presented. The key instruments for deriving such Taylor expansions are the semigroup approach, i.e. to understand the SPDE as a mild integral equation, and an appropriate recursion technique.

  • David Speer (Bielefeld): Directing Brownian Motion in Periodic Potentials
    2009/07/03, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider a single Brownian particle subjected to periodic and symmetric potentials. Directed particle transport can be achieved only by breaking symmetry, such as applying a constant force. Usually, particle transport is in the direction of that force, in line with the second law of thermodynamics. In non-equilibrium, this behaviour may be drastically different due to a subtle interplay of deterministic chaos, symmetry and stochastic forces. This may be exploited to achieve almost complete control of transport direction, even transport directly against that force (negative absolute mobility) [1]. Recently, the effect was observed experimentally for Josephson junctions [2].
    [1] D. Speer et al., Europhys. Lett. 79, 10005 (2007), D Speer et al., Phys. Rev. Lett. 102, 124101 (2009)
    [2] J. Nagel et al., Phys. Rev. Lett. 100, 217001 (2008)

  • Lutz Angermann (Clausthal): Rosenbrock-Verfahren fü PDAEs
    2009/06/19, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Mathematical modelling of various problems in science, engineering, medicine etc. lead to (in general nonlinear) equation systems consisting of coupled equations of different kinds, e.g. parabolic, elliptic and ordinary differential equations and algebraic equations. Such systems are called partial differential equations (PDAEs).
    In the development of numerical methods for the solution of such systems via the vertical line method (spatial discretization first) a set of new questions arise when we try to transfer methods for differential algebraic equation of moderate size to the evolving (ordinary) differential algebraic equations (called MOL-DAEs). In this talk these questions - together with adequate solution approaches - will be presented. Die mathematische Modellierung zahlreicher Probleme aus Naturwissenschaft, Technik, Medizin etc. fürt auf (i. Allg. nichtlineare) Gleichungssysteme, die aus gekoppelten Gleichungen unterschiedlichen Typs bestehen, zum Beispiel aus parabolischen, elliptischen, gewöhnlichen Differentialgleichungen und aus algebraischen Gleichungen. Derartige Systeme heißen partielle differentiell-algebraische Gleichungen (PDA-Systeme, engl.: partial differential algebraic equations, PDAEs).
    Die Entwicklung numerischer Approximationsverfahren für die Lösung dieser Systeme mittels der vertikalen Linienmethode (Diskretisierung zuerst im Raum) wirft bei dem Versuch der Übertragung bekannter, für differentiell-algebraische Gleichungen moderater Größe entworfener Methoden auf die entstehenden (gewöhnlichen) differentiell-algebraischen Gleichungen (sog. MOL-DAEs) eine Reihe zusätzlicher oder neuer Fragestellungen auf, die - einschließlich entsprechender Lösungsansätze -- im Vortrag erörtert werden sollen.

  • Mario Ohlberger (Münster): Reduzierte Basis Techniken für parametrisierte nichtlineare Evolutionsgleichungen
    2009/05/15, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: In this talk we will present the reduced basis method for the efficient computation of parametrized evolution equations. The method allows an offline/online partition of the solution process. First in an offline phase a reduced basis space, adapted to the concrete problem, is generated by means of Finite Element or Finite Volume methods. After that we can derive - independent of the complexity of the underlying Finite Element of Finite Volume method - very fast simulation results for any parameter variations. In diesem Vortrag werden wir die Reduzierte Basis Methode zur effizienten Lösung parametrisierter Evolutionsgleichungen vorstellen. Die Methode erlaubt eine offline/online Zerlegung des Lösungsprozesses. Zunächst wird in einer Offline-Phase mit Hilfe von Finite Elemente oder Finite Volumen Verfahren ein reduzierter Basisraum generiert, der auf die konkrete Problemstellung angepasst ist. In einer Online-Phase können dann - unabhängig von der Komplexität der zugrundeliegenden Finite Elemente oder Finite Volumen Verfahren - sehr schnell Simulationsergebnisse für beliebige Parametervariationen berechnet werden.

  • Hans-Georg Purwins (Münster): Lokalisierte Lösungen der erweiterten FitzHugh-Nagumo-Gleichung
    2009/05/07, 12:30, V3-201, FSPM Colloquium
    Abstract: Self-organized dissipative structures play in important role in nature and engineering. Many scientists believe that the understanding and application of them is one of the biggest challenges of modern science. This talk deals with such structures in form of solitary localized spots, also called 'dissipative solitons' (DS). These objects show particle-like behavior in many ways and can be observed both in experiments and as solutions of reaction diffusion systems of FitzHugh-Nagumo type.
    In the first part of the talk the occurence of DS is presented by means of experimental electrical transport systems, e.g. as stationary and travelling isolated pulses, as stationary travelling and rotating 'molecules' and as 'cristalline', 'fluid' and 'gaseous' multi-particle systems. The occuring interaction phenomena cover spreading and clustering as well as generation and annihilation. Numerical experiments show that all experimental observations can be described by the generalized FitzHugh-Nagumo equation. It turns out that this equation can be seen as a 'normal form' for a bigger universality class of DS carrying systems.
    In the second part it will be shown how under certain assumptions particle equations, which describe the dynamical behavior of weak interacting DS very well, can be derived from the generalized FitzHugh-Nagumo equation. Selbstorganisierte dissipative Strukturen sind in Natur und Technik weit verbreitet und nicht wenige Wissenschaftler sind der Ansicht, dass deren Verstîndnis und Anwendung eine der ganz großen Herausforderungen der modernen Naturwissenschaften darstellen. Der vorliegende Vortrag beschäftigt sich mit derartigen Strukturen in der Form von solitären lokalisierten Spots, die auch „Dissipative Solitonen” (DSen) genannt werden. Diese Objekte zeigen in vieler Hinsicht teilchenhaftes Verhalten und werden sowohl experimentell als auch als Lösungen von Reaktions-Diffusions-Systemen vom FitzHugh-Nagumo-Typ beobachtet.
    Im ersten Teil des Vortrags wird an Hand von experimentellen elektrischen Transportsystemen dargelegt, dass DSen z.B. als stationäre und laufende isolierte Pulse, als stationäre, laufende und rotierende „Moleküle” und als „kristalline”, „flüssige” und „gasförmige” Vielteilchensysteme auftreten. Die dabei entdeckten Wechselwirkungsphänomen umfassen sowohl Streuung und Clusterbildung als auch Generation und Annihilation. Numerische Untersuchungen zeigen, dass sich alle experimentellen Beobachtungen qualitativ durch die verallgemeinerte FitzHugh-Nagumo- Gleichung beschreiben lassen. Es erweist sich, dass diese Gleichung als eine Art „Normalform” für eine grüßere Universalitätsklasse DSen tragender Systeme betrachtet werden kann.
    Im zweiten Teil des Vortrags wird besprochen, wie sich unter bestimmten Voraussetzungen aus der verallgemeinerten FitzHugh-Nagumo-Gleichung Teilchengleichungen ableiten lassen, die das dynamische Verhalten schwach wechselwirkender DSen sehr gut beschreiben.

  • Christian Wieners (Karlsruhe): Effiziente numerische Methoden in der Elasto-Plastizität
    2008/07/11, 14:15, V2-210, Seminar Numerical Analysis
    Abstract: The talk begins with illustrating the classical model of Prandtl Reuß plasticity and efficient numerical methods for the solution of the discrete finite element problem will be presented. As these problems are in general ill-conditioned or even ill-posed, robust methods are needed. In particular modern methods will be discussed based on optimization concepts (non-smooth newton methods, SQP methods).
    In the second part of the talk recent applications are presented, The classical models can be regularized using infinitesimal rotations or gradients of plastic distortion (joint work with P. Neff, Darmstadt). It will be shown, that numerical methods can be applied to the extended models. Finally the efficience of the methods will be demonstrated in a parallel simulation of an elastoplastic soil mechanic model (joint work with W. Ehlers, Stuttgart). Im Vortrag wird zunächst das klassische Modell der Prandtl-Reuß-Plastizität erläutert, und es werden effiziente numerische Methoden zur Lösung des diskreten Finite-Elemente-Problems vorgestellt. Da diese Probleme in der Regel schlecht konditioniert oder sogar schlecht gestellt sind, werden robuste Verfahren benötigt. Insbesondere werden moderne Methoden diskutiert, die auf Konzepten der Optimierung beruhen (nicht-glatte Newton-Verfahren, SQP-Verfahren).
    Im zweiten Teil des Vortrags werden neuere Anwendungen vorgestellt. Die klassischen Modelle lassen sich unter Zunahme von infinitesimalen Rotationen oder Gradienten der plastischen Verzerrungen regularisieren (Zusammenarbeit mit P. Neff, Darmstadt). Es wird gezeigt, dass sich die numerischen Lösungsverfahren auf die erweiterten Modelle übertragen lassen. Schließlich wird die Effizienz der Methoden an einer parallelen Simulation eines elasto--plastischen bodenmechanischen Modells demonstriert (Zusammenarbeit mit W. Ehlers, Stuttgart).

  • Peter Giesl (Sussex): Determination of the Basin of Attraction of Equilibria and Periodic Orbits
    2008/07/10, 14:15, V2-210, Seminar Numerical Analysis
    Abstract: The basin of attraction of equilibria or periodic orbits of an autonomous ODE can be determined through sublevel sets of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation itself is a Lyapunov function.
    For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position is Borg's criterion. The sufficiency and necessity of this criterion in different settings will be discussed.

  • Fritz Colonius (Augsburg): Near Invariance and Local Transience for Perturbed Systems
    2008/06/27, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Nearly invariant subsets of the state space of a dynamical system are subsets which can only be left after long time. For families of random diffeomorphisms one can characterize these subsets via an associated discrete-time control system and, also using the Perron-Frobenius operator, one can show that the exit times are positive and polynomially unbounded.

  • Günther Grün (Erlangen): Energiemethoden zur Analyse von Benetzungsph"anomenen
    2008/06/20, 14:15, V5-148, Seminar Numerical Analysis

  • Jens Lorenz (Albuquerque, New Mexico): The Brenner-Klimontovich Modifications of the Navier-Stokes-Fourier System
    2008/06/16, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: The classical Navier-Stokes-Fourier equations for heat conducting compressible flows form a coupled hyperbolic-parabolic system. The system has been criticized, on principle grounds, as being inconsistent with non-equilibrium thermodynamics. In this talk I consider modified systems, suggested by Brenner and Klimontovich, which are essentially parabolic. The modified systems distinguish between a mass velocity and a volume velocity.
    I also show some crude numerical results.

  • Gerhard Starke (Hannover): Eine adaptive gemischte Finite-Elemente-Methode für elastische Kontaktprobleme
    2008/05/23, 14:15, V5-148, Seminar Numerical Analysis

  • Wolf-Jürgen Beyn: Localization and continuation of nonlinear eigenvalues
    2008/05/16, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Nonlinear eigenvalue problems are ubiquitous in the stability analysis of nonlinear systems, such as vibrating systems or systems with delay. Numerical discretizations then lead to large and sparse parameterized nonlinear eigenvalue problems
    beyn1
    where the matrix family beyn2 depends smoothly on the real parameter beyn3 and analytically on the eigenvalue parameter beyn4. We aim at an algorithm that detects a small swarm of eigenvalues λ within a prescribed complex domain and that continues the swarm with respect to the parameter s.
    A new localization procedure is presented that determines the eigenvalues (and eigenvectors) in the interior of a smooth contour of the complex plane. The method builds on Cauchy's integral formula and on a theorem of Keldysh. Then we discuss a continuation method that pursues the swarm of eigenvalues with the parameter and that deflates and inflates the swarm when collisions with outside eigenvalues occur.

  • Tycho van Noorden (Eindhoven): Crystal dissolution and precipitation in porous media: formal homogenization and numerical experiments
    2008/05/09, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We investigate a two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium. The model contains a free boundary and allows for changes in the pore volume. Using a level-set formulation of the free boundary, we apply a formal homogenization procedure to obtain upscaled equations. For general micro--scale geometries, the homogenized model that we obtain falls in the class of distributed microstructure models. For circular initial inclusions the distributed microstructure model reduces to system of partial differential equations coupled with an ordinary differential equation. In order to investigate how well the upscaled equations describe the behavior of the micro-scale model, we perform numerical computations for a test problem.

  • Christof Eck (Bielefeld): Asymptotics and numerical approximation of phase field models
    2008/04/18, 14:15, V5-148, Seminar Numerical Analysis

  • Etienne Emmrich (TU Berlin): Analysis of the time discretization of evolution equations with a monotone operator by the BDF2
    2008/02/20, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Zeitabhängige Prozesse in Natur und Technik önnen oft durch das Anfangswertproblem für eine Evolutionsgleichung rster Ordnung mit monotonem Hauptteil modelliert werden. Zur Zeitdiskretisierung stehen eine Vielzahl von Methoden zur Verfügung, wobei sich die zweischrittige Formel der rückwärtigen Differenzen (Backward differentiation formula 2) durch ihre Eigenschaften auszeichnet.
    Der Vortrag gibt zunächst einen Überblick über bekannte Resultate für lineare und semilineare Probleme bei äquidistantem als auch variablem Zeitgitter. Alsdann werden Wohlgestelltheit, A-priori-Abschätzungen, Konvergenz stückweise polynomialer Prolongationen, Stabilität und Fehlerabschätzungen für den Fall konstanter Zeitschrittweite bei Anwendung auf eine Evolutionsgleichung mit monotonem Operator und verstärkt stetiger Störung studiert. Als fundamental stellt sich dabei eine algebraische Identität heraus, die zugleich die G-Stabilität des Verfahrens nach sich zieht. Schließlich wird der Fall variabler Zeitschritte diskutiert.Time-dependent processes in nature and science can often be modelled by an initial value problem for an evolution equation of first order with monotone main term. There are plenty of methods for the time discretization whereas the Backward differentiation formula 2 excels by its properties.
    First the talk will give an overview of known results for linear and semilinear problems on both equidistant and variable time grids. Then well-posedness, a-priori estimates, convergence of piecewise polynomial prolongations, stability and error estimates in case of constant time stepsize are studied for an evolution equation with monotone operator and reinforced continous noise. A special algebraic identity turns out to be fundamental, from which also the G-stability of the method follows. In the end the case of variable stepsizes will be discussed.

  • Peter Benner (TU Chemnitz): Control-oriented model reduction for parabolic systems
    2008/01/25, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We will discuss model reduction techniques for the control of dynamical processes described by parabolic partial differential equations from a system-theoretic point of view.
    The methods considered here are based on spatial semi-discretization of the PDE followed by balanced truncation techniques applied to the resulting large-scale system of ordinary differential equations. Several choices of the system Gramians that are used for balancing will be presented.
    We will discuss open-loop and closed-loop techniques that allow to preserve system properties important for controller design. Furthermore we will discuss an error estimate based on a combination of FEM and model reduction error bounds. We will also discuss how the state of the full-order system can be recovered from the reduced-order model. Several numerical examples will be used to demonstrate the proposed model reduction techniques.

  • Barnabas Garay (TU Budapest): Chaos Detection by Computer
    2008/01/21, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations for certain mappings associated with the dynamics and thus to check certain sufficient conditions for chaotic behaviour in a rigorous way.
    Our proof of the underlying abstract theorem avoids of referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem.
    The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach.
    Abstract results and computational methods are demonstrated by finding planar subsets with chaotic behaviour for iterates of the classical Henon mapping as well as for the time-T-map of the solution operator to a damped pendulum equation with T-periodic forcing.

  • Alexander Dressel (Stuttgart): Existence, uniqueness and time-asymptotic behaviour of weak solutions for a viscoelastic two-phase model with nonlocal capillarity
    2008/01/11, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: The aim of this talk is to study the existence, uniqueness and time-asymptotic behaviour of solutions of an initial-boundary value for a viscoelastic two-phase material with capillarity in one space dimension. Therein, the capillarity is modelled via a nonlocal interaction potential. The existence proof relies on uniform energy estimates for a family of difference approximations: with these estimates at hand we show the existence of a global weak solution. By means of a nontrivial variant of existing arguments in the literature (the so-called "Andrews-trick") , uniqueness and further regularity are proven.Then, based on the existence and regularity results, we prove the time-asymptotic convergence of the strain-velocity field.

  • Anton Arnold (TU Wien): Offene Randbedingungen für Wellenausbreitungsprobleme in unbeschränkten Gebieten
    2007/12/19, 15:00, V5-148, Seminar Numerical Analysis
    Abstract: Partielle Differentialgleichungen auf unbeschränkten Gebieten treten ganz natürlich in akustischen, quantenmechanischen und strömungsmechanischen Problemen auf. Die numerische Simulation muß daher meist auf ein endliches Teilgebiet beschränkt werden - durch die Einführung von (künstlichen) absorbierenden Randbedingungen.
    Im Vortrag werden wir einen Überblick über solche Strategien geben und insbes. Anwendungen auf die zeitabhängige Schrödinger Gleichung in 1D und 2D (Streifen und Kreisgeometrie) diskutieren.Partial differential equations on unbounded domains occur in a natural way in acoustic, quantum mechanic and fluid mechanic problems. But the numerical simulation has to constrain on finite subdomains - by the introduction of (artificial) absorbing boundary conditions.
    In this talk we will give an overview of such strategies, and will discuss especially their application to the time-dependent Schrödinger equation in 1D and 2D (stripes and circular geometry).

  • Heinrich Voß (TU Hamburg-Harburg): Numerical methods for sparse nonlinear eigenvalue problems
    2007/12/14, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider the nonlinear eigenvalue problem
    voss1
    where voss2 is a family of sparse matrices. Problems of this type arise in damped vibrations of structures, conservative gyroscopic systems, lateral buckling problems, fluid-solid vibrations, and the electronic behaviour of quantum dot heterostructures, to name just a few.
    We discuss iterative projection methods of Jacobi-Davidson and Arnoldi type which are particularly efficient if the eigenvalues of (1) satisfy a minmax property. Moreover, we present a variant of the automated multi-level substructuring for nonlinear problems.

  • Olaf Steinbach (TU Graz): Boundary Integral Equations: Analysis and Applications
    2007/12/07, 14:15, V5-148, Seminar Numerical Analysis

  • Kathrin Schreiber (TU Berlin): Nonlinear Rayleigh functionals
    2007/11/23, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: After a short introduction on nonlinear eigenvalue problems, defined byschreiber1 where schreiber2 is a matrix-valued mapping, we review Rayleigh quotients for Hermitian and general matrices and introduce appropriate {\em Rayleigh functionals} p(u) and p(u, v) defined by
    schreiber3
    for nonlinear eigenvalue problems, where u, v are approximations for right and left eigenvectors. Local existence and uniqueness of p is shown as well as 'stationarity' (technically p is not differentiable). Bounds for the distance of p and the exact eigenvalue are provided, which are of the same order as in the linear case.
    The last part of the presentation includes numerical results, where the emphasis lies on complex symmetric problems, where the application of the symmetric Rayleigh functional gives considerably better results associated with an Jacobi-Davidson type method compared to the standard Jacobi-Davidson method.

  • Helmut Podhaisky (Halle): Two-step peer methods for time-dependent partial differential equations
    2007/10/19, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Für die numerische Lösung von Anfangswertaufgaben gibt es zwei populäre Verfahrensklassen: lineare Mehrschrittformeln, z.B. vom BDF-Typ, auf der einen Seite und Runge-Kutta-Verfahren auf der anderen. Vor- und Nachteile sind gut bekannt: Mehrschritt-Verfahren haben einen sehr geringen Aufwand pro Schritt, Runge-Kutta-Verfahren habe exzellente Stabilitätseigenschaften. Kann man die Vorteile kombinieren? Die Antwort ist 'ja' (bzw. 'ja, eventuell'), man muss nur allgemeine lineare Verfahren (engl.: \emph{general linear methods}, GLMs), also mehrstufige Mehrschrittverfahren, betrachten.
    Im Vortrag werden Peer-Methoden vorgestellt, die s Approximationen podhaisky2, i=1,...,s in einem diagonal impliziten Schema
    podhaisky1
    berechnen. Die Ordnungsbedingungen ergeben sich aus Taylorreihenentwicklung. Schwieriger ist, Stabilität, insbesondere A-Stabilität, zu sichern und die verbleibenden freien Parameter so zu optimieren, dass die Methoden robust arbeiten.
    Am Ende des Vortrags werden Peer-Methoden bis zur Ordnung 4 als Zeitintegrationsverfahren im FEM-Code KADOS zur Lösung partieller Differentialgleichungen angewendet. There are two classes of methods for the numerical solution of initial value problems: linear multi-step methods, e.g. of the BDF type, on the one hand and Runge-Kutta methods on the other hand. Advantages and disadvantages are well-known: small costs per step for multi-step methods, excellent stability properties for Runge-Kutta methods. Is it possible to combine both advantages? The answer is 'yes' (or 'yes, possibly'), we just have to look at general linear methods (GLMs), i.e. multi-level multi-step methods.
    In this talk we present peer methods which compute s approximations podhaisky2, i=1,...,s in a diagonal implicit scheme
    podhaisky1.
    The order conditions are given by the Taylor expansion. The more difficult task is to ensure stability, particularly A-stability, and to optimize the remaining free parameters in such a way that the methods work robust.
    In the end of the talk peer methods up to order 4 are used in FEM code KADOS to solve partial differential equations.

  • Arnd Scheel (Minnesota): Periodic patterns: perturbation, modulation and bifurcation
    2007/10/12, 14:15, V5-148, Seminar Numerical Analysis

  • Thorsten Hüls: Numerical approximation of homoclinic trajectories for non-autonomous maps
    2007/05/09, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: For time-dependent dynamical systems of the form
    huels1
    homoclinic trajectories are the non-autonomous analog of homoclinic orbits from the autonomous world.
    More precisely, two trajectories (xn)n ∈ Z, (yn)n ∈ Z of (1) are called homoclinic to each other, if
    huels2
    We introduce two boundary value problems, the solution of which yield finite approximations of these trajectories. Under certain dichotomy and transversality assumptions, we prove existence, uniqueness and error estimates. Finally, the method and the error estimates are illustrated by an example.

  • Sergei Pilyugin (St. Petersburg): Dynamics of some mappings determined by piecewise linear functions
    2007/04/25, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We study dynamics of multidimensional mappings that are determined by scalar functions. Such mappings arise, for example, when we discretize a semilinear parabolic equation. If the determining scalar function is piecewise-linear (with a finite number of "corner" points), then the dynamics is described by a finite number of parameters, and approaches of "discrete nature" are applicable.

  • Jörg Härterich (FU Berlin): Convergence to Rotating Waves in Spatially Inhomogeneous Balance Laws
    2007/04/11, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: I will study the long-time behaviour of scalar balance laws where the source term is space-dependent. It turns out that under some assumptions solutions exist which converge to rotating waves. After explaining the proof of this statement I will discuss which role these rotating waves play within the global attractor. In addition, some remarks concerning the effect of small viscosity and the case where the assumptions are violated, will be presented.

  • Daniel Holtz (Bielefeld): Überlappender Schwarz-Algorithmus für nichtlineare Konvektions-Probleme
    2006/12/20, 12:30, V3-201, Seminar Numerical Analysis
    Abstract: Der Schwarz-Algorithmus wird verwendet, um Randwertprobleme mit Hilfe von Gebietszerlegungen numerisch parallel zu berechnen. Der Vortrag stellt eine Realisierung dieses Algorithmus vor. Im eindimensionalen Fall wird für parabolische nichtlineare Differentialgleichungen Konvergenz des Algorithmus bewiesen und numerisch überprüft. Viele Parameter des Algorithmus sind vom Anwender frei wählbar. Basierend auf der Wahl der Randwerte für die Teilgebiete wird eine Verbesserung des Algorithmus vorgestellt.

  • Marcel Oliver (Bremen): Subgrid closures for passive advection through nonreflecting boundary conditions in Fourier space
    2006/12/13, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a time-independent velocity field and for a time-dependent synthetic velocity field whose Fourier coefficients follow independent Ornstein-Uhlenbeck stochastic processes.

  • Martin Rasmussen (Augsburg): Morse partitions of non-autonomous dynamical systems
    2006/12/06, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Das globale asymptotische Verhalten dynamischer Systeme auf kompakten metrischen Räumen läßt sich mittels Morse-Zerlegungen beschreiben. Deren Komponenten, die so genannten Morse-Mengen, erhält man als Schnitte von Attraktoren und Repeller. In diesem Vortrag werden spezielle Begriffe von Attraktoren und Repeller für nichtautonome dynamische Systeme eingeführt, die geeignet für eine nichtautonome Verallgemeinerung der Morse-Zerlegungen sind. Die dynamischen Eigenschaften dieser Morse-Zerlegungen werden diskutiert; besonderes Augenmerk wird hierbei auf eindimensionale und lineare Systeme gelegt.

  • Malte Samtenschnieder: Periodische Orbits zeitdiskreter nicht-autonomer dynamischer Systeme und ihre Stabilitätseigenschaften
    2006/11/08, 14:15, U2-205, Seminar Numerical Analysis
    Abstract: Ausgehend vom autonomen Fall, stehen in diesem Vortrag periodische Orbits nicht-autonomer dynamischer Systeme im Mittelpunkt. Für natürliche Zahlen 2 ≤ r < k betrachten wir - statt für festes F - für eine zeitlich veränderliche k-periodische Funktionenfamilie {Fn}n ∈ Z so genannte r-Zykel cr, für die wir eine Stabilitätsanalyse durchführen. Mit Hilfe eines Satzes, der im nicht-autonomen Fall den Zusammenhang zwischen der Stabilität eines Fixpunkts und der Stabilität eines Zykels herstellt, zeigen wir: Wenn cr für das betrachtete nicht-autonome dynamische System global asymptotisch stabil ist, folgt, dass r ein Teiler von k ist. In dieser speziellen Situation approximieren wir r-periodische Orbits k-periodischer Funktionenfamilien mit Hilfe eines auf einer Fixpunktgleichung basierenden Lösungsverfahrens. Ausgehend vom periodischen Beverton-Holt-Modell und der periodischen Stiletto-Abbildung geben wir numerische Ergebnisse an, deren Stabilität wir ebenfalls untersuchen. Abschließend skizzieren wir, welche Probleme auftreten, wenn wir die Bedingung, dass r ein Teiler von k ist, fallen lassen. Wir begründen, warum wir dann allgemein keine Lösung angeben können.

  • Anke Mayer-Bäse (Florida): Challenges in Computational Intelligence: From Neurodynamics to Medical Imaging
    2006/07/12, 14:15, V5-148, Seminar Numerical Analysis

  • Clarence Rowley (Princeton): Template-based methods for model reduction and control of systems with symmetry
    2006/07/05, 14:15, V5-148, Seminar Numerical Analysis

  • Thorsten Hüls: Non-autonomous difference equations and bifurcations
    2006/06/28, 14:15, V5-148, Seminar Numerical Analysis

  • Janosch Rieger: Numerical grid methods for differential inclusions
    2006/05/31, 14:15, V5-148, Seminar Numerical Analysis

  • Renate Winkler (HU Berlin): Stochastic DAEs in Circuit Simulation
    2006/05/24, 14:15, V5-148, Seminar Numerical Analysis

  • Jens Rademacher (WIAS Berlin): Computing absolute and essential spectra using continuation
    2006/05/17, 14:15, V5-148, Seminar Numerical Analysis

  • Daniel Kressner (TU Berlin): Structured eigenvalue problems
    2006/05/10, 14:15, V5-148, Seminar Numerical Analysis

  • Vera Thümmler: Wie man wandernde Wellen einfriert,ohne ihre Stabilität zu zerstören
    2006/05/03, 14:15, V5-148, Seminar Numerical Analysis

  • Alexander Lust: Eine hybride Methode zur Berechnung von Liapunow-Exponenten.
    2006/04/26, 14:15, V5-148, Seminar Numerical Analysis

  • Sergei Pilyugin (St. Petersburg): Sets of dynamical systems with various limit shadowing properties
    2006/04/12, 14:15, V5-148, Seminar Numerical Analysis

  • Simon Malham (Edinburgh): Efficient strong integrators for linear stochastic systems
    2006/03/17, 14:15, V5-148, Seminar Numerical Analysis

  • Sergey Piskarev (Bielefeld, Moskau): Maximal regularity for parabolic and elliptic problems
    2006/01/25, 14:15, V5-148, Seminar Numerical Analysis

  • Andreas Münch (HU Berlin): Non-classical shock solutions and other issues in thin film problems
    2006/01/11, 14:15, V5-148, Seminar Numerical Analysis

  • Alexander Dressel (Heidelberg): Existence of smooth shock profiles for hyperbolic balance laws
    2005/12/14, 14:15, V5-148, Seminar Numerical Analysis

  • Jens Rottmann-Matthes: Spektrale Eigenschaften gemischt hyperbolisch-parabolischer Systeme
    2005/07/22, 14:15, V5-148, Seminar Numerical Analysis

  • Abigail Wacher (Frankfurt): Lösung partieller Differentialgleichungen mit gewichteten, beweglichen finiten Elementen
    2005/06/08, 14:15, V5-148, Seminar Numerical Analysis

  • Caren Tischendorf (TU Berlin): Stabilitätserhaltende Integration von DAEs
    2005/02/02, 14:15, V5-148, Seminar Numerical Analysis

  • Sergey Piskarev (Twente,Moskau): On the approximation of attractors
    2005/01/26, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: We consider semilinear problems of the form u' = Au + f(u), where A generates an exponentially decaying compact analytic semigroup in a Banach space E and f is globally Lipschitz and bounded map from Eα into E (Eα=D((-A)α) with the graph norm). These assumptions ensure that the problem has a global attractor. Under a very general approximation scheme we prove that the dynamics of such problem behaves upper semicontinuously.
    We also prove that, if all equilibrium solutions of this problem are hyperbolic, then there is an odd number of such equilibrium solutions. Additionally, if we impose that every global solution converges as t → ± ∞, (e.g. gradient semigroups with isolated equilibria), then we prove that under this approximation scheme the attractors also behave lower semicontinuously.
    This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents which may be applied to many other problems not related to discretization.

  • Wilhelm Huisinga (FU Berlin): Metastability and Dominant Eigenvalues of Transfer Operators
    2004/11/25, 10:15, W0-135, Seminar study group pattern formation
    Abstract: There are many problems in physics, chemistry and biology where the length and time scales corresponding to the microscopic descriptions (given in terms of some stochastic or deterministic dynamical system), and the resulting macroscopic effects differ many orders of magnitude. Rather than resolving all microscopic details, often one is interested in characteristic features on a macroscopic level (e.g., phase transitions, conformational changes of bio-molecules, climate changes etc.). In this setting, metastability is important macroscopic characteristic which is related to the long time behavior of the dynamical system. It refers to the property that the dynamics is likely to remain within a certain part of the state space for a long period of time, until it eventually exits and transits to some other part of the state space. In this talk we introduce the concept of metastability in the setting of Markov processes, and prove upper and lower bounds for a decomposition of the state space into metastable subsets in terms of dominant eigenvalues and eigenvectors of a corresponding transfer operator. The bounds are explicitly computable and sharp. The results do not rely on any asymptotic expansions in terms of some smallness parameter, but rather hold for arbitrary transfer operators satisfying a reasonable spectral condition.

  • G. Grammel (TU München): Approximation bei Differentialinklusionen
    2004/11/24, 14:15, V5-148, Seminar Numerical Analysis

  • Barnabas Garay (TU Budapest): The Miranda approach: a framework for computer-assisted proofs of chaos
    2004/11/17, 14:15, V5-148, Seminar Numerical Analysis

  • Julia Nolting (Scheifler): Bifurkationen periodischer Orbits und ihre numerische Berechnung
    2004/07/21, 14:15, V5-148, Seminar Numerical Analysis

  • Stefan Siegmund (Frankfurt/Main): Zeitvariante lineare Systeme
    2004/07/15, 14:15, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics
    Abstract: Will man nichtlineare Phänomene lokal verstehen, so kann dies bestenfalls nur so gut gelingen wie man die lineare Theorie verstanden hat. Das Verhalten einer linearen Differentialgleichung xt = Ax wird vollständig durch die Eigenwerte und Eigenräume der n x n-Matrix A beschrieben. Welche mathematischen Konzepte übernehmen die Rolle der Eigenwerte und Eigenräume, falls A sich durch zufällige, kontrollierte oder allgemeine deterministische Einflüsse zeitlich ändert? Im Vortrag werden die historische Entwicklung über Floquet, Lyapunov, Bohl, Osedelets, Sacker und Sell nachgezeichnet, Beispiele gegeben und Zusammenhänge und aktuelle Resultate beschrieben. When studying nonlinear phenomena locally one can expect at most as good understanding as in the linear theory. The behavior of a linear differential equation xt = Ax is completely characterized by the eigenvalues and eigenspaces of the n x n matrix A. What mathematical concepts take place for the eigenvalues and eigenspaces if A varies over time caused by random, controlled or general deterministic influences? In this talk we will follow the historical development via Floquet, Lyapunov, Bohl, Osedelets, Sacker and Sell, give some examples and describe the correlations and some actual results.

  • Jerrold E. Marsden (CalTech, Pasadena): The Euler-Poincare Equations
    2004/07/01, 17:15, V3-201, Mathematical Colloquium
    Abstract: The Euler-Poincare equations were born in 1901 when Poincare made a sweeping generalization of the classical Euler equations for the rigid body and ideal fluids. He did this by formulating the equations on a general Lie algebra, the rigid body being associated with the rotation Lie algebra and fluids with the Lie algebra of divergence free vector fields. Since then, this setting has been used for many other situations, such as the KdV equation, shallow water waves, averaged fluid equations, and the template matching equations of computer vision to name just a few. This talk will give an overview of Euler-Poincare and Lie-Poisson reduction theory (from the tangent and cotangent bundles of a Lie Group to its Lie algebra or dual) and then will focus on the specifics for the case of the algebra of all vector fields. Special singular solutions will be described which generalize the peakon (soliton) solutions of the (Camassa-Holm-Fokas-Fuchsteiner) shallow water equations from one to higher dimensions; the manner in which momentum maps (in the sense of Noether's theorem from mechanics) play an important role in these special singular solutions will be presented. (Joint work with Darryl Holm)

  • Giovanni Samaey (Leuven): Connecting orbits in delay differential equations: computation and application to traveling waves in delay PDEs
    2004/06/02, 14:15, V5-148, Seminar Numerical Analysis
    Abstract: Connecting orbits in delay differential equations (DDEs) are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state. However, in contrast with ODEs, the stable manifold of a steady state of a DDE is infinite-dimensional. We circumvent this problem by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved in the Matlab package DDE-BIFTOOL using a collocation method. We show numerical convergence results in terms of discretization and truncation errors, and compare these to the ODE case.
    Besides their importance for the bifurcation analysis of DDEs, homoclinic and heteroclinic orbits arise naturally when looking for traveling waves ofdelay partial differential equations (delay PDEs). We show that, using DDE-BIFTOOL, we can compute these traveling waves, as well as the rightmost part of their spectrum. The numerical results suggest that the spectral properties of traveling waves in delay PDEs are comparable to properties that were proved by Sandstede for the PDE case.

  • Imre Bozi (Budapest): Multiplicity results for the one-dimensional p-Laplacian
    2004/05/26, 14:15, V5-148, Seminar Numerical Analysis

  • Bernd Krauskopf (Bristol): The saddle-node Hopf bifurcation with global reinjection
    2004/02/05, 12:30, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics

  • Christian Poetzsche (Augsburg): Nonautonomous Dynamics, Time Scales and Discretization
    2003/12/10, 14:15, V5-148, Seminar Numerical Analysis

  • Tobias Gayer (Augsburg): Almost-invariance of diffusion processes and parameter dependent control problems - results and numerical methods
    2003/11/05, 14:15, V5-148, Seminar Numerical Analysis

  • Nils Wagner (Stuttgart): Multiple eigenvalues of parameter dependent matrix polynomials
    2003/10/29, 14:15, V5-148, Seminar Numerical Analysis

  • Lars Grüne (Bayreuth): Numerics and applications of stochastic optimal control
    2003/05/28, 14:15, V5-148, Seminar Numerical Analysis

  • Zou Yong-Kui (Changchun): Generalized Hopf bifurcation for non-smooth planar dynamical systems
    2003/05/21, 14:15, V5-148, Seminar Numerical Analysis

  • Jens Kemper: Attraktoren und invariante Maße in Reaktions-Diffusions-Gleichungen
    2003/01/29, 14:15, V5-148, Seminar Numerical Analysis

  • Andrei Afendikov (Keldysh Institut): Numerical exterior algebra in spectral problems of stability
    2002/10/24, 12:30, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics
    Abstract: Spectral problems of stability on a finite or infinite interval often lead to stiff problems that are difficult to handle numerically. Using as an example the problem of stability for the pulse solution to the complex Ginzburg-Landau equation it is supposed to demonstrate how classical shooting methods can be transformed to a modern exterior algebra approach which leads to the investigation of the so-called Evans function.
    The numerical algorithm for evaluating the Evans function uses explicitly the matrix representation of the Hodge star operator and the numerical integrator that respects the Plücker imbedding of the Grassman manifold into the space of exterior forms.

  • Eusebius J. Doedel (Concordia Univ., Montreal): Continuation of Periodic Solutions in Conservative Systems with Application to the N-Body Problem
    2002/10/02, 14:15, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics
    Abstract: I will show how boundary value continuation software can be used to compute families of stable and unstable periodic solutions of conservative systems. A simple example will be used to illustrate the main idea. I will show how the computational approach can be used to follow the recently discovered figure-8 orbit of Montgomery, Chenciner, and Simo, as the mass of one of the bodies is varied. The numerical results show, among other things, that there exists a continuous path from the figure-8 orbit to periodic solutions of the restricted three body problem.
    Various aspects of this work are done in cooperation with Andre Vanderbauwhede (Gent), Don Dichmann (Aerospace Corporation), Jorge Galan (Sevilla), and Herb Keller and Randy Paffenroth (Caltech).

  • Björn Sandstede (Ohio State Univ., Columbus): Spectral properties of spiral waves
    2002/07/11, 12:30, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics

  • Andreas Keese (Braunschweig): Numerical solution of systems with stochastic operator
    2002/06/20, 12:30, V2-216, Seminar of the Research Group Spectral analysis, asymptotic distributions and stochastic dynamics

  • Werner Vogt (TU Ilmenau): Numerical approximation of invariant tori and quasi-periodic solutions of dynamical systems
    2002/06/07, 14:15, V5-148, Seminar Numerical Analysis

  • Kurt Lust (Leuven): Accurate computation of Floquet multipliers in multiple shooting and Gauss-Legendre codes
    2002/02/13, 14:15, V5-148, Seminar Numerical Analysis

  • Arno F. Münster (Würzburg): Strukturbildung in chemischen Reaktionen mit Ionen
    2001/11/22, 10:15, W9-109, Seminar study group pattern formation

  • Qin Mengzhao (Peking): Multisymplectic methods for infinite-dimensional Hamiltonian systems
    2001/11/16, 14:15, U5-133, Seminar Numerical Analysis

  • Alexander Lust: Numerische Berechnung von Liapunow-Exponenten
    2001/11/07, 14:15, V5-148, Seminar Numerical Analysis

  • Thorsten Hüls: Heterokline Orbits zwischen nichthyperbolischen Fixpunkten
    2001/10/24, 14:15, V5-148, Seminar Numerical Analysis

  • Sergey Pilyugin (St. Petersburg): New results on shadowing
    2001/07/05, 12:30, V2-205, Seminar Numerical Analysis