Oberseminar Angewandte Analysis und Numerik

Oberseminar Angewandte Analysis und Numerik

*Donnerstag, 14 Uhr ct, Raum 110

 

Prof. Dr. P. E. Kloeden
Prof. Dr. T. Weth
Prof. Dr. J. Baumeister
aProf. Dr. H. Crauel
Prof. Dr. J. Bliedtner
Prof. Dr. B. von Harrach


 

Aktuelle Vorträge

Sommersemester 2017


Ab sofort finden die Vorträge in Raum 110 statt.





13.07.2017      Vortrag Masterarbeit von Stefan Rümmler

Titel:
Erweiterte dissipative Systeme und ihre Anwendungen

Abstract




06.07.2017      Huyuan Chen
(Nanjing University/China, z Zt. Frankfurt am Main)


Titel:  
Liouville theorem for fractional semilinear equations in unbounded domains

Abstract



 

29.06.2017       Vortrag von Dr. Janosch Rieger, Monash University

(im Anschluss an Bachelorvortrag von Herrn Groß)

Title: Recent advances in domain reconstruction from electrical impedance
tomography data

Abstract:
Electrical impedance tomography is an emerging budget-priced,
non-invasive medical imaging technique that is very likely to
complement computerised tomography in important applications
such as pulmonary function control and breast cancer screening
in the future. The main difficulty associated with this technology
is that the arising inverse problem is strongly ill-posed.

In this talk, I will discuss an alternative approach to domain
reconstruction from electrical impedance tomography data, which
is based on the concept of the convex source support introduced
by Kusiak and Sylvester, as well as an appropriate numerical
discretisation of the resulting problem.



29.06.2017       Bachelorvortrag von Sebastian Groß (14:00 c.t.)

Thema: Bewertung von Barrier-Optionen mit Finiten Elementen



22.06.2017     Tolga Yesil (Frankfurt am Main)

Titel:
Dual ground state solutions for the critical nonlinear Helmholtz equation

Abstract:
Using a dual variational approach, we obtain real-valued solutions of a
weighted critical nonlinear Helmholtz equation. The weight function is
assumed to be bounded, positive, asymptotically periodic and to satisfy
a certain flatness condition at one of its maximum points. The solutions
obtained are so-called dual ground states, i.e. solutions arising from
critical points of the dual functional with the property of having
minimal energy among all nontrivial critical points. This is a joint
work with Gilles Evequoz.





Sondertermin Fr. 09.06. um 15 Uhr, Raum 711 klein

Selina Müller

Titel:
"Variationelle Methoden für nicht differenzierbare Funktionale und deren Anwendung auf partielle Differentialgleichungen"

(Vortrag Bachelorarbeit)

Zusammenfassung:
Betrachtet wird ein nicht lineares Randwertproblem, wobei die rechte Seite eine
nicht stetige Nichtlinearität darstellt. Für dieses Problem wird eine nicht triviale Lösung in einem geeigneten
Sinne gesucht. Dazu wird eine Ausweitung der variationellen Methoden auf nicht differenzierbare Funktionale
benötigt, da das zugehörige Funktional dieses Randwertproblems nicht notwendigerweise differenzierbar ist.
Hierfür werden lokal Lipschitz-stetige Funktionen betrachtet, die nicht notwendigerweise differenzierbar sind.






Sondertermin: Dienstag, 04.04.2017, 11:00 Uhr (c.t.) in Raum 110 (RMS 10)

Vortrag von Minh Nguyet Mach, Ph.d. (University of Helsinki)

Title:"Convergence of the current-to-voltage measurements of the Shunt Electrode Model to those of the Continuum Model"


Sondertermin: Dienstag 28.03.2017, 15:00 Uhr (c.t.) in Raum 110 (RMS 10)

Bachelorvortrag von Janin Heuer (Goethe-Universität Frankfurt)

Thema: Anwendungen der Optimalsteuerung in der Raumfahrt


Sondertermin: Dienstag 21.02.2017, 14:00 Uhr (c.t.) in Raum 110 (RMS 10)

Mastervorträge Numerik (Goethe-Universität Frankfurt)

Jessica Stein:  Regularisiertes Newton-Verfahren für ein Parameteridentifikationsproblem

Steffen Ebert: Nichtlineare Tikhonov Regularisierung eines Paramteridentifikationsproblems






Archiv




Wintersemester 2016/17



09.02.2017    Sven Jarohs (Frankfurt am Main)

Titel:
Starshape of superlevel sets of solutions to equation involving the fractional Laplacian

Abstract: 
In this talk, I will present a general framework to analyze the geometry of solutions to equations
involving the fractional Laplacian in starshaped rings. By analyzing the difference of the solution with a scaled
version of this solution, using the scaling properties of the fractional Laplacian and different versions of
maximum principles, we show that under rather general assumptions on the right-hand side the solution
has starshaped superlevel sets. I will also present some examples in which this result can be applied.
The talk is based on a joint work with Tadeusz Kulczycki and Paolo Salani.




02.02.2017    Sebastian Becker (Goethe-Universität Frankfurt)

Titel:
Nicht-autonome und zufällige dynamische Systeme, die durch deterministische und stochastische
Differentialgleichungen generiert sind (Masterarbeit)

Abstract: Im Vortrag werden autonome, nicht autonome und zufällige dynamische Systeme vorgestellt,
miteinander verglichen und erweitert. Anhand von zunächst skalaren Differentialgleichungen werden die
induzierten dynamischen Systeme mit den zuvor vorgestellten Definitionen in Verbindung gebracht, um
so den nicht autonomen Einfluss besser untersuchen zu können. Des weiteren werden gemeinsam
wirkender nicht-autonomer deterministischer und stochastischer Einfluss betrachtet und so generierte
Systeme analysiert.




19.1.2017    Oscar Agudelo  (University of West Bohemia, Pilsen)

Titel: Boundary concentration phenomena for the higher-dimensional Keller-Segel system

Abstract:
We study the existence of steady states to the Keller-Segel system with linear chemotactical
sensitivity function on a smooth bounded domain in RN, N ≥ 3, having rotational symmetry. We find three
different types of chemoattractant concentration which concentrate along suitable (N−2)−dimensional
minimal submanifolds of the boundary. The corresponding
density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on
those boundary submanifolds.
This is a joint work with Angela Pistoia from La Sapienza, Università degli Studi di Roma.





8.12.2016    Nils Ackermann (UNAM, Mexico Stadt, derzeit Goethe-Universität Frankfurt)

Titel: Ground states for irregular and indefinite superlinear Schrödinger equations

Abstract:  We consider the existence of a ground state for the subcritical stationary
semilinear Schrödinger equation −∆u + u = a(x)|u| p−2 u in H 1 , where a ∈
L ∞ (R N ) may change sign. Our focus is on the case where loss of compactness
occurs at the ground state energy. By providing a new variant of the Splitting
Lemma we do not need to assume the existence of a limit problem at infinity, be
it in the form of a pointwise limit for a as |x| → ∞ or of asymptotic periodicity.
That is, our problem may be irregular at infinity. In addition, we allow a to
change sign near infinity, a case that has never been treated before.




27.10.2016     Marcello di Biase (Goethe-Universität Frankfurt)

Titel:
Über Stochastische Bifurkation

Abstract:
Große Anstrengungen galten in den letzten Jahrzehnten den Versuchen, eine Bifurkationstheorie für zufällige dynamische Systeme zu schaffen, die als Verallgemeinerung der gut verstandenen Bifurkationsszenarien deterministischer Dynamik verstanden werden kann. Die Ergebnisse sind so vielfältig wie die Klassen der untersuchten zufälligen dynamischen Systeme selbst. Einige ausgewählte Ansätze werden umrissen, Beispiele diskutiert und auf ansatzspezifische Eigenheiten aufmerksam gemacht.





20.10.2016    Maximilian Engel (Imperial College London)


Titel: "Bifurcation analysis of stochastically driven limit cycles"

Abstract: We investigate bifurcations from an attractive random equilibrium to shear-induced chaos for stochastically driven limit cycles, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Lai-Sang Young and co-workers, extending results on periodically kicked limit cycles to the stochastic context. We also apply concepts from ergodic theory, like entropy and the SRB property of the invariant random measure, to describe the random attractors in the chaotic case.



6.10.2016   14 ct, Raum 110     

Nicola Abatangelo (Brüssel)

Title:
A very weak theory for fractional Dirichlet problems

Abstract:
It is long since known that s-harmonic functions (namely, functions whose fractional Laplacian is zero) on a domain can show an explosive behaviour at the boundary, in sharp contrast with classical harmonic functions. We will present a theory of Stampacchia's sort for solutions to fractional elliptic Dirichlet problems which can deal with such singular behaviour. In order to gain uniqueness and, more generally, well-posedness, these boundary problems are set by prescribing two different types of Dirichlet datum at the same time, one of which is describing the asymptotic explosive profile of the solution at the boundary.







Sommersemester 2016

8.9.2016 14 ct, Raum 110


Vortrag Bachelorarbeit (tba)



Sondertermin:

27.07.2016   11:00 Uhr st, Raum 110     

Sven Jarohs (Goethe-Universität Frankfurt)

Titel: 
On the maximum principle for the fractional polylaplacian

Abstract: It is well-known that in general operators of order four do not satisfy
a maximum principle for supersolutions. Since maximum principles are an
important tool in the analysis of partial differential equations, the
question arises why and when this property is lost for operators of
order between 2 and 4. In this talk we will analyze real, positive
powers of the Laplacian and show that whenever the power is in an
Intervall starting with an odd number, then the maximum principle fails.
By the structure of the explicit counterexample it follows that such
powers of the fractional Laplacian may satisfy a maximum principle only
for solutions in certain connected sets. One of such sets is given by
the ball, where the maximum principle follows from an explicit solution
formula given by Boggio's formula. The talk is based on a joint work
with Nicola Abatangelo and Alberto Saldana.




14.07.2016   14 Uhr ct    Rohit Kumar Mishra (Bangalore/Indien)


Titel: On Inversion of Some Integral Transforms in R^n

Abstract


07.07.2016   14 Uhr c.t.      Andreas Hauptmann (Universität Helsinki/Finnland)

Titel: Direct reconstructions from partial-boundary data in electrical impedance tomography

Abstrakt: In electrical impedance tomography a body is probed with an electrical
current to obtain information about the inner conductivity distribution. In this
application full-boundary measurements are not always possible. Therefore, we
present the partial-boundary inverse conductivity problem in a realistic setting
and analyze the error that partial-boundary measurements introduce. Computational
convergence results and reconstructions are presented for medical motivated simulated data.



07.07.2016, 15 Uhr c.t.
    Daniel Roth (Frankfurt am Main)

Titel:
Monte-Carlo Verfahren für die Elektrische Impedanz-Tomografie

Abstract: Das direkte Problem der elektrischen Impedanz-Tomografie
beschäftigt sich damit, für eine bekannte Leitfähigkeit durch angelegten
Strom das zugehörige Potential zu berechnen. Dieses Problem kann
beispielsweise mit einem Finite-Differenzen Verfahren gelöst werden. Der
Vortrag widmet sich einem alternativen Verfahren zum Lösen des direkten
Problems. Dieses Verfahren, welches auf auf einem Monte-Carlo
Algorithmus basiert, löst mittels Zufallszahlen das Gleichungssystem der
Finiten Differenzen, ohne das Gleichungssystem explizit aufzustellen.



02.06.2016   14 Uhr ct     Elias Polak (Frankfurt)

Titel: Die Grundzustandsenergie eines N-Fermionen-Systems (Bachelorarbeit)



04.02. 2016   Fabian Rücker (Frankfurt)

Titel:
Anwendung des Minimax-Theorems von Nikaido auf Probleme der
Kontrolltheorie



Sondertermin, Mittwoch, den 03.02.2016  16-18 Uhr, Raum 404

16:15 Uhr   Prof. Susanna Terracini (Turin)

Titel:
Entire solutions and spiralling asymptotic profiles of competition diffusion sytems

Abstract

17:15 Uhr   Prof. Gianmaria Verzini (Milano)

Titel:
Strong competition versus fractional diffusion

Abstract




21.01.2016     Dr. Alberto Saldana (Brüssel)

Titel:
On the extended Allen-Cahn equation.

Abstract: Nonlinear fourth-order PDEs usually have a richer and more complex set of solutions
when compared to its second-order counterpart. In this sense, many models exhibit behaviors
that could be better described with fourth-order equations, like ocean and atmosphere dynamics,
bridges, and pattern formation, just to mention some of them. The theory for higher-order nonlinear
problems, however, is far less developed than its second-order analogue and many basic questions
remain open. Lack of maximum principles, oscillatory behavior of solutions, and regularity issues
are some of the main difficulties in the study of such problems.
I this talk I consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and
Navier boundary conditions. I present results on existence, uniqueness, positivity, stability, a priori
estimates, and symmetry. As an application, we construction a saddle solution in the whole space.
The proofs rely on variational and bifurcation methods. Some numerical approximations of solutions
will also be discussed.



14.01.2016     Linda Lintz (Frankfurt)

Titel: Der fraktionale p-Laplace-Operator




05.11.2015     Dr. Stefanie Hollborn (Universität Mainz)

Titel:
Ein schnelles Prüfverfahren der elektrischen Impedanztomographie

Abstract: Die elektrische Impedanztomographie erzeugt Bilder des unsichtbaren Körperinneren
eines Untersuchungsobjekts, indem sie die Werte der elektrischen Leitfähigkeit im Inneren aus
Strom-Spannungsmessungen an der Körperoberfläche (mathematisch) ermittelt. In vielen
Anwendungen muss diese Leitfähigkeitsverteilung allerdings nicht vollständig rekonstruiert
werden, sondern es genügt zu überprüfen, ob und wo die Leitfähigkeit von einem erwarteten
Wert abweicht. Diese Regionen - sogenannte Inhomogenitäten - weisen bei Materialprüfverfahren
beispielsweise auf Schadstellen hin.

Ich werde ein Verfahren vorstellen, das aus dem Vergleich einer einzigen Strom-Spannungsmessung
mit einem einwandfreien Referenzzustand Informationen über Lage, Größe und Gestalt der
Verunreinigung liefert. Das Verfahren nutzt aus, dass elektrische Potentiale in homogenem
Material als harmonische Funktionen modelliert werden. So kann ein konvexes Gebiet bestimmt
werden, in das die Messdaten nicht harmonisch fortsetzbar sind und das vorhandene Verunreinigungen anzeigt.




29.10.2015     Max Weidemann (Uni Frankfurt)
Achtung, geänderte Zeit und Raum: Beginn ist 16:30 Uhr, Raum 404

Titel:
Monotoner Transport von Wahrscheinlichkeitsmaßen (Bachelor-Arbeit)



22.10.2015     Friedrich Schäufele

Titel:
Sequential Quadratic Programming: Theorie, Implementierung und Anwendung




15.10.2015     Thomas Varnay (Frankfurt)

Titel: Konstruktion von Frames




08.10.2015    Zhitao Zhang (Chinese Academy of Sciences, Bejing)

Beginn: 15 Uhr ct

Titel: Existence, symmetry and bifurcation of solutions for Schrödinger systems                                       

Abstract: We are concerned with the important system of nonlinear Schrödinger equations with linear
and (or) nonlinear couplings which arises from Bose-Einstein condensates, we prove Terracini’s conjecture
for the phase segregation of the limit competition case, we use variational methods and bifurcation
theory to prove the existence of ground state and bound state solutions of the systems, structure of
and the (partial) symmetry of solutions of the systems.



13.8.2015     Joel Kübler (Frankfurt)

Titel:
Charakterisierung von Herglotz-Wellen


30. Juli 2015    Marcel Freitag
(Univ. Paderborn)


Titel: Finite speed of propagation in a fourth-order degenerate parabolic equation modeling
Bose-Einstein condensation


Abstract


7. Mai 2015  Matthias Gundlach (Technische Hochschule Mittelhessen, Gießen)


Titel: Chaos für die Raumklimatisierung

Abstract: Raumluftströmungen lassen sich mit Differenzialgleichungssystemen beschreiben, die dem
aus der Chaostheorie bekannten Lorenz-System ähneln und entsprechende Phänomene aufweisen.
Letztere können in der Klimatisierung von Räumen zur Energieeffizienzsteigerung genutzt werden.
In dem Vortrag werden die Modelle zur Beschreibung von Raumluftströmungen vorgestellt, die zugehörige
Dynamik samt ihrer Attraktoren vorgestellt und erläutert, wie diese Attraktoren für chaotische Strömungen
in Räumen auch auf der Grundlage von experimentell ermittelten Daten nachgewiesen werden können


13. November 2014     Sebastian Becker
(Institut für Mathematik, Goethe-Universität)

Titel: "Stochastische Differentialgleichungen, die kein zufälliges dynamisches System erzeugen"

Abstract: Es wird die Vollständigkeit von Lösungsflüssen von stochastischen Differentialgleichungen untersucht.
Der Existenz- und Eindeutigkeitssatz für stochastische Differentialgleichungen liefert eine fast sichere Vollständigkeit
der Flüsse. Allerdings reicht diese nicht aus, um ein zufälliges dynamisches System zu erzeugen. Ein weiterer
Vollständigkeitsbegriff wird daher vorgestellt, der dieses Nullmengenproblem aufgreift.



4. Dezember 2014     Yulia Abdalova (St. Petersburg State University)

Titel: Local bifurcation for discrete-time non-autonomous systems connected with a heart model

Abstract:  Results in bifurcation theory for autonomous differential and difference
equations are well-known. For the investigation of such equations one typically
applies a theorem of Shoshitaishvili using the Jacobi matrix of the system. But
in the non-autonomous case, this theorem is not applicable. For non-
autonomous differential equations first bifurcation results are proved in [1], [2],
[3].
          In the present work we introduce a modified principle of reduction for
systems depending on a parameter. Using this principle we consider three basic
types of bifurcations in discrete-time non-autonomous systems: the saddle-node,
pitchfork and period-doubling bifurcations. Stability properties of the solutions
of these systems are shown and analogies with autonomous discrete-time
systems under certain conditions on the coefficients are demonstrated.
          As an example we consider a two-dimensional non-autonomous
discrete-time system which is used as a conduction model of the heart. Under
certain typical assumptions for this system, we consider the equation expressing
the dependence of the current conduction time and from a previous recovery
time in a simplified form. We observe the behavior of this equation in the case
of a constant control parameter which leads to so-called alternation which is
connected with asphyxia, a heart diseases, described in [4].
                                      References
     1.   Kloeden P.E., Siegmund S. // Bifurcation and continious transitions of
          attractors in autonomous and nonautonomus systems. International
          Journal of Bifurcation and Chaos, v. 15, No 3, p. 743-762 (2005).
     2.   Nguyen V. M. // A reduction principle for topological classification of
          nonautonomous differential equation Proceeding of the Royal Society
          of Edinburg, v. 123A, p. 621-632 (1993).
     3.   Langa J., Robinson J.C. and Suarez A. // Stability, instability, and
          bifurcation phenomena in non-autonomous differential equations.
          Nonlinearity, v. 15, p. 1-17 (2002).
     4.   Sun J., Amellal F., Glass L. and Billette J. // Alternans and Period-
          doubling Bifurcation in Atrioventricular Nodal Conduction. J. theor.
          Biol., v. 173,p. 79-91 (1995).




29. Januar 2015
   Dr. Nicola Soave (Universität Gießen)

Title: Liouville-type theorems for an elliptic system modelling
          phase-separation and optimal partition problems

Abstract




05. Februar 2015     Sven Jarohs (Frankfurt am Main)

Titel: Maximumprinzipien für nichtlokale Operatoren.

Abstract: Nach einer kurzen Einführung zu nichtlokalen Operatoren und der Präsentation einiger
grundlegender Eigenschaften wird das schwache Maximumprinzip gezeigt. Aufbauend hierauf
wird das starke Maximumprinzip für nichtlokale Operatoren bewiesen. Ich werde kurz den
wichtigsten Unterschied zum lokalen Fall erläutern und einige Anwendung zeigen.
Insbesondere werden als eine Anwendung antisymmetrische Lösungen eines zeitabhängigen
Problems betrachtet.



12. Februar 2015    Robert Grigo (Frankfurt am Main)

Titel:  "Stochastische Navier-Stokes-Gleichungen auf dünnen Gebieten"

Abstract: Um qualitative Aussagen über die Dynamik von Flüssigkeiten zu machen, ist die
globale Existenz der Navier-Stokes-Gleichungen von zentraler Bedeutung. Da dies im
dreidimensionalen Fall zu einen ungelösten Millenium-Problem führt, kann man relativ
große Mengen an Anfangswerten und rechte Seiten bestimmen, indem man die Gebiete
in einer Dimension durch einen Parameter beschränkt. In den 90er wurde diese Methode von
Raugel und Sell etabliert, daraufhin wurde im letzten Jahrzehnt diese Thematik für Stochastische
Navier-Stokes-Glechungen von Chueshov und Kuksin in einen Random-Kick-Force Model
aufgegriffen. In diesen Vortrag soll das Rauschen nicht mit Dirac-Distributionen auf diskreten
Zeitpunkten sitzen, sondern durch ein allgemeineres additives Rauschen modelliert werden.




24. Juli 2014      Metin Tapirdamaz



Titel: Hausdorff-Dimension zufälliger Attraktoren


17. Juli 2014     Anne Heppner (Frankfurt)



Titel: Qualitative Analyse von Rossby-Wellen


Abstract: Rossby-Wellen bezeichnen großräumige Wellenbewegungen in der Atmosphäre oder
dem Ozean, die sich horizontal ausbreiten. Diese spielen eine wichtige Rolle für die Entwicklung
von Hoch- und Tiefdruckgebieten. Im Vortrag wird ein mathematisches Modells auf der Grundlage
partieller Differentialgleichungen hergeleitet, dessen Lösungen die Wellen beschreiben. Ferner wird
ein klassisches Resultat von Kloeden über die Eindeutigkeit einer Familie antisymmetrischer Lösungen
präsentiert. Der wesentliche Schritt in der Herleitung  dieses Resultats ist der Beweis der  radialen
Symmetrie einer in meridionaler Richtung skalierten Lösung.




3. Juli 2014     Prof. Nils Ackermann (UNAM, Mexiko-Stadt)

Titel: Growth estimates for Laplacian eigenvalues under partial symmetries and applications to Bahri-Lions type results


Abstract: We prove new estimates for the growth of the eigenvalues of the Dirichlet Laplacian on a
bounded domain that is partially symmetric. These are reminiscent of Lieb-Cwikel-Rosenbljum type
results. We apply these estimates to a perturbed Lane-Emden equation on a partially symmetric bounded
domain to obtain results in the Spirit of Bahri and Lions. To achieve this we use an existence result of Tanaka
for critical points in a symmetric mountain pass setting with prescribed lower bounds for the Morse indices.
Together with our spectral estimates these yield improved growth rates for the respective partially symmetric
critical levels of the unperturbed problem. An application of Bolle's perturbation method yields the existence of
an infinity of solutions for the perturbed problem under weaker conditions on the exponents than known before.




8. Mai 2014    Nikolaos Sfakianakis  (Mainz)

Title : "A finite element method for the simulation of motility of living cells"

Mittwoch, 19.02.14,  Raum 110,    Prof. Jacson Simsen – UNIFEI - Brazil

Titel:  On global attractors for parabolic problems with variable exponents

Abstract:
In this talk I will give an overview on the results which we have obtained during the last 5 years about
existence  and upper semicontinuity of global atractors for parabolic problems with variable exponents.


06.02.2014      Dr. Gilles Evequoz (Frankfurt)

Titel: Reelle L^p-Lösungen der nichtlinearen Helmholtz-Gleichung

Abstract:

Die Helmholtz-Gleichung wird u.a. zur Modellierung der Ausbreitung von akustischen Wellen verwendet.
In diesem Vortrag wird zunächst ein Überblick über die Lösungstheorie im linearen Fall gegeben. Darauf
aufbauend werden neue, auf variationellen Methoden beruhende Resultate über die Existenz und die
Fernfeldentwicklung von reellen L^p-Lösungen im nichtlinearen Fall vorgestellt. Diese Lösungen
korrespondieren zu zeitperiodischen Lösungen der zugehörigen Wellengleichung mit zeitlich konstanter
Energiedichte (stehende Wellen).

Mittwoch, 27.11.2013, 16 Uhr, Raum 711 groß      Stephan Dahlke (Universität Marburg)

Titel: Shearlet Coorbit-Räume:  Gruppentheoretische Grundlagen, Konstruktion und Eigenschaften.


Abstract: Eine der zentralen Aufgaben der Angewandten Analysis ist die Analyse von Signalen. Der erste
Schritt ist stets die Zerlegung des Signals mittels geeigneter  Basisfunktionen,  welche es erlauben, die
interessierende Information effizient aus dem Signal zu extrahieren.  In den letzten Jahren ist insbesondere
die Detektion von Richtungsinformationen in den Mittelpunkt des Interesses gerückt, und viele verschiedene
Ansätze wie Curvelets, Ridgelets, Contourlets, Shearlets  usw. sind vorgeschlagen worden.  Hierbei nehmen
Shearlets eine gewisse Sonderstellung ein, denn die stetige Shearlet-Transformation  kann mittels einer
quadratintegrablen Darstellung einer lokalkompakten, topologischen Gruppe, der Shearlet-Gruppe,  hergeleitet
werden.  Dieser rein gruppentheoretische Zugang hat viele Vorteile. So können etwa  neue, mit der stetigen Shearlet-
Transformation assoziierte, kanonische Glattheitsräume  konstruiert werden.  Das Hilfsmittel hierzu ist die Coorbit--Theorie
,  die von H. Feichtinger und K. Gröchenig in einer Reihe von Arbeiten entwickelt wurde.  In diesem Vortrag sollen zunächst
  die gruppentheoretischen Grundlagen der Shearlet-Transformation diskutiert werden. Anschließend wird  ein kurzer
Überblick über die Grundprinzipien  der Coorbit-Theorie gegeben und gezeigt, wie sich diese Theorie auf den Shearlet
-Fall anwenden lässt.  Nachdem somit die  Shearlet Coorbit-Räume etabliert sind, werden wir deren strukturelle
Eigenschaften analysieren,  das heißt, wir studieren  Dichtheits-und Einbettungseigenschaften dieser Räume und
diskutieren  erste Spursätze .

.

. 06.06.2013   Professor Eduardo Colorado Heras (University of Madrid, Spanien)

 

Titel:       A classical critical problem in a non-classical fractional setting

 

Abstract

 

 

23.05.2013   Arno Berger  (University of Alberta, Canada)

 Titel: Digit distributions in dynamics

Abstract: The study of numbers generated in one way or another by dynamical
systems, be they continuous or discrete, deterministic or stochastic, is
a classical, multifaceted field. A notorious gem in this field is the wide-
spread, unexpected emergence of a particular logarithmic distribution,
commonly referred to as Benford's Law (BL). This talk will focus on two
recently established characterisations of BL for a wide class of deterministic
and stochastic processes. For example, every finite-dimensional linear flow
generically obeys BL in a very strong sense, as does, with probability one,
every path of geometric Brownian motion. The talk will describe the main
ideas behind these results and also present some of the many challenges
that remain.

 

 

27.02.2013 (Raum 711 klein)  13 Uhr

Prof. Dr. Jens Starke (Technical University of Denmark)

 

Titel: Multiscale analysis of traveling waves and oscillations in particle models

 

21.02.2013   Dr. Mouhamed Moustapha Fall (AIMS Senegal)

Titel: Hardy's inequality with singularity on the boundary

Abstract: In this talk, we discuss the Hardy inequality with one point  singularity on the boundary
of a bounded domain  within the framework of Brezis-Marcus. We will see how the geometry (not local)
of the domain inters in obtaining existence and nonexistence of a minimizer.

 

 

24.01.2013    Frau Bahareh Akhtari (Sharif University, Teheran, Iran, z. Zt. Universität Mannheim)

Titel: Numerical Methods for Stochastic Delay Differential Equations under Locally Lipschitz Coefficients


 

Archiv

 

01.11.2012 Dr. Enea Parini (Université Aix-Marseille)

 

Titel: Optimal constants for higher-order functional embeddings


08.11.12   Dr. Martin Hutzenthaler (LMU München)


Titel
: Numerical solution of stochastic differential equations

Abstract: The classical numerical discretization scheme for differential equations
is Euler's method. For stochastic differential equations, however, this
method (in combination with multilevel Monte Carlo methods) turns out 
to fail as soon as the coefficient functions of the SDE are superlinearly 
growing (so that the classical global Lipschitz assumption is not satisfied).
In this talk, we explain this deficiency of Euler's method in some 
detail. Moreover we show how to modify Euler's method to obtain a properly working numerical
disretization scheme for SDEs.

 

15.11.12   Alberto Saldana (Goethe-Universität Frankfurt)

 

Title: Asymptotic axial symmetry of solutions of nonlinear parabolic equations.

Abstract: In general, it is a good rule of thumb that the parabolic flow reduce the complexity of a solution
of a PDE. In other words, one would expect that the symmetry improves through time. With this in mind,
I will present some results regarding the asymptotic (in time) symmetry of solutions of some nonlinear
parabolic boundary value problems in radial bounded domains whose initial profile satisfy a
reflection inequality with respect to a hyperplane containing the origin.

Sign changing solutions and nonlinearities that depend both on the time and the spatial variable are
considered, so only a partial symmetry is to be expected, namely, axial symmetry together with some
monotonicity properties (also called foliated Schwarz symmetry ).
The extension of these results and the methods involved in the proofs to unbounded domains presents
some challenging difficulties. I will also talk about some of them.
This is a joint work with Tobias Weth.

 

29.11.12    Sven Jarohs (Goethe-Universität Frankfurt)

Titel: Asymptotic Symmetry for nonlocal parabolic equations

Abstract: Asymptotic symmetry has been proven in very general settings for local equations.
I will recall shortly the proof ideas. Afterwards I will present the notion of nonlocal equations,
namely a parabolic problem involving the fractional Laplacian. After giving a suitable setup,
I will try to explain the new difficulties that arise in the nonlocal setting in comparison with the
local case and give ideas how we were able to overcome them. Finally I will present our results
for the parabolic problem involving the fractional Laplacian. This is a joint work with Tobias Weth.

 

 

Archiv 2012

 

Professor Navaratnam Sri Namachchivaya,(Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign,  28. Juni 2012,  14.45 Uhr


Title: From Random to Data Driven Dynamical Systems

Abstract:
My presentation will focus on stability, dimensional reduction and
filtering techniques of nonlinear dynamical systems with uncertainties.
I shall outline a collection of problems that combine techniques of
model reduction and filtering. When the rates of change of different
variables differ by orders of magnitude, efficient data assimilation
can be accomplished  by constructing nonlinear filtering equations for
the coarse-grained signal.  In particular, we study how scaling
interacts with filtering via stochastic homogenization.
Finally, if time permits, I shall present some research highlights on
nonlinear control of under-actuated systems and delay differential
equations with fluctuating delay.
This is a joint work with graduate students Vishal Chikkerur,
Ying Tian Jiang, Nishanth Lingala, Mahmoud Mamlouk, Hoong Chieh Yeong,
Nicolas Perkowski (HU-B), and Christian Rapp (TU-H).

 

Jan-Erik Stecher, Universität Heidelberg, 9. Februar 2012

 

Titel: Dirac Concentrations in Lotka-Volterra parabolic PDE

Abstract: In this presentation a diploma thesis based on a paper written by Benoit Perthame
and Guy Barles will be discussed.
It deals with a parabolic nonlinear PDE which describes the evolution of species, including small mutations.
We will illustrate existence and uniqueness results and present convergence results by letting mutations vanish.
This leads to a specific structure of the solution. It can be written as a sum of Dirac measures, which can
be interpreted as a mathematical way to support Darwin's Law of Evolution.

 

Dr. Georg Schöchtel, Fachbereich Mathematik, TU Darmstadt, 26. Januar 2012

Title: Motion of inertial particles in Gaussian fields driven by an infinite-dimensional fractional Brownian motion


Abstract: The motion of an inertial particle in a fractional Gaussian random field is studied.
The motion is described by Newton's second law for a particle on a 2D torus, with force
proportional to the difference between a background fluid velocity and the particle velocity itself.
The fluid velocity satisfies a linear stochastic PDE driven by an infinite-dimensional fractional
Brownian motion. The usefulness of such random velocity fields in simulations is that we can
generate random velocity fields with a given energy spectrum, thus creating caricatures of realistic
turbulent flows. This model captures also the clustering phenomenon of preferential concentration
observed in real world and numerical experiments, i.e. particles cluster in regions of low vorticity and
high strain rate. We prove almost sure existence and uniqueness of particle paths and give sufficient
conditions to rewrite this system as a random dynamical system with a global random pullback
attractor. Further we give upper bounds of the almost sure constant Hausdorff dimension of the
random attractor. Finally numerical investigations are considered.

 


Prof. Hamid Zangeneh, Isfahan University of Technology,
              z.Zt. Goethe-Universität, 19. Januar 2012

Titel: "Pattern Selection on the Growing Tips of Plants:
             Bifurcations on Spherical Cap"

 

Nils Ackermann, Universidad Nacional Autonoma de Mexico, 7. Juli 2011


Titel: 
Multi-soliton standing waves in expanding waveguides

Abstract: Consider a compact submanifold M without boundary of
N-dimensional Euclidean space and, for R>0 large enough, the tubular
neighborhood U_R of radius 1 of the expanded manifold RM.  Also
consider the time-independent Schrödinger equation with a cubic power
nonlinearity on U_R and Dirichlet boundary conditions.  Solutions to
this equation yield standing light waves for the waveguide U_R, filled
with a self-focusing medium.  We prove, for any natural number n and R
large enough, the existence of a positive solution with n bumps.  If M
is one dimensional, i.e. U_R is a tubular guide, we may allow M to
have boundary and prove existence of multi-bump solutions with
alternating signs along the tube.  The proof rests on Lyapunov-Schmidt
reduction to the configuration space of M and on asymptotic estimates
of the variational functional.

 

Maria Anguiano, University of Sevilla, 14. April 2011

Titel: Existence of pullback attractors for nonlinear and nonautonomous parabolic PDEs

Abstract: "The understanding of the asymptotic behaviour of dynamical systems
is one of the most important problems of modern mathematical physics.
One way to treat this problem for systems having some dissipativity properties
is to analyze the existence and structure of its global attractor.
On some occasions, some phenomena are modelled by nonlinear evolutionary
equations which do not take into account all the relevant information of the real
systems. Instead some neglected quantities can be modelled as an external force
which in general becomes time-dependent. For this reason, non-autonomous
systems are of great importance and interest. In this talk I will analyze several
models of reaction-diffusion equations in some bounded and unbounded domains.
I will use the pullback theory so much for single-valued as for multi-valued non-
autonomous dynamical systems, since this allows for more generality in the
non-autonomous terms, to prove the existence of pullback attractors for our models
of reaction-diffusion equations."

Juan Carlos de los Reyes, TU Berlin, 17. Februar 2011

Titel: On semi-smooth Newton methods for the numerical solution of viscoplastic
fluid flow and related variational inequalities

Abstract: In this talk we focus on numerical optimization techniques for the solution
of Bingham viscoplastic fluid flow and other related variational inequalities. Bingham
materials are characterized by the presence of a so-called yield stress: they behave like
solids in regions where the stresses are small and like incompressible fluids where
the stresses are larger than a plasticity threshold. Based on both primal and dual formulations
of the problem, two regularization strategies will be presented. The well-posedness of
each regularized problem is verified and convergence of the regularized solution towards
the original one is studied.
For the solution of each regularized system, generalized Newton algorithms are constructed.
We will present results on global and local superlinear convergence in a function space
setting and/or in finite dimensional spaces, after discretization of the systems. Also
continuation strategies, based on the properties of the path-value functional, are designed and numerically tested.
Finally, since a similar yield behavior is also found in problems arising from other application
areas like elastic contact, plasticity or image denoising, the applicability of the approach
to other variational inequalities will be discussed.

 

Univ.Prof.Dr. Ronny Ramlau (Universität Linz), 10. Februar 2011

 

Dr. Matthias Kurzke (Universität Bonn),  13. Januar 2011

Titel: Vortex motion in the Landau-Lifshitz-Gilbert equations

Abstract: The Landau-Lifshitz-Gilbert equations (modeling the evolution of ferromagnets) are a hybrid
of harmonic map heat flow and the Schrödinger map flow. In certain geometries, vortex-like singularitie
s are naturally observed. We derive the equation of motion for these singularities.

(This is joint work with C. Melcher, R. Moser, D. Spirn.)

Archiv

 

Denis Bonheure (Universite Libre de Bruxelles), 10. Juni 2010

Symmetry and multiplicity of least energy solutions for semilinear elliptic boundary value problems.

Abstract : We review some well-known as well as recent results about symmetry, uniqueness or
multiplicity of positive or nodal solutions of homogeneous Dirichlet boundary value problems.
We then discuss analogous problems with Neumann boundary conditions and we emphasize
the drastic differences and open questions.

Dr. Arno Berger (University of Alberta), 20. Mai 2010

Dynamics in finite time - Some thoughts on concepts and applications

Abstract: Motivated strongly by applications, notably in geophysical
fluid dynamics, finite-time dynamics aims at identifying and systematically studying dynamical roperties
of systems that are defined only over a bounded interval of time. Many classical symptotic concepts do not
apply in this situation and have to be modified or replaced altogether. Quite a number of modified or new
concepts have recently been proposed in this regard, and finite-time dynamics has become an active and
diverse field. Much attention is being focused on the development of finite-time stability, spectral and bifurcation
theories that are both practicable and consistent with their classical counterparts. A key ingredient in these
(and other) areas is hyperbolicity which, ever since the dawning of dynamical systems theory, has been
recognised as a fundamental concept. Not too surprisingly, when reformulated appropriately hyperbolicity
emerges as a key notion in the finite-time context as well.

After providing a brief overview of finite-time dynamics in general, and hyperbolicity in the classical and the
finite-time settings in particular, this talk will motivate and present several results that generalise and unify
earlier work. Specifically, the existence, non-uniqueness and robustness of finite-time (un)stable manifolds
will be discussed, as well as the basic problem of detecting  Lagrangian) hyperbolicity from (Eulerian) data
encoded in a dynamic partition of the extended phase space. Some of the fundamental challenges inherent
to finite-time dynamics, both practical and conceptual, will become apparent.

 

Prof. Ken Palmer (National University of Taiwan, z. Zt Universitaet Tuebingen), 6. Mai 2010

Lyapunov Exponents and Sensitive Dependence

In general if an orbit of a one-dimensional map has positive Lyapunov exponent, it need not exhibit sensitive
dependence. However if the map is smooth and the orbit stays away from critical points, then positive Lyapunov
exponent does imply sensitive dependence. Also we exhibit a class of maps for which positive Lyapunov exponent
does imply sensitive dependence even for orbits which have critical points in their omega limit set.

 

Hugo Tavares (University of Lisbon, z.Z. Frankfurt), 22. April 2010

Asymptotic study of a reaction diffusion system with competition terms

Abstract: We take a system of partial differential equations with competition terms which arises in the phenomenon
of Bose-Einstein condensation. For such system, we study the asymptotics of its solutions as the competition term
goes to infinity. The limiting profiles are segregated, and we study the properties of their nodal sets. We present a
general regularity theory which can be applied in other situations.

(joint works with B. Noris, G. Verzini and S. Terracini)

 

Martin Riedler (Heriot Watt University, Edinburg), 11. Februar 2010

A stochastic model for voltage potential transport in passive neuronal membranes

Abstract: In my talk I will give a short introduction to the typical approaches of modelling inneuroscience
and characteristic problems and questions that arise. Particularly, I will focus on hybrid stochastic models
which represent the class of models for single neurons or neuronal membranes, respectively, closest to
the biophysical realityand still analytically tractable and practically relevant. To this end I will introduce aa
class of models employing Piecewise Deterministic Processes which give a closedand mathematically
precise description of a stochastic neuronal membranes based onfundamental and biophysical accurate premises.
Further, I will present algorithms for the (approximate) simulation of the neuron models we are considering
. I will also present error estimates and sketch the proof of convergence for the algorithms.

 

Christophe Troestler (Université de Mons-Hainaut, Belgien), 28. Januar 2010

Oddness of least energy nodal solutions of the Lane-Emden problem.

Abstract: We will discuss the symmetries of least energy, sign changing, solutions of Lane-Emden type of problems.
  In particular, we will be interested in the oddness of these solutions on spherically symmetric domains.
We will outline generalizations to more general domains and non-linearities.

 

Dr. Gilles Evequoz (Goethe-Universität Frankfurt), 10. Dezember2009

Gap solitons for the discrete nonlinear Schrödinger equation with an interface


Abstract: In this talk, we shall discuss a special case of the discrete nonlinear Schrödinger equation
on the whole space where the potential and the nonlinearity are both periodic on each side of a given
interface, but not globally periodic. We study the existence of ground-state solutions (i.e. solutions having
the least energy among all) of the corresponding stationary problem, using a variational method.
This is joint work with Prof. Wolfgang Reichel (Universität Karlsruhe)

 

Prof. Dr. Andreas Rieder (Universität Karlsruhe) 19. November 2009

Ein Newton-Löser für die Elektrische Impedanztomographie

 

Abstract: Wir präsentieren ein inexaktes Newton-Verfahren zur effizienten Lösung des inversen Problems
der zweidimensionalen Elektrischen Impedanztomographie, wobei wir das sogenannte 'vollständige
Elektrodenmodell' zugrunde legen. Im Fokus des Vortrags stehen die Herleitung des Verfahrens, seine Konvergenz
und numerische Effizienz. Letztere verdeutlichen wir durch  numerische Resultate.

 

PD Dr. Thomas Lorenz (Goethe-Universität Frankfurt) 29. Oktober 2009

Mutational analyis: for evolutions, think beyond vector spaces

 

Prof. Dr. Oliver Dorn (University of Manchester), 24. September 2009

"A level set approach for structural inversion from indirect data"

Abstract: In many real world applications of science and engineering the task arises to invert for material
profiles inside a given domain from indirectly obtained data. These situations occur for example in medical
imaging, remote sensing, geophysical tomography or nondestructive testing of materials. The regions of
interest are probed in these applications by some kind of fields which propagate inside these regions
according to a given physical law, which typically is described by a partial differential equation (PDE).
The inverse problem consists of finding a map of the local distribution of the parameters entering in the
PDE from data measured outside of the domain of interest. This inverse problem is almost always ill-posed,
and stable solutions are only available by adding prior information to the problem. A classical way of doing so
is to use so-called Tikhonov Philips functionals, yielding smooth representations of the solutions. When it is
known that there are interfaces in the domain, then these classical Tikhonov Philips solutions are suboptimal,
and alternative techniques need to be developed. We will present in this talk a novel technique for structural
inversion which uses a level set technique for inverting for parameter profiles with interfaces. We demonstrate
this quite novel technology for two important applications, namely the early detection of breast cancer using
microwaves, and for a crack-detection problem. Numerical simulations in 2D are presented for these two
applications which show that the level set technique shows great promise as an alternative tool to classical
Tikhonov Philips inversion schemes.

 

Prof. Dr. J. Zubelli (IMPA, Rio de Janeiro), 6. August 200914 Uhr c.t., 711 gr.

"Inverse Problems in Finances: A Short Survey of Calibration Methods"

Abstract: We survey the problem of calibrating the volatility of securities and asset prices by using
quoted prices of financial instruments, such as derivatives. This is done within the framework
of regularization provided by inverse problem theory and leads to very interesting mathematical
and computational problems.We briefly review a number of recent contributions to the field and
highlight its activity.

 

Patrick Grüning (Goethe-Universität Frankfurt)

18. Juni 2009, 14:15 uhr, R. 711gr.

"Multilevel Monte Carlo Simulationen von Optionspreisen und Sensitivitäten des Underlyings"

 

Sven Jarohs (Goethe-Universität Frankfurt)

28. Mai 2009, 14.15 Uhr, R. 711gr.

"Hausdorffmaß von Mengen unter Linearen Operatoren"

 

Isabella Ianni (Sissa, Trieste)

    Thursday, 14 May, 14.15 in room 711 gr.

"Semiclassical standing waves for the nonlinear Schrodinger-Maxwell equation"

Abstract:  We study the semiclassical limit for a nonlinear stationary
Schrodinger-Maxwell system of equations in R3.
This system has been introduced as a model which describes standing waves
for the nonlinear Schrodinger equation interacting with the electrostatic
field.
By using a Lyapunov-Schmidt reduction in a variational setting we prove
the existence of families of solutions which exhibit a concentration behavior.
Precisely we find solutions concentrating pointwise and also radially
symmetric solutions exhibiting concentration around a sphere.
Some stability results are also discussed.

 

Arnulf Jentzen (Doktorand bei Prof. Dr. P. E. Kloeden)

    Tuesday, 5 May 2009, 16.15 in room 110

"Taylor expansions for stochastic partial differential equations"

 

Dr. Gilles Evequoz (Universität Karlsruhe)

    Thursday, 30 April 2009, 14.15 in room 711 gr.

"Hadamard differentiability and bifurcation for some nonlinear elliptic equations"

 

Mouhamed Moustapha Fall (SISSA, Trieste)

        Thursday, 23 April 2009, 14.15 in room 711 gr.

"The Free Boundary Plateau Problem for large H-surfaces"

 

Dr. Herbert Egger (RWTH Aachen)

       Tuesday, 7 April 2009, 14.15 in room 110

 "An Inverse Problem in Computational Finance: Analysis, Regularization and Numerical Solution"