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Kolloquium für Angewandte Mathematik

Durch das Institut für Mathematik der TUHH (E-10) wird ein Kolloquium für Angewandte Mathematik veranstaltet, in dem mehrfach pro Semester Referenten über neuere Forschungsergebnisse berichten. Die Vorträge werden vorwiegend Themen der Hauptarbeitsgebiete des Arbeitsbereiches (Numerische Lösung linearer und nichtlinearer Gleichungssysteme, numerische Behandlung großer Eigenwertaufgaben, numerische Behandlung von Differentialgleichungen, nichtlineare Optimierung, Fredholm- und Spektraltheorie) betreffen.

Wenn Sie per E-Mail über die Vorträge informiert werden wollen, können Sie sich in der Mailing-Liste "mathe-kolloquium" eintragen.

Vorträge

| 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 |

2017

Bi-stetige Halbgruppen

Jan Meichsner

Institut fuer Mathematik, Lehrstuhl angewandte Analysis, TUHH,

15.06.2017, 14:45 Uhr

Raum H - SBC5 H0.03 (noch unbestaetigt)

Zusammenfassung:

In dem Vortrag wird es um bi-stetige Halbgruppen gehen. Das Konzept geht auf die Dissertation
'Bi–Continuous Semigroups on Spaces with Two Topologies: Theory and Applications' von
F. Kühnemund aus dem Jahre 2001 zurueck. Betrachtet werden Halbgruppen auf einem
Banachraum, welche nicht stark-stetig sind. Das wird behoben, indem man sich eine groebere
Topologie betrachtet.
Da ich Anfaenger auf dem Feld bin, wird der Vortrag eine Einfuehrung enthalten und die Nuetzlichkeit
des Konzepts an einigen Beispielen illustriert.

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The need for linear system solvers in dispersive wave modeling

Jörn Behrens

UHH,

26.01.2017, 14:00 Uhr

Am Schwarzenberg-Campus 3 (E), Raum 3.074

Zusammenfassung:

Tsunami modeling is - to first (and very accurate) approximation - performed with the help of shallow water theory and equations. This is still the method of choice for many applications, including forecasting, hazard assessment and inundation modeling. However, for long propagation distances as well as highly nonuniform topographies dispersive effects become important. While truly dispersive model equations are fully three-dimensional and therefore expensive with respect to computational requirements, a common approach to dispersive modeling comprises a non-hydrostatic correction of shallow water equations. In order to derive this correction term, a linear system of equations needs to be solved in each time step - even when the time-stepping scheme is explicit.

In the presentation we will introduce the basic modeling concepts for tsunami simulation, will show the derivation of non-hydrostatic correction terms and motivate further research on solvers for linear systems of equations.

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2016

Nuklearität und Tensorprodukte

Karsten Kruse

12.12.2016, 15:00 Uhr

Am Schwarzenberg-Campus 3 (E), Raum 3.074

Zusammenfassung:

Im Vortrag wird es darum gehen, wie man eine vektorwertige Gleichung löst, wenn man die entsprechende Gleichung schon einmal skalarwertig gelöst hat. Typische Beispiele hierfür sind elliptische Differentialgleichungen. Hierbei geht es dann weniger darum, den Differentialoperator selbst zu untersuchen, sondern die Eigenschaften der Räume, auf denen er lebt.

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Fractional Powers of Linear Operators

Jan Meichsner

24.11.2016, 14:00 Uhr

Am Schwarzenberg-Campus 3 (E), Raum 3.074

Zusammenfassung:

Im wesentlichen ein 60 bis 90 minütiger Arbeitsstandbericht. Es werden grundlagen der Theorie fraktionaler Operatoren erläutert und danach auf die Problematik der Einführung durch harmonische Erweiterung eingegangen.

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Vier konkrete Anwendungen von Toeplitzoperatoren

Albrecht Böttcher

TU Chemnitz,

02.11.2016, 13:30 Uhr

TUHH, Gebäude A, Raum A0.19

Zusammenfassung:

Vier konkrete Anwendungen von Toeplitzoperatoren

Es werden vier konkrete und sehr unterschiedliche Anwendungen von Toeplitzoperatoren vorgestellt. Diese sind (1) ein Problem aus der optimalen ell-eins-Kontrolle, (2) Spektralfaktorisierung von Polynomen vom Grad 20000, (3) Berechnung des Volumens der Fundamentalgebiete gewisser hochdimensionaler Gitter, und (4) Bestimmung der Grenzmenge der Nullstellen von Polynomen vom Fibonacci-Typ in der Hausdorffmetrik. Der Vortrag erlaubt es, viermal abzuschalten und ebenso oft wieder einzusteigen.

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Four concrete applications of Toeplitz operators

I present four concrete and very different applications of Toeplitz operators. These applications are (1) a problem in optimal ell-one control, (2) spectral factorization of polynomials of degree 20000, (3) computation of the volume of the fundamental domains of some high-dimensional lattices, and (4) the determination of the Hausdorff limit of the zero set of polynomials of the Fibonacci type. The talk allows you to switch off four times and to re-enter the same number of times.

Zusätzliche Informationen zur Person:

214 Paper
9 Bücher
- weiß *alles* über Toeplitzmatrizen und -operatoren
- hält großartige Vorträge

https://www-user.tu-chemnitz.de/~aboettch/

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Oscillation in a posteriori error estimation

Andreas Veeser

Dipartimento di Matematica, Universita degli Studi di Milano,

04.07.2016, 16:15 Uhr

Am Schwarzenberg-Campus 3 (A), Raum A 1.19.1

Zusammenfassung:

The goal of an a posteriori error analysis for an approximate PDE
solution is to establish the equivalence of error and a posteriori
estimator. Unfortunately, this equivalence is often only up to so-
called oscillation terms.

In this talk we shall clarify the reasons for the presence of
oscillation. Moreover, we propose a new approach to a posteriori error
estimation, where oscillation can be bounded by the error and so does
not longer spoil the aforementioned equivalence.

This is joint work with Christian Kreuzer (Bochum).

Zusätzliche Informationen zur Person:

Andreas Veeser graduated from Freiburg University and obtained his Phd in 1998 from the same university. After several Postdoctoral positions at Freiburg University and the University of Milano, he took a permanent researcher position at the University of Milano, where he became Associate Professor in 2005.
He is world-wide expert of Numerical Analysis, Finite Elements and Adaptivity, and has done substantial breakthrough contributions in all these fields. He is frequently invited as plenary speaker at conferences and workshops and also to deliver specialized courses all over the world.

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Trefftz discontinuous Galerkin methods for wave problems

Dr Andrea Moiola

University of Reading,

24.06.2016, 10:30 Uhr

Am Schwarzenberg-Campus 3 Building A Raum A.1.19.1

Zusammenfassung:

We present a space-time discontinuous Galerkin (DG) method for linear
wave propagation problems.
The special feature of the scheme is that it is a Trefftz method,
namely that trial and test functions are solution of the partial
differential equation to be discretised in each element of the
(space-time) mesh.
The DG scheme is defined for unstructured meshes whose internal faces
need not be aligned to the space-time axes.
The Trefftz approach can be used to improve and ease the
implementation of explicit schemes based on ``tent-pitched'' meshes.
We show that the scheme is well-posed, quasi-optimal and dissipative,
and prove a priori error bounds for general Trefftz discrete spaces.
A concrete discretisation can be obtained using piecewise polynomials
that satisfy the wave equation elementwise, for which we show high
orders of convergence.
If time allows, we will describe a similar Trefftz-DG method for the
Helmholtz equation, i.e. wave equation in time-harmonic regime, for
which non-polynomial basis functions are used and quite a complete
theory has been established.

Zusätzliche Informationen zur Person:

After graduating from University of Pavia, from 2008 to 2011, Dr. Andrea Moiola carried his Phd at the Seminar for Applied Mathematics (SAM) - ETH Zürich, under the supervision of Ralf Hiptmair (ETH) and Ilaria Perugia (University of Vienna/Pavia).
From March 2012 to February 2013, He held a ''Fellowship for prospective researchers'' on ''Computational wave propagation'', funded by the Swiss National Science Foundation, based in Reading and supervised by Simon N. Chandler-Wilde.
Since March 2013, he holds a ''senior research fellow'' position at the Department of Mathematics and Statistics of the University of Reading. He is an expert on the numerical approximation of Helmholtz and wave propagation problems.

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Solving the Vlasov equation in low-rank tensor format

Dr. Katharina Kormann

Technische Universität München, Zentrum Mathematik - M16, Boltzmannstraße 3, 85747 Garching, Germany

26.04.2016, 16:15 Uhr

Am Schwarzenberg-Campus 3, Gebäude A, Raum A.0.01 und A.3.31

Zusammenfassung:

The evolution of a plasma in external and self-consistent fields is modelled by the Vlasov equation for the distribution function in six dimensional phase space. Due to the high dimensionality and the development of small structures the numerical solution is very challenging. Grid-based methods
for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two or four dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the hierarchical Tucker format provides efficient basic linear algebra routines for low-rank representations of tensors.

In this talk, I will present a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Interpolation formulas for the low-parametric tensor format as well as efficient implementations will be discussed. Numerical simulations for the Vlasov-Poisson equation are shown for the Landau damping test case in two, four, and six dimensional phase space as well as simulations with a constant magnetic field. Depending on the test case, the memory
requirements reduce by a factor $10^2$-$10^3$ in four and a factor $10^5$-$10^6$ in six dimensions compared to the full-grid method.

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Auxiliary Space Methods for Variational Problems in H{curl)

Ralf Hiptmair

ETH Zürich,

28.01.2016, 15:30 Uhr

Am Schwarzenberg-Campus 1 (A), A1.20

Zusammenfassung:

Auxiliary space preconditioning targets elliptic boundary value problems discetized by means of finite elements. The idea is to use a related discrete boundary value problem, for which efficient solvers are available, as a preconditioner. The connection between both problems is established by means of a suitable prolongation operator.

We apply this strategy to variational problems for the bilinear form $(\alpha(x)\cdot,\cdot)_0+(\beta(x)curl\cdot,curl\cdot)_0$ ($\alpha,\beta$ uniformly positive coefficient functions) posed on the function space $H(curl)$ (or $H_0(curl)$).
These are commonly encountered in magneto-quasistatic models for electromagnetic phenomena (eddy current models). Finite element Galerkin discretization usually relies on Nedelec's $H(curl)$-conforming edge elements, but discontinuous Galerkin (DG) methods are a viable option, too. In any case, one faces large sparse linear systems of equations, for which efficient preconditioners are badly needed. Three settings will be discussed:

I) When edge elements are used on a single unstructured mesh, coarser meshes needed for the application of geometric multigrid solvers may not be available. They may be easy to construct, however, for a semi-structured mesh, suggesting the use of an auxiliary edge element space on that mesh.
II) In the same setting as (I), algebraic multigrid methods (AMG) could look promising. Alas, AMG schemes for edge finite element discretizations that match the performance of those for $H^{1}$-conforming finite elements are not available. To harness standard nodal AMG schemes one may use an auxiliary space of continuous piecewise polynomial vectorfields.
III) Using a DG discretization on a standard triangulation, which may be required in the context of magneto-hydrodynamics, an edge element space may serve as auxiliary space.

For all these cases we present theoretical results about the performance of the preconditioner with focus on $h$-independence and robustness with respect to jumps of the coefficients. The main ideas needed to verify the abstract assumptions of the theory of auxiliary space preconditioning will be outlined.

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2015

Rational Arnoldi methods

Prof. Lothar Reichel

Department of Mathematical Sciences, Kent State University, Ohio, USA

18.09.2015, 14:00 Uhr

Am Schwarzenberg-Campus 3 (E), Raum 3.074

Zusammenfassung:

The standard Arnoldi method is one of the most popular schemes for reducing a large matrix A to a small one. The method requires the evaluation of matrix-vector products with A. Rational Arnoldi methods reduce the matrix A by both evaluating matrix-vector products and solving linear systems of equations with A. Rational Arnoldi methods are attractive to use when A has a structure that allows efficient solution linear systems of equations with A. They are commonly applied to the computation of an invariant subspace of A and to the approximation of matrix functions. We discuss implementations of rational Arnoldi methods and compares their properties.

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Interpolationsbasierte Reduzierte-Basis-Modellierung von Lösungskurven mit Umkehrpunkten

Hagen Eichel

Eröffnung des Promotionsverfahrens,

03.09.2015, 10:00 Uhr

Am Schwarzenberg-Campus 3 (E), Raum 3.074

Zusammenfassung:

Bei der numerischen Simulation physikalischer Prozesse treten häufig große parameterabhängige nichtlineare Gleichungssysteme auf. Zur Verringerung des Rechenaufwands werden oft Reduzierte-Basis-Methoden verwendet, die sich in lokale und globale Methoden unterscheiden lassen, wobei letztere Umkehrpunkte bezüglich des Parameters gewöhnlich nicht zulassen. In dieser Arbeit wird ein globaler, interpolationsbasierter Ansatz für Probleme mit Umkehrpunkten entwickelt und es werden die Vorteile und Grenzen dieser Methode aufgezeigt.

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On the spectrum of certain random operators: A link to Julia sets

Raffael Hagger

09.04.2015, 16:00 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

After the introduction of random matrices to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity. Wigner's idea was to consider families of Hamiltonians that underlie a certain probability distribution to describe overly complicated systems. Of particular interest are, of course, the spectra of these Hamiltonians. In this talk we consider random, in general non-self-adjoint, tridiagonal operators on the Hilbert space of square-summable sequences. To model randomness, we use an approach by Davies that eliminates all probabilistic arguments.

Despite the rising interest, not much is known about the spectra of non-self-adjoint random operators. The Feinberg-Zee random hopping matrix reveals this in a beautiful manner. The boundary of its spectrum appears to be fractal, but a proof has not been found yet. While we can not give a proof either, we present a reason why this is very plausible. Certain tridiagonal operators share remarkable symmetries that allow us to enlarge known subsets of the spectrum by sizeable amounts. In some cases like the Feinberg-Zee random hopping matrix, this implies that the spectrum contains an infinite sequence of Julia sets.

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Orthogonalization with a non-standard inner product and approximate inverse preconditioning

Miro Rozložník

Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic

19.03.2015, 15:00 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

One of the most important and frequently used preconditioning techniques for solving symmetric positive definite systems is based on computing the approximate inverse factorizations. It is also a well-known fact that such factors can be computed column-wise by the orthogonalization process applied to the unit basis vectors provided that we use a non-standard inner product induced by the positive definite system matrix A. In this contribution we consider the classical Gram-Schmidt algorithm (CGS), the modified Gram-Schmidt algorithm (MGS) and also yet another variant of sequential orthogonalization, which is motivated originally by the AINV preconditioner and which uses oblique projections.

The orthogonality between computed vectors is crucial for the quality of the preconditioner constructed in the approximate inverse factorization. While for the case of the standard inner product there exists a complete rounding error analysis for all main orthogonalization schemes, the numerical properties of the schemes with a non-standard inner product are much less understood. We will formulate results on the loss of orthogonality and on the factorization error for all previously mentioned orthogonalization schemes.

This contribution is joint work with Jiří Kopal (Technical University Liberec), Miroslav Tůma and Alicja Smoktunowicz (Warsaw University of Technology).

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2014

H²-matrix methods for boundary integral equations

Steffen Börm

Christian-Albrechts-Universität Kiel,

05.12.2014, 14:00 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

Boundary integral equations are an important tool for analyzing elliptic partial differential equations arising, e.g., in structural mechanics or the simulation of acoustic or electromagnetic fields. Standard discretization techniques lead to large and densely populated matrices that require special algorithms.

The H²-matrix method offers efficient compression schemes for large matrices and can also perform algebraic operations like multiplication, inversion or factorization directly on the compressed matrices.

This talk gives an introduction to the basic concepts of H²-matrices and routlines two recent results: the Green hybrid compression scheme can be used to construct compressed approximations of discretized boundary element systems. Preconditioners for these systems can be constructed by applying a sequence of local low-rank updates to H²-matrices.

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Recursive Low-Rank Truncation

Wolfgang Hackbusch

Max-Planck-Institut für Mathematik in den Naturwissenschaften,

13.11.2014, 15:30 Uhr

Schwarzenbergstrasse 93, Raum A1.20

Zusammenfassung:

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the smaller
block matrices by low-rank matrices and agglomerating them into a new, coarser
block decomposition, one obtains a recursive method. The required computation work is O(rnm) where r is the desired rank and n x m is the size of the matrix. New estimates are presented for the errors A-B and M-A,
where A is the result of the recursive truncation applied to M, while B is the best approximation.

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Immer wieder Hurwitz Neues über unendliche, total nichtnegative Matrizen und eine alte Bemerkung B.Riemanns

Dr. Prashant Batra

Institut für Rechnertechnologie, Schwarzenbergstrasse 95E, Raum 3.074

01.07.2014, 15:45 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

In Zusammenhang mit der Nullstellenlage von Polynomen welche ausschließlich nichtnegative Koeffizienten aufweisen wurden von Holtz und Tyaglov (SIAM Review, 2012) speziell strukturierte, unendliche Matrizen betrachtet, deren Minoren sämtlich nicht-negativ sind genau dann, wenn das Polynom nur negative Nullstellen besitzt.

Wir werden zum einen diese aufwendige Charakterisierung der
Nullstellenlage von Polynomen deutlich vereinfachen, desweiteren den Satz von Holtz und Tyaglov auf eine Klasse ganzer Funktionen ausweiten sowie den Bezug zu bekannten Klassen total nichtnegativer Matrizen herstellen.

Als mathematische Anwendungen erhalten wir einen einfachen, unabhängigen Beweis der Charakterisierung von Holtz-Tyaglov, eine neue Verknüpungseigenschaft der betrachteten Matrizen sowie eine Charakterisierung der Nullstellenlage spezieller ganzer Funktionen.

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Domain Decomposition for elliptic PDE eigenvalue problems

Lars Grasedyck

RWTH Aachen,

30.06.2014, 15:00 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

We consider the solution of a rather simple class of eigenvalue problems $Ax=\lambda{Mx}$ for symmetric positive definite matrices $A$,$M$ that stem, e.g., from the discretisation of a PDE eigenvalue problem. Thus, the problem is in principle simple, but the matrices $A$ and $M$ are large-scale and we would like to compute all relevant eigenvalues, where relevant is to be understood in the sense that all eigenvalues should be computed that can be captured by the discretisation of the continuous PDE eigenvalue problem.

We propose a new method for the solution of such eigenvalue problems.
The new method combines ideas of domain decomposition, as in the automated multi-level substructuring (short AMLS) or component mode synthesis, with the concept of hierarchical matrices (short $\cal{H}$-matrices) in order to obtain a solver that scales almost linearly (linear up to logarithmic factors) in the size of the discrete space, i.e. the size $N$ of the linear system times the number of sought eigenvectors. Whereas the classical AMLS method is very effective for PDEs posed in two dimensions, it is getting very expensive in the three-dimensional case, due to the fact that the interface coupling in the domain decomposition requires dense matrix operations. We resolve this problem by use of data-sparse hierarchical matrices. In addition to the discretisation error our new approach involves a projection error due to AMLS and an arithmetic error due to $\cal{H}$-matrix approximation. We will shortly analyse the complexity in theory and practice, and consider several numerical examples that underline the performance of the solver.

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A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier-Stokes equations

Leo Rebholz

24.06.2014, 15:30 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter $\gamma$, converge to the associated coupled method solution with rate $\gamma^{-1}$ as $\gamma\rightarrow \infty$. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.

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Preconditioners for time-dependent PDE-constrained optimization

Martin Stoll

MPI Magdeburg,

24.04.2014, 16:00 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

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2013

On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime

Alexander Linke

WIAS Berlin,

17.10.2013, 14:15 Uhr

Schwarzenbergstrasse 95E, Raum 3.074

Zusammenfassung:

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix-Raviart element is proposed, where divergence-free, lowest-order Raviart-Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

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Numerical Treatment of Tensors

Wolfgang Hackbusch

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

04.07.2013, 14:00 Uhr

Schwarzenbergstraße 95H, Raum H0.03

Zusammenfassung:

The numerical treatment of tensors and the use of tensors for various numerical problem has rapidly increased in the last time. It is now applied to many fields in analysis (treatment of pdes, representation of multivariate functions, etc.). The key for an efficient numerical treatment is a suitable format. We discuss the various formats, their properties, and operations with tensors.

Literature: W. H.: Tensor spaces and numerical tensor calculus. Springer 2012

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H-Matrizen für Finite-Differenzen Matrizen

Dominik Enseleit

UHH, UHH

29.05.2013, 13:30 Uhr

Schwarzenbergstrasse 95E, Raum 1.050

Zusammenfassung:

Die Technik der Hierarchischen Matrizen H-Matrizen) ermöglicht die Berechnung einer approximativen H-Inversen oder H-LU-Zerlegung in fast linearer Komplexität und kann auf diese Weise zur effizienten Lösung linearer Gleichungssysteme eingesetzt werden. Vor der Verwendung der H-Matrix-Technik ist zu untersuchen, ob eine H-Matrix Approximation der Inversen bzw. der Faktoren der LU-Zerlegung existiert.
Resultate dieser Form konnten bereits für diverse Matrizen (z.B. Finite-Element-Matrizen) gezeigt werden, im Finite-Differenzen-Kontext sind jedoch keine Veröffentlichungen zum Einsatz der H-Matrix-Technik bekannt. Mit der Zielsetzung die Anwendbarkeit der H-Matrix-Technik für eine Finite-Differenzen-Matrix aus dem meteorologischen Transport- und Strömungsmodell METRAS zu untersuchen, wird in diesem Vortrag ein Resultat für Finite-Differenzen-Matrizen vorgestellt. Aufbauend auf dem methodischen Ansatz für Finite-Element-Matrizen wird die Existenz einer H-Matrix Approximation der Inversen von Finite-Differenzen-Matrizen gezeigt.
Die Ergebnisse können mittels numerischer Tests bestätigt werden. Bei Testproblemen, die in Anlehnung an das Gleichungssystem aus dem Modell METRAS aufgestellt werden, lässt sich im Einklang mit den theoretischen Ergebnissen jedoch eine Verschlechterung des Fehlerverlaufs in Abhängigkeit von einem Parameter feststellen. Für diese Fälle wird eine modifizierte Partitionierungsstrategie vorgestellt, deren Verwendung zu deutlich besseren Ergebnissen führt.

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Darstellung und Approximation von Tensoren im Hierarchischen Format

Stefan Kühn

MPI, Leipzig

18.04.2013, 14:00 Uhr

Schwarzenbergstrasse 95E, Raum 1.050

Zusammenfassung:

Die effiziente Darstellung und Approximation von Tensoren gewinnt in vielen Anwendungsbereichen der Mathematik wie Quantenchemie und -physik und auch generell innerhalb der Numerik immer mehr an Bedeutung. In diesem Vortrag werden wir ein neues Format zur Darstellung von hochdimensionalen
Tensoren vorstellen - das sogenannte Hierarchische Format oder auch Hierarchische Tucker-Format - und die grundlegende Arbeitsweise einer darauf basierenden inexakten Arithmetik erläutern. Der Schwerpunkt liegt auf der Approximation von Tensoren, sowie den Vorteilen des neuen Formates im Vergleich zu Standardformaten wie dem kanonischen Format oder der
Unterraum-/Tucker-Darstellung.

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Titchmarsh-Weyl theory for elliptic differential operators on unbounded domains

Jussi Behrndt

TU Graz, Österreich

22.01.2013, 15:00 Uhr

Schwarzenbergstrasse 95E, Raum 1.050

Zusammenfassung:

In this talk we describe the spectral properties of selfadjoint Schrödinger operators on unbounded domains with
an associated Dirichlet-to-Neumann map. In particular, a
characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from singular
Sturm–Liouville theory.

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2012

Robust successive computation of eigenpairs for nonlinear eigenvalue problems

Cedric Effenberger

École polytechnique fédérale de Lausanne EPFL, Lausanne

12.12.2012, 15:00 Uhr

Schwarzenbergstrasse 95E, Raum 1.050

Zusammenfassung:

We consider eigenvalue problems which are nonlinear in the eigenvalue
parameter. Newton-based methods are well-established techniques for determining individual eigenpairs of such nonlinear eigenvalue problems. If a larger number of eigenpairs is sought, however, the tendency of these methods to re-converge to previously discovered eigenpairs is a hindrance. In this talk, a deflation strategy for nonlinear eigenvalue problems will be presented, which overcomes this limitation in a natural way. Furthermore, we will comment on how this deflation approach can be implemented in a Jacobi-Davidson framework with only minimal overhead.

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Invariant pairs for nonlinear eigenvalue problems

Prof. Dr. Daniel Kressner

École polytechnique fédérale de Lausanne EPFL, Lausanne

28.11.2012, 15:00 Uhr

Schwarzenbergstrasse 95E, Raum 1.050

Zusammenfassung:

The concept of invariant subspaces is fundamental to linear eigenvalue problems and provides an important theoretical foundation in the design of numerical eigenvalue solvers. It turns out that there is no straightforward extension of this concept to eigenvalue problems that are nonlinear in the eigenvalue parameter. One obstacle is that eigenvectors belonging to different eigenvalues may become linearly dependent in the nonlinear case. Invariant pairs offer an elegant way to avoid this obstacle and appear to be the most natural extension of invariant subspaces. In this talk, we give an overview of the properties of invariant pairs and explain how they can be used in the design of numerical algorithms for nonlinear eigenvalue problems, as they arise for example in band diagram calculations for photonic crystals and fluid-structure interaction problems.

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Schrödinger-Operatoren mit kompakter Resolvente

Peter Stollmann

TU Chemnitz, TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz

24.10.2012, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 1.050

Zusammenfassung:

Ein klassischer Satz von Friedrichs besagt, dass Schrödingeroperatoren kompakte Resolvente besitzen, wenn das zugrundeliegende Potential bei Unendlich gegen Unendlich geht. In diesem Vortrag werden wir einen einfachen Beweis einer Verallgemeinerung präsentieren, basierend auf einer gemeinsamen Arbeit mit D. Lenz (Jena) und D. Wingert.

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The Lanczos algorithms and their relations to formal orthogonal polynomials, Padé approximation, continued fractions, and the qd algorithm

Martin Gutknecht

ETH Zurich; Seminar for Applied Mathematics, LEO D3 (Leonhardstrasse 27), 8092 Zurich, Switzerland

14.03.2012, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

In their seminal 1952 paper on the conjugate gradient (CG) method Hestenes and Stiefel pointed out that their method, which is applicable to linear systems of equations with symmetric positive definite matrix only, is closely related to certain orthogonal polynomials, the corresponding Gauss quadrature formulas, certain continued fractions, and their convergents (or `partial sums'). The latter can be seen to be Padé approximants of a function that involves the resolvent of the matrix.

Around the same time, in 1950 and 1952, Cornelius Lanczos published two related articles, of which the second one introduced a precursor of the biconjugate gradient (BCG or BiCG) method, which generalizes CG to the case of a nonsymmetric system. Here, the residual polynomials are formal orthogonal polynomials only, but the connections to continued fractions and Padé approximants persist. Moreover, there is a relation to the qd algorithm of Rutishauser (1954). The understanding of all these connections became probably the key to Rutishauser's discovery of the LR algorithm (1955, 1958), which was later enhanced by John G.F. Francis to the ubiquitous QR algorithm (1961/62).

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Solving large nonsymmetric linear systems with IDR(s) on a geographically separated cluster of parallel computers

Martin van Gijzen

Delft University of Technology; Faculty of Electrical Engineering, Mathematics and Computer Science, Mekelweg 4; 2628 CD Delft; The Netherlands

29.02.2012, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The IDR(s) method is a family of fast algorithms for iteratively solving large nonsymmetric linear systems. In the talk we will discuss an IDR(s) variant that is specifically tuned for parallel and grid computing. In particular in grid computing the inner product is a bottleneck operation. We will discuss three techniques that we have used to alleviate this bottleneck in IDR(s). Firstly, the efficient and stable IDR(s)-biortho method is reformulated in such a way that it has a single global synchronisation point per iteration step. Secondly, the so-called test matrix is chosen so that the work, communication, and storage involving this matrix is minimised in multi-cluster environments. Finally, a methodology is presented for a-priori estimation of the optimal value of s using only problem and machine--based parameters. We will also discuss a preconditioned version of IDR(s) that is particularly suited for grid computing. We will illustrate our results with numerical experiments on the DAS--3 Grid computer, which consists of five cluster computers located at geographically separated places in the Netherlands.

This is joint work with Tijmen Collignon.

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2011

The Lanczos Algorithm in Finite-Precision Arithmetic

Ivo Panayotov

Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, England

16.03.2011, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The Lanczos algorithm was introduced in 1950 as means of solving eigenvalue problems. Despite its apparent elegance, the algorithm was initially neglected by the scientific community because it was observed to depart from its theoretical properties due to the effects of finite-precision computer arithmetic. The algorithm regained popularity several decades later when it was shown that despite its departure from theory, it nevertheless produces highly accurate eigenvalue estimates.

In my talk, I will briefly introduce the Lanczos algorithm and will present bounds characterizing the quality of eigenvalue estimates generated by the algorithm in exact arithmetic. Then, I will describe the difficulties of producing similar bounds in finite-precision arithmetic, and will present rounding error results, including recent ones, which overcome these difficulties.

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2010

Singular optimal control, Lur'e equations and even matrix pencils

Prof. Dr. Timo Reis

Institut für Numerische Simulation, Technische Universität Hamburg-Harburg

08.12.2010, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

Lur'e equations are a generalization of algebraic Riccati equations and they arise in linear-quadratic optimal control with cost functional being singular in the input.
For Riccati equations, it is well-known that there is a one-to-one correspondence between set of solutions and certain Lagrangian eigenspaces of a Hamiltonian matrix.
The aim of this talk is to generalize this concept to Lur'e equations. We are led to the consideration of deflating subspaces of even matrix pencils.

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Demands of modal reanalysis techniques in Engineering Design

Jiacong Yin

Peking University, China

24.11.2010, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

1. A brief introduction about our group in Peking University
2. Seismic design of buildings with accidental eccentricity
3. Structural design of wind turbine blades

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Inverse Iteration, Newton-Abschätzungen und Anwendung auf Rayleigh-Quotienten-Iterationen bei nichtlinearen Eigenwertproblemen

Prof. Hubert Schwetlick

TU Dresden, Institut für Numerische Mathematik

14.04.2010, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

Bekanntlich liefert ein Schriitt $(u,\theta) \mapsto u_+^{InvIt}$ der Inversen Iteration für das nichtlineare Eigenwertproblem $T(\lambda)x=0$ dieselbe Richtung wie ein Schritt $(u,\theta) \mapsto (u_+^{Newt},\theta_+^{Newt})$ des Newtonverfahrens für das erweiterte System $T(\lambda)x=0,\;w^Hx=1$ mit einem geeigneten Skalierungsvektor $w$, d.h., es gilt $\mbox{span}\,\{u_+^{InvIt}\}=\mbox{span}\,\{u_+^{Newt}\}$. Es liegt daher nahe, zur Abschätzung der Verbesserung der Eigenvektorapproximation $u$ durch die Inverse Iteration Newton-Techniken zu verwenden. Es wird gezeigt, dass dies zu genauen Abschätzungen führt, wenn explizit mit dem Restglied zweiter Ordnung gearbeitet und dessen spezielle Produktstruktur berücksichtigt wird wie das von \textsc{Heinz Unger} [50] erstmalig (und ohne publizierten Beweis) für das lineare Problem $T(\lambda)=A-\lambda I$ getan worden ist.

Durch Kombination mit neuen Abschätzunegn für das nichtlineare klassische bzw. verallgemeinerte Rayleigh-Funktional läßt sich dann einfach die quadratische Konvergenz
der nichtlinearen Rayleigh-Funktional-Iteration wie auch die kubische Konvergenz der nichtlinearen Verallgemeinerung der zweiseitigen Ostrowskischen Rayleigh-Quotienten-Iteration herleiten.

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On the motion of several rigid bodies in an incompressible non-Newtonian fluid

Prof. Sarka Necasova

Institute of Mathematics of the Academy of Sciences, Praha, Czech Republic

03.02.2010, 13:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The motion of one or several rigid bodies in a viscous fluid occupying a bounded domain ­$\Omega in R^3$ represents an interesting theoretical problem featuring, among others, possible contacts of two or more solid objects. We consider the motion of several rigid bodies in a non-Newtonian fluid of a power-law type. Our main result establishes the existence of global-in-time solutions of the associated evolutionary system, when collisions of two or more rigid objects do not appear in a finite time unless they were present initially.

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A self-similar solution for the porous medium equation in a two-component domain

Prof. Jan Filo

Comenius University, Bratislava, Slovak Republic

27.01.2010, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition
\[
w^\prime =h^\prime \;,\qquad h=\psi(w)\qquad\text{at the point $\,x=0\,$}.
\]
We show that, for a prescribed power-law nonlinearity $\psi$ and using the solution $(w,h)$, a self-similar solution to the porous medium equation in the two-component domain can be constructed.

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2009

Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf Bifurcations in large scale problems

Prof. Dr. Karl Meerbergen

Katholieke Universiteit, Leuven

16.12.2009, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer et. al. (SINUM, 34, (1997) pp. 1-21) proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilises a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearisation process). The problem is formulated as an inexact inverse iteration method that requires the solution of a sequence of Lyapunov equations with low rank right hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on numerical examples.

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The generalized Riemann problem (GRP) method for compressible fluid flows

Prof. Jiequan Li

School of Mathematics, Capital Normal University, Beijing, China

02.09.2009, 16:15 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

In this talk I will briefly review the generalized Riemann problem (GRP) method for compressible fluid flows. There were originally two versions of this method:
Lagrangian and Eulerian. The latter is always derived via a passage from the former. In our recent efforts, we developed a direct Eulerian GRP method using the ingredient of Riemann invariants. The main advantage is (1) to avoid the passage from the Lagrangian to Eulerian and thus easily to be extended into multidimensional cases; (2) treat sonic cases easily; and (3) conveniently combine with other techniques such as adaptive meshes.
We will also report some stability, convergence properties, and applications to shallow water equations on the sphere (earth).

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ON THE CONTROL OF NUMERICAL EFFECTS OF DISPERSION AND DISSIPATION PREVAILING IN FINITE DIFFERENCE SCHEMES

Dr. Bippine Appadu

University of Mauritius, Reduit, Mauritius

02.09.2009, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

In CFD, Atmospheric Sciences and Computational Aeroacoustics, many problems involve regions of discontinuity. When used to solve problems involving regions of shocks, dispersive schemes give rise to oscillations while dissipative schemes cause smearing, close to these regions of sharp gradients.

Based on the results of the 1-D shallow water problem, when solved using MCLF2, we observe that different cfl numbers yield results with different amount of dispersion and dissipation. This led us to devise a technique in order to locate the cfl number at which we can obtain results with efficient shock-capturing properties. This new technique involves the control of numerical effects of dispersion and dissipation in numerical schemes. We baptise this technique as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES. The cfl number at which dissipation curbs dispersion optimally is then located. It is termed as the optimal cfl.

We extend the concept of CDDES to that of Minimised Integrated Square Difference Error,(MISDE). The latter is an improved technique over the CDDES technique since it can be used to obtain two optimal parameters which are generally the cfl number and another variable, for efficient-shock capturing. Another technique of optimisation is devised which enables better control over the grade and balance of oscillation and dissipation to optimise parameters which regulate dispersion and dissipation effects. This technique is baptised as Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation, (MIEELDLD) and has advantages over the previous technique, MISDE.

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Discrete Empirical Interpolation for Nonlinear Model Reduction

Prof. D. C. Sorensen

Rice University, Houston, Texas

10.07.2009, 10:00 Uhr

Schwarzenbergstrasse 95, Gebäude D, Raum D1025

Zusammenfassung:

A dimension reduction method called Discrete Empirical Interpolation (DEIM) will be presented and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reduced-order models for unsteady and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem.

I will describe DEIM as a modification of POD that reduces the complexity as well as the dimension of general nonlinear systems of ordinary differential equations (ODEs). It is, in particular, applicable to ODEs arising from finite difference discretization of unsteady time dependent PDE and/or parametrically dependent steady state problems. Our contribution is a greatly simplified description of Empirical Interpolation in a finite dimensional setting. The method possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the 1-D FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures non-linear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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On numerical simulation of flow in time-dependent domains

Prof. Miloslav Feistauer

Karls-Universität Prag, Department of Numerical Mathematics

13.05.2009, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The lecture will be concerned with the simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. The system of the governing equations is discretized in space by the discontinous Galerkin method. The time discretization is based on a semi-implicit linearized time stepping scheme, which leads to the solution of a linear algebraic system on each time level. As a result we get an efficient and robust numerical process. The applicability of the developed method will be demonstrated by some computational results obtained for flow in a channel with a moving wall and past an oscillating airfoil.

These results were obtained in cooperation with Vaclav Kucera and Jaroslava Prokopova from Charles University in Prague, Faculty of Mathematics and Physics, and Jaromir Horacek from Institute of Thermomechanics of Academy of Sciences of the Czech Republic.

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IDR in variations

Prof. Martin Gutknecht

Seminar for Applied Mathematics, ETH Zurich

28.01.2009, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was first developed by Sonneveld around 1979 and documented on three and a half pages of a 1980 proceedings paper by Wesseling and Sonneveld. Soon after IDR, Sonneveld introduced his widely applied Conjugate Gradient Squared (CGS) algorithm. Then, in 1990, van der Vorst suggested Bi-CGSTAB that he claimed to improve both those methods.

Bi-CGSTAB has become a method of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k.

In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to BiCGSTAB if s = 1 and to ML(s)BiCGSTAB if s > 1. In 2008, a new, particularly ingenious and elegant variant of IDR(s) has been proposed by the same authors.

In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections.

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2008

Non-Oscillatory Central Schemes -- a Powerful Black-Box-Solver for Hyperbolic PDE's

Prof. Alexander Kurganow

Tulane University, New Orleans, USA

17.12.2008, 14:30 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

I will first give a brief description of finite-volume, Godunov-type methods for hyperbolic systems of conservation laws. These methods consist of two types of schemes: upwind and central. My lecture will focus on the second type -- non-oscillatory central schemes.

Godunov-type schemes are projection-evolution methods. In these methods, the solution, at each time step, is interpolated by a (discontinuous) piecewise polynomial interpolant, which is then evolved to the next time level using the integral form of conservation laws. Therefore, in order to design an upwind scheme, (generalized) Riemann problems have to be (approximately) solved at each cell interface. This however may be hard or even impossible.

The main idea in the derivation of central schemes is to avoid solving Riemann problems by averaging over the wave fans generated at cell interfaces. This strategy leads to a family of universal numerical methods that can be applied as a black-box-solver to a wide variety of hyperbolic PDEs and related problems. At the same time, central schemes suffer from (relatively) high numerical viscosity, which can be reduced by incorporating of some upwinding information into the scheme derivation -- this leads to central-upwind schemes, which will be presented in the lecture.

During the talk, I will show a number of recent applications of the central schemes.

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Numerical entropy production as a regularity/error indicator

Prof. Gabriella Puppo

Dipartimento di Matematica, Politecnico di Torino,Italy

03.12.2008, 16:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

Uniqueness for weak solutions of conservation laws is based on the sign of the entropy production across discontinuos solutions. Although the entropy plays a fundamental role in the theory of hyperbolic systems, it is generally not used as a computational tool.
In this talk I describe how the numerical production of entropy induced by the discretization of the equations is a reliable indicator of the quality of the numerical solution. Thus the entropy production can be used as a regularity indicator, identifying the cells in which non linear limiters must be used to prevent the onset of spurious oscillations.
More quantitatively, when the solution is smooth, the entropy production has the same size of the local truncation error and can therefore be used as an a-posteriori error indicator to drive the construction of adaptive grids.

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Numerical Solution of Conservation Laws over Non-Uniform, Adaptively Redefined Meshes

Dr. Sfakianakis Nikos

University of Heraklion, Greece

20.11.2008, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

We start with a brief introduction to Conservation Laws and to their numerical solutions. Then we discuss the construction and manipulation of non-uniform meshes, using geometric properties of the numerical solution under consideration. Next, we examine properties (such as consistency, stability and order of accuracy) of numerical schemes over both uniform and non-uniform meshes. Finally, we combine a proper mesh selection mechanism with Entropy Conservative or oscillatory numerical schemes for the evolution step.

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On the multiscale rodlike model in polymeric fluids

Hui Zhang

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China

02.04.2008, 15:00 Uhr

Schwarzenbergstrasse 95, Raum 3.053

Zusammenfassung:

We will show the new rigid rod-like model in a polymeric fluid. The constitutive relations considered are motivated by the kinetic theory. The micro equation has five spatial freedom variables, two of them are in the configuration domain and the others are in the macro flow domain. It is obtained the local existence of the solution with large initial data and global existence of the solution with small Deborah and Reynolds constants in periodic domains. For the case of no flow we will give the structure of stationary solutions to the micro equation with Maier-Saupe potential on the sphere. The stationary solutions are shown to be necessarily a set of axially symmetric functions, and a complete classification of parameters for phase transitions to these stationary solutions is obtained. It is shown that the number of stationary solutions hinges on whether the potential intensity crosses two critical values 6.731393 and 7.5. Furthermore, we present explicit formulas for all stationary solutions. It is first theoretically proven that there is a hysteresis phenomenon when the non-dimensional potential intensity among particles changes. In the weak shear flow, we show that there exist many stable dynamic states: flow-aligning, tumbling, log-rolling and kayaking, which depend on the initial concentrated orientation of liquid crystal particles. Theoretical analysis is reported the first time that the Kayaking state does not circulate around a fixed direction but the asymmetric axis will periodically change.

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