# Forschungsgruppe ''Numerische Mathematik und Wissenschaftliches Rechnen''

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## Seminar Numerische Mathematik / Numerical mathematics seminars aktuelles Programm / current program

Donnerstag, 19. 12. 2013, 14:00 Uhr (ESH)

F. Litzinger   (Freie Universität Berlin)
Diskretisierung der stationären inkompressiblen Navier-Stokes-Gleichungen in 3D auf unstrukturierten Tetraedergittern

Das nichtkonforme Crouzeix-Raviart-Element zur numerischen Lösung der stationären inkompressiblen Navier-Stokes-Gleichungen besitzt eine nichtphysikalische Abhängigkeit des Geschwindigkeitsfehlers vom Druckterm, die typisch für nicht divergenzfreie gemischte Finite-Elemente-Methoden ist. Mithilfe eines Rekonstruktionsoperators, der diskret divergenzfreie auf echt divergenzfreie Vektorfelder abbildet, wird dieses Problem beseitigt [1, 2]. Im Stokes-Fall muss für diese Vorgehensweise nur die rechte Seite des Gleichungssystems geändert werden. Die Implementierung beider Verfahren für das Problem in 3D wird vorgestellt und die Ergebnisse anhand einiger Testbeispiele präsentiert, die den Vorteil des modifizierten Verfahrens deutlich machen.
REFERENCES
: Linke, A.: On the Role of the Helmholtz-Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime. In: Computer Methods in Applied Mechanics and Engineering (accepted, 2013).
: Brennecke, Ch.: Eine divergenzfreie Rekonstruktion für eine nicht-konforme Diskretisierung der inkompressiblen Stokes-Gleichungen.

und 15:00 Uhr (ESH)

M. Hoffmann   (Freie Universität Berlin)
Das Navier-Stokes-Darcy Problem

Das Navier-Stokes-Darcy Problem ist das gekoppelte Problem freier Strömung mit Strömung durch poröse Medien. Ein Anwendungsbeispiel ist etwa das Modell eines Flusses mit seinem Flussbett. Der Fokus meiner Arbeit liegt darauf, wie genau man die Navier-Stokes Gleichungen für die freie Strömung und die Darcy Gleichungen für die Strömung durch poröse Medien koppeln kann. Einige der resultierenden Probleme werden besprochen und es werden Vorschläge zur Lösung dieser gemacht. Die generelle Idee zur numerischen Lösung des gekoppelten Problems ist die Betrachtung von getrennten Problemen, welche durch einen gemeinsamen Teil ihres Randes, genannt Interface, verbunden werden. Diese getrennten Probleme können nun iterativ gelöst werden, wobei nach jedem Lösungsschritt die Randdaten am Interface des jeweils anderen Problems erneuert werden. Die in einem der Iterationsschritte vorliegenden Navier-Stokes Gleichungen machen es notwendig, sich mit ihrer Nichtlinearität zu beschäftigen, welche zusätzliche Fixpunktiterationen erforderlich macht. In diesem Kontext kommt die Frage auf, wie man die Anzahl der Fixpunktiterationen zur Lösung der Navier-Stokes Gleichungen pro Iteration einschränken sollte, um beste Ergebnisse zu erzielen. Diese und andere Fragen werden anhand von zwei numerischen Beispielen diskutiert. Ein Beispiel vergleicht die numerischen Ergebnisse des Navier-Stokes-Darcy Problems mit einer analytischen Lösung, das andere ist das eingangs erwähnte Modell eines Flusses mit seinem Flussbett und vergleicht die erhaltenen Ergebnisse mit Ergebnissen aus der Literatur.

Donnerstag, 7. 11. 2013, 14:00 Uhr (ESH)

F. Dassi   (Politecnico di Milano)
Surface mesh simplification strategy for saptial regression analysis of cortical surface data

We present a new mesh simplication technique specically developed for a statistical data analysis over the cortical surface. The aim of this approach is to produce a simplied mesh which does not distort the original data distribution so that the statistical estimates computed over the new mesh exhibit good inferential properties. To achieve this goal, we propose an iterative technique such that, for each iteration, we contract the edge of the mesh with the lowest value of a cost function that takes into account both the geometry of the surface and the distribution of the data on it. After the data are associated with the simplied mesh, they are analyzed via the spatial regression model for non-planar domains developed in . The approach proposed here resorts to a suitable penalized regression method and uses a conformal attening map to generalize existing spatial smoothing techniques for planar domains to non-planar surfaces by suitably including the conformal parameterization. The eectiveness of this technique is numerically investigated via a simulation study and an application to real cortical surface thickness data.
REFERENCES
 Chung, M.K., Robbins, S.M., Dalton, K.M., Davidson, R.J., Alexander, A.L. and Evans, A.C. (2005), Cortical thickness analysis in autism with heat kernel smoothing. NeuroImage 25:1256-1265
 Dassi, F., Ettinger, B., Perotto S. and Sangalli L.M. (2013), A mesh simplication strategy for a spatial regression analysis over the cortical surface of the brain. In preparation.
 Ettinger, B., Perotto S. and Sangalli L.M. (2012), Spatial regres-sion models for data over two-dimensional manifolds. Technical report. N. 54/2012, MOX, Dipartimento di Matematica ``F. Brioschi,'' Politecnico di Milano, http://mox.polimi.it/it/progetti/pubblicazioni.
 Garland, M. and Heckbert, P.S. (1997), Surface simplication using quadric error metrics, in Proceedings of the 24th annual conference on Computer Graphics and Interactive Techniques, SIGGRAPH '97, pp. 209-216.

Donnerstag, 24. 10. 2013, 14:00 Uhr (ESH)

Prof. V. A. Galkin   (Surgut State University, Russia)
Generalized solutions of the smolukhovskii semilinear system of equations and their approximations Dienstag, 24. 09. 2013, 13:30 Uhr (ESH)

F. Dassi   (Politecnico di Milano)
A curvature-adapted anisotropic surface remeshing method

We present a new method for remeshing surfaces that exploits the intrinsic anisotropic natural of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a high dimensional space (here we use 6 dimensions). This allow us to form a simple isotropic mesh optimization problem in this embedded space. Then the mesh can be optimized by any standard mesh optimization techniques, such as edge flips, edge contraction, vertex smoothing, and vertex insertion. This method results a curvature-adapted mesh. We present examples of remeshed surfaces from implicit functions and CAD models.

Donnerstag, 05. 09. 2013, 14:00 Uhr (ESH)

Some recent results on the convergence order of finite volume methods for evolution equations on general nonconforming multidimensional spatial meshes

Recently a finite volume method is introduced and developed in Eymard et al. (2010), originally applied to stationary elliptic problems. Such method can be applied and analyzed on very general nonconforming meshes with star-shaped polyhedral elements. Another feature is that the meshes are not necessarily satisfying the classical orthogonality property. The aim of this talk is to present some recent results concerning the convergence order of some implicit finite volume schemes, using the spatial meshes introduced in Eymard et al. (2010), for parabolic and hyperbolic equations. We will also investigate the case of linear Schrödinger evolution equation. In the nonlinear case of Schrödinger evolution equation, we will set an implicit linear finite volume scheme and we sketch, as perspectives, some results on the convergence order to be proved.
References:
 Bradji, A.: An analysis of a second order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes. Applied Mathematics and Computation, 219/11, 6354--6371, 2013
 Bradji, A.: Convergence analysis of some high--order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes. Numerical Methods for Partial Differential Equations, 29/4, 1278--1321, 2013
 Bradji, A.: A theoretical analysis of a new finite volume scheme for second order hyperbolic equations on general nonconforming multidimensional spatial meshes. Numerical Methods for Partial Differential Equations , 29/1, 1--39, 2013
 Bradji, A. and Fuhrmann, J.: Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nconforming multidimensional spatial meshes. Applications of Mathematics, Praha, 58/1, 1--38, 2013
 Bradji, A. and Fuhrmann, J.: Error estimates of the discretization of linear parabolic equations on general nonconforming spatial grids. Comptes rendus - Mathematique 348/19-20 , 1119--1122, 2010
 Eymard, R.; Gallouet, T.; Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30, No. 4, 1009-1043 (2010).
 Eymard, R.; Gallouet, T.; Herbin, R.: Finite volume methods. Handbook of Numerical Analysis. P. G. Ciarlet and J. L. Lions (eds.), North-Holland, Amsterdam VII (2000), 723--1020.
 Koprucki, T, Eymard, R. and Fuhrmann, J.: Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator. WIAS Preprint NO. 1260 (2007).

Dienstag, 20. 08. 2013, 13:30 Uhr (ESH)

F. Dassi   (Politecnico di Milano)
Anisotropic error estimates for pde defined on surfaces

Donnerstag, 11. 07. 2013, 14:00 Uhr (ESH)

Prof. J. Shewchuck   (University of California at Berkeley)
Fast segment insertion and incremental construction of constrained Delaunay triangulations

The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has an O(k n^2) worst-case running time, where n is the number of input vertices and k is the number of input segments. We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses, and demonstrate with a performance comparison that it is faster than gift-wrapping for segments that cross many edges. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(n log n + n log^2 k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(n log n) time, incremental CDT construction is easier to program and competitive in practice. Moreover, the ability to incrementally update a CDT by inserting a segment is useful in itself.

und 15:00 Uhr (ESH)

J. Neumann   (WIAS Berlin)
Multi-level Monte-Carlo methods for SPDE

A simple model for ground water flow applies the Darcy equation. In practice the data inputs for this model are hardly known and need to be estimated. Stochastic approaches can characterise the resulting uncertainties. Several methods exist to solve these stochastic partial differential equations (SPDE). The multi-level Monte-Carlo method can treat the underlying model as a black box and is thus independent of the problem at hand. Furthermore it does not suffer from the curse of dimensionality as other approaches do. This makes it especially promising for models which include random fields as uncertainties. The generation of these random fields is another important aspect of such SPDE.

Mittwoch, 10. 07. 2013, 9:00 Uhr (Raum 406)

C. Suciu   (WIAS Berlin)
Direct discretizations of uni- and bi-variate population balance systems on tensor-product meshes

und 11:00 Uhr (HVP 11A, Raum 3.13)

V. Travnikov   (TU Dresden)
Forschungsarbeiten an der Technischen Universität Dresden

und 13:00 Uhr (Raum 406)

Ch. Merdon   (HU Berlin)
Aspects of guaranteed error control for finite element approximations in computations for PDEs

Donnerstag, 04. 07. 2013, 14:30 Uhr (ESH)

J. Dannberg   (GFZ Potsdam)
Modelling mantle plumes with ASPECT: an advanced solver for problems in earth's convection

On long timescales, the rocks in the Earth's mantle deform like a highly viscous fluid. Therefore, their behavior can be described by the theory of fluid dynamics, more precisely, the Navier-Stokes equations. The ASPECT code is designed mainly by computational scientists of the Texas A&M University to solve this system of partial differential equations for pressure, velocity, temperature and composition in a 2D or 3D domain, neglecting inertia. Building on several libraries, it uses modern numerical methods, such as adaptive mesh refinement, and is both highly parallelized and easily extensible. These features make the programme ideal for modelling convection in the Earth' mantle: movement driven by buoyancy forces due to density differences caused by temperature and chemical composition. We use ASPECT to investigate the ascent of hot, positively buoyant material, so-called mantle plumes, from the core-mantle boundary to the Earth's surface.

Dienstag, 02. 07. 2013, 13:00 Uhr (ESH)

M. Arias-Chao   (Alstom Power Ltd., Switzerland)
Calibration and uncertainty quantification for gas turbines models

Calibration and uncertainty quantification for GT Models is a key activity for GT manufactures. The deviation of the numerical model to the real GT is obtained with a calibration technique. Since both, numerical models and the measurement data are uncertain the calibration process is intrinsically stochastic. Traditional approaches for calibration of a numerical GT model are deterministic. Therefore, quantification of the remaining uncertainty of the calibrated GT model is not clearly derived. However, there is the business need to know the probability of the GT performance predictions at tested or untested conditions. Furthermore, a GT performance prediction might be required for a GT new model when no experimental data are available. In this case, quantification of the uncertainty of the baseline GT, in which the new development is based on, and propagation of the design uncertainty for the new GT is required for decision making reasons. In this context the GT Thermal Integration Department in ALSTOM Schweiz (Baden) has a technology project in collaboration with WIAS to develop an automated method for calibration and uncertainty quantification for GT Models. The project is named Heat Balance Optimization Toolbox. This presentation attempts to provide an overview of the Heat Balance Optimization Toolbox project. How far have we gone, what technical challenges remain, and what are the future activities are questions that will be address during the meeting. The following mathematical topics will be covered: weighted least squares, Bayesian inference, Markov Chain Monte Carlo, clustering, Gaussian Process regression.

Donnerstag, 27. 06. 2013, 14:00 Uhr (ESH)

St. Moore   (RICAM, Linz, Austria)
Discontinuous Galerkin isogeometric analysis for elliptic PDEs on manifolds

Isogeometric analysis uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This patch representation corresponds to a domain decomposition. In this talk, we will present a Discontinuous Galerkin (DG) Method that allows for discontinuities only along the subdomain (patch) boundaries. The required smoothness is obtained by the DG terms associated with the boundary of the subdomains. The construction and corresponding discretization error analysis of such DG scheme will be presented for Elliptic PDEs living in plane domains as well as on open and closed surfaces.
This is a joint talk with Ulrich Langer (RICAM, Linz, Austria).

Donnerstag, 30.05. 2013, 14:00 Uhr (ESH)

L. Schumacher   (Humboldt Universität zu Berlin)
Isogeometric analysis for scalar convection-diffusion equations

The talk presents the main ideas of IGA, important results from the error analysis, and numerical examples.

Freitag, 24. 05. 2013, 10:30 Uhr (ESH)

Prof. H.-J. Starkloff   (Westsächsische Hochschule Zwickau)
On some questions related to generalized polynomial chaos expansions

Generalized polynomial chaos expansions are series expansions of random variables or random functions using orthogonal polynomials in certain basic random variables. They are a versatile tool for efficiently working with stochastic quantities. So, recently they are widely used for the investigation of stochastic models, for uncertainty quantification methods or stochastic simulation purposes. In particular, recent approaches for the solution of ordinary or partial differential equations with random parameters often use certain versions of generalized polynomial chaos expansions. In the talk basic definitions and properties of generalized polynomial chaos expansions are presented. Furthermore some problems are considered which are related to convergence properties or algebraic transformations of generalized polynomial chaos expansions.
The talk is based on joint work with A. Mugler (BTU Cottbus) and O. Ernst and B. Sprungk (TU Chemnitz).

Dienstag, 23. 04. 2013, 11:00 Uhr (HVP 11a, 4.13)

M. Rother   (Otto-von-Guericke-Universität Magdeburg)
Eine Reise ins Ich und zurück in 20 Minuten

Der vorgesehene Vortrag soll den Bewerber vorstellen und das Aufgabengebiet seiner Tätigkeit während der ersten Arbeitsstelle nach dem Studium umreißen. Im ersten Teil der Präsentation stellt sich der Bewerber kurz vor und beschreibt persönliche Interessen für fachliche Vertiefungsgebiete. Der fachliche und zweite Vortragsteil wird sich mit der mathematischen Handhabung von nicht-lokalen Evolutionen von Populationsbilanzen beschäftigen. Solche Gleichungen findet man zum Beispiel in der Biolaborforschung. Dort kann man verteilte Größen messen, die wiederum modellhaft die Virusausinfektion in Zellkulturen beschreiben. Das Hauptproblem liegt in der Entwicklung einer stabilen und effizienten numerischen Methode zur Approximation der Gleichungen in mehr als 3 Dimensionen (angestrebt sind 6 bis 10).

und 13:30 Uhr (ESH)

Prof. J. Wensch   (TU Dresden)
Efficient time integration methods for the compressible Euler equations in atmospheric dynamics

The simulation of atmospheric dynamics is an important issue in Numerical Weather Prediction. It relies on the numerical solution of the Euler equations. These equations exhibit phenomena on different temporal scales. In the lower troposphere sound waves propagate approximately ten times faster than the advective waves. We use a spatial discretization based on finite volumes. An approach to overcome the CFL restriction caused by sound waves are split-explicit methods. By multirate techniques the terms relevant for sound waves are integrated by small time steps with a cheap time integration procedure, whereas the slow processes are solved by an underlying Runge-Kutta method using a larger macro step size. The analysis of these methods is based on the interpretation as an exponential or Lie group integrator. We construct methods based on TVD-RK schemes and discuss order and stability properties. Numerical simulation results for established benchmark problems are given.

Donnerstag, 18. 04. 2013, 14:00 Uhr (ESH)

M. Schmuck   (Imperial College London)
Upscaling of ionic transport equations in strongly heterogeneous media and finite element approximations

We consider the well-accepted Nernst-Planck-Poisson equations  for the description of ionic transport and electrokinetic phenomena such as electro-phoresis and -osmosis. Applications range from designing microfluidic devices, energy storage devices, and semiconductors to emulating communication in biological cells by synthetic nanopores. Based on this classical description, we derive a new effective macroscopic set of equations [1,2,3] which describe binary symmetric electrolytes in porous and strongly heterogeneous media. Heterogeneous materials naturally induce corrected transport parameters which we call ''material tensors''. Our systematic and well-accepted upscaling strategy by homogenization gives a reliable understanding on the influence of the micro-geometry on the macroscale. The new equations provide an essential computational advantage by a strong reduction of the degrees of freedom in comparison to the classical description which requires a full resolution of the pore geometry and hence a high-dimensional resolution of the problem. For the classical descritpion, we briefly present a linear finite element scheme [4,5] that is reliable in the sense that discrete solutions conserve mass, are non-negative and bounded, and converge towards weak solutions. Finally, we can qualitatively and quantitatively characterize the suitability of the new upscaled equations for specific applications by error estimates  which compare the exact microscopic solution with the solution of the new effective transport equations.
References:
 M. Schmuck, A new upscaled Poisson-Nernst-Planck system for strongly oscillating potentials, J. Math. Phys. 54:021504 (2013).
 M. Schmuck, First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations, Z. angew. Math. Mech. 92:304-319 (2012).
 M. Schmuck and P. Berg, Homogenization of a catalyst layer model for periodically distributed pore geometries in PEM fuel cells, Appl. Math. Res. Express. 2013(1):57-78 (2012).
 A. Prohl and M. Schmuck, Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system, ESAIM, Math. Model. Numer. Anal. 44(3):531-571 (2010).
 A. Prohl, and M. Schmuck, Convergent discretizations for the Nernst-Planck-Poisson system, Num. Math. 111 (4):591-630 (2009).
 M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Mod. Meth. Appl. S. 19(6):993- 1015 (2009).

und 15:30 Uhr (Raum: 406)

Dr. L. Kamenski (The University of Kansas, USA)
Anisotrope Gitter: Den Zusammenhang zwischen Gittern und Eigenschaften der Diskretisierungsverfahren richtig verstehen

Dienstag, 02.04.2013, 9:00 Uhr (ESH)

G. Müller   (TU Berlin)
Beispiele numerischer Verfahren für PDEs und Optimalsteuerungsprobleme

und 12:30 Uhr (ESH)

Ch. Merdon   (HU Berlin)
Garantierte Fehlerkontrolle bei PDGLen mit aktuellen Equilibrierungstechniken

und 14:00 Uhr (ESH)

R. Richter   (Bundesanstalt für Materialforschung und -prüfung (BAM) )
Numerische Verfahren für die aktive Thermografie zur Untersuchung von Rückwandgeometrien und Singularitäten einer geometrischen Evolutionsgleichung (MCF)

Donnerstag, 14. 02. 2013, 14:00 Uhr (ESH)

Dr. L. Kamenski   (The University of Kansas)
Adaptive finite elements with anisotropic meshes

In this talk I will try to give an overview over anisotropic mesh adaptation for the finite element method, advantages and disadvantages of anisotropic meshes and how to construct them. The commonly used methods to obtain much needed directional information are either using error estimates or approximating derivatives of the true solution (post-processing). Both methods perform well in practice although they both still have open theoretical questions. I will try to give an explanation on how error estimates can be used for the mesh adaptation and why an inaccurate Hessian recovery is still good enough for the mesh adaptation.
I will also give some recent results on estimating the conditioning of finite element equations with anisotropic meshes. Interestingly, the conditioning with non-isotropic meshes is not as bad as generally assumed. In particular, we will see that, if the number of anisotropic elements is relatively small, the condition number of the preconditioned system is basically the same as with uniform meshes.

Donnerstag, 07. 02. 2013, 14:00 Uhr (Raum: 406)

Prof. G. Barrenechea   (University of Strathclude, UK)
Eigenvalue enclosures for the Maxwell operator

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the remaining part of the talk we formulate a general framework which is free from spectral pollution, predicts correct multiplicities and allows estimation of eigenfunctions. We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

und 15:00 Uhr (Raum: 406)

Dr. P. Knobloch   (Charles University, Institute of Numerical Mathematics, Czech Republic)
Mesh optimization for convection-diffusion problems

Numerical solution of convection-dominated problems requires the use of layer-adapted anisotropic meshes. Since a priori construction of such meshes is difficult for complex problems, it is proposed to generate them in an adaptive way by moving the node positions in the mesh such that an a posteriori error estimator of the overall error of the approximate solution is reduced. This approach is formulated for a SUPG finite element discretisation of a stationary convection-diffusion problem defined in a two-dimensional polygonal domain. The optimisation procedure is based on the discrete adjoint technique and a SQP method using the BFGS update. The optimisation of node positions is applied to a coarse grid only and the resulting anisotropic mesh is then refined by standard adaptive red-greed refinement. Four error estimators based on the solution of local Dirichlet problems are tested and it is demonstrated that an L2 norm based error estimator is the most robust one. The efficiency of the proposed approach is demonstrated on several model problems whose solutions contain typical boundary and interior layers.

Donnerstag, 31. 01. 2013, 14:00 Uhr (ESH)

M. Bessemoulin-Chatard   (Universite Blaise Pascal, Clermont-Ferrand II)
A finite volume scheme for a Patlak-Keller-Segel model with cross-diffusion

In this talk, I will analyse a finite volume scheme for a 2D Keller-Segel model with cross-diffusion, studied by S. Hittmeir et A. Jüngel in an article written in 2011. We consider the parabolic-elliptic model with an additional cross-diffusion term in the elliptic equation. This diffusion term avoids the blow-up and leads to the global-in-time existence of weak solutions. We consider an implicit in time and finite volume in space scheme. After proving the existence of a solution to the implicit scheme, we obtain an entropy inequality by using discrete versions of Sobolev inequalities. Then we can deduce some a priori estimates which allow to obtain the convergence of the discrete solution to the continuous one when the approximation parameters tend to zero. If the cross-diffusion parameter is sufficiently large, the approximate solution converges to the homogeneous steady-state. Thanks to a discrete logarithmic Sobolev inequality, we obtain an estimate of the rate of convergence.

Donnerstag, 24. 01. 2013, 14:00 Uhr (ESH)