|Verantwortliche Dozenten:||Alle Professoren der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen|
|Termine:||Di 16-18 in MA 313 oder n.V.|
|Inhalt:||Vorträge von Gästen und Mitarbeitern- zu aktuellen Forschungsthemen|
|Di 4.10.2005||16:15||MA 313|| Prof. Dr. Ivan L. Sofronov
(Keldysh Institut, Moskau, Russia)
|On numerical generation of low-reflecting boundary conditions for anisotropic media (Abstract)|
|Di 11.10.2005||16:15||MA 313|| Prof. Dr.
(Technion, Haifa, Israel)
|The exponent of a K-primitive matrix (Abstract)|
|Di 18.10.2005||16:15||MA 313|| Prof.
Dr. Michael Hintermüller
|Feasible and Non-Interior Path-following in constrained minimization with low multiplier regularity (Abstract)|
|Di 25.10.2005||16:15||MA 313|| Dr. Thomas Slawig
|Algorithmisches Differenzieren in Simulation und Optimalsteuerung von Differentialgleichungen (universitätsöffentlicher Habilitationsvortrag)|
|Di 1.11.2005||16:15||MA 313|| Prof. Dr. Yvan Notay
(Université Libre, Brüssel, Belgien)
|Multilevel block factorization with aggregation and coarsening by dynamic MILU (Abstract)|
|Di 8.11.2005||16:15||MA 313|| Dr. Kees
(TU Delft, Niederlande)
|Efficient preconditioners for the Helmholtz equations (Abstract)|
|Di 15.11.2005||16:15||MA 313|| Steffen Bauer
|Modeling electrical and magnetical activity of cardiac tissue (Abstract)|
|Di 22.11.2005||16:15||MA 313|| Prof. Dr. Lutz Tobiska
|Finite Element Methods for Incompressible Flows in Time-Dependent Domains (Abstract)|
|Di 29.11.2005||16:15||MA 313|| Prof. Dr. Khakim Ikramov
(U. Moskau, Russland)
|The coneigenvalues of a matrix as the second set of its eigenvalues (Abstract)|
|Mi 30.11.2005||16:00||MA 313|| Dr.
(U. Oslo, Norwegen)
|Hölder continuity of weak solutions for a degenerate chemotaxis model (Abstract)|
|Di 13.12.2005||16:15||MA 313|| Prof. Dr. Vincent Heuveline
|Neue Trends bei der Modellierung und Simulation komplexer Strömungsvorgänge (Abstract)|
|Di 3.1.2006||16:15||MA 313|| Dr. Etienne Emmrich
|Mathematical Analysis and Numerical Methods for the peridynamic equation of motion in non-local elasticity theory (Abstract)|
|Di 10.1.2006||16:15||MA 313|| Timo
|A Systems Theoretic Approach to PDAEs and Applications to Electrical Circuits (Abstract)|
|Di 31.1.2006||16:15||MA 313|| Prof.
Dr. Eugene Tyrtyshnikov
|A unifying approach to construction of the best structured preconditioners (Abstract)|
|Di 14.2.2006||16:15||MA 313|| Prof.
Dr. Ruth Curtain
(U. Groningen, Niederlande)
|Stabilization of almost impedance passive system by feedback (Abstract)|
|Di 28.2.2006||16:15||MA 313|| Prof.
Dr. Michel Pierre
(ENS Cachan, Frankreich)
|About regularity of optimal shapes (Abstract)|
|Mi 1.3.2006||16:00||MA 313|| Prof.
Dr. Michel Pierre
(ENS Cachan, Frankreich)
|Weak supersolutions and solutions for reaction-diffusion systems with L^1 a priori estimates (Abstract)|
|Di 7.3.2006||16:15||MA 313|| Prof.
Dr. Philippe Souplet
(U. Paris-Nord, Frankreich)
|Nonlinear Liouville theorems and singularity estimates in elliptic and parabolic problems (Abstract)|
|Di 14.3.2006||16:15||MA 313|| Maike Schulte
|Numerische Lösung der Schrödinger Gleichung auf unbeschränkten Gebieten (Abstract)|
|Mi 15.3.2006||16:15||MA 313|| Prof.
Dr. Ulrich Langer
(U. Linz, Österreich)
|Computational Electromagnetics: From the Simulation to the Optimization (Abstract)|
|Di 21.3.2006||16:15||MA 313|| Prof. Dr. Daniela
(Cleveland, OH, USA)
|Large scale statistical parameter estimation in complex systems with an application to metabolic models (Abstract)|
|Mi 22.3.2006||16:15||MA 313|| Prof. Dr. Alessio
|On a class of viscous Hamilton-Jacobi equations with superlinear terms and unbounded solutions (Abstract)|
Interessenten sind herzlich eingeladen!
Weitere Vorträge siehe auch:
Rückblick auf das Numerik-Oberseminar, den Vorläufer dieses Kolloquiums:
Abstracts zu den Vorträgen:
Recently a spectral approach of constructing low-reflecting boundary conditions for the wave equation in anisotropic media has been proposed , . The conditions are generated by two successive stages: firstly an exact operator of boundary conditions is numerically obtained for a discrete counterpart of governing equations; afterwards this boundary operator is approximated by a "cheaper" one in order to achieve reasonable computational costs for its application. In other words the second stage consists of a sharp compression of original huge matrix of the exact non-reflecting discrete operator. We use, in particular, spatial Fourier finite series, as well as sum-of-exponentials representation for occurring temporal kernels while performing such compression. The approach is implemented for the example of wave propagation in media with two different speeds of sound in half-spaces. It is shown that non-local discrete operators of obtained low-reflecting boundary conditions do provide required accuracy without any enormous computational efforts . This is joint work with Olga Podgornova.  I. L. Sofronov, O. V. Podgornova; Spectral nonlocal boundary conditions for the wave equation in moving media, Keldysh Inst. Appl. Math., preprint No. 53, 2004  I. Sofronov and O. Podgornova; A spectral approach for generating nonlocal boundary conditions for external wave problems in anisotropic media. Accepted for publication in J. Scient. Comp., 2005
Primal-dual path-following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed. Regularity properties of the path are proved. The path structure allows to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. The Moreau-Yosida regularized subproblems of the new path-following technique are solved efficiently by semismooth Newton methods. The overall algorithmic concept is provided and numerical tests (including a comparison with primal-dual path-following interior point methods) for state constrained optimal control problems show the efficiency of the new concept.
We consider linear systems arising from scalar elliptic PDEs, and discuss "algebraic'' multilevel methods, i.e. methods that are applicable in a "black box'' fashion, and nevertheless target (near) optimal convergence properties by setting an appropriate multilevel hierarchy.
We first present preconditioning by multilevel approximate block factorization and its basic convergence theory, giving some comparison with the convergence theory of more classical algebraic multigrid methods. We then discuss how this approach may be combined with coarsening by aggregation, which we illustrate with a double pairwise aggregation scheme. The latter tends to form aggregates with four nodes in, which ensures fast coarsening and small set up cost.
We next consider the improvement of this coarsening by so-called dynamic modified ILU (MILU). One performs here a tentative MILU factorization of the matrix block connecting the fine grid nodes, and the nodes for which the corresponding pivot is too small are moved to the coarse grid. This approach may be justified by the conditioning analysis of MILU factorizations, and we illustrate why it may be more efficient than the more classical compatible relaxation approach. We eventually discuss how coarsening may be obtained directly (i.e., from scratch) by dynamic MILU, and present some issues related to the integration of this approach in an efficient multilevel algorithm.
In this paper, the time-harmonic wave equation in heterogeneous media is solved numerically. The underlying equation governs wave propagations and scattering phenomena arising in many area's, e.g. in aeronautics, geophysics, and optical problems. In particular, we look for solutions of the Helmholtz equation discretized by finite difference discretizations. Since the number of grid points per wavelength should be sufficiently large to result in acceptable solutions, for very high wavenumbers the discrete problem becomes extremely large, prohibiting the use of direct solvers.
Since the coefficient matrix is sparse, iterative solvers are an interesting alternative. In this paper we present a novel preconditioner for high wavenumbers. The preconditioner is based on the inverse of the Helmholtz operator, where an imaginary term is added. This preconditioner can be approximated by multigrid. This is somewhat surprising as multigrid, without enhancements, has convergence troubles for the original Helmholtz operator at high wavenumbers.
In recent years, computer simulations entered the field of basic medical research in order to test hypotheses on functional relations, usefulness of therapies or assessment of diagnostic techniques. Models of the cardiovascular system mainly cover electrophysiological questions and try to reconstruct the spatial and temporal variations of the electrical potentials on the myocardium and the surrounding tissue. Unfortunately, detailed and satisfactory models of myocardial tissue still need enormous computing power; the use of advanced mathematical methods is therefore essential.
In this talk an overview will be given on the basic physiological relations of the heart and how they can be reflected numerically. The electrophysiological behavior of myocardial tissue is described by a bidomain approach which characterizes the myocardium as a continuum consisting of two domains which share the same spatial regions. This approach leads to a system of coupled PDEs. Our solution method consists of a semi-implicit operator splitting technique to decouple this system of PDEs which leads to an elliptical PDE, a parabolic PDE and a system of ODE which can be solved separately. As the elliptical part usually takes most computing power, our focus was on optimizing this part. We found good results for this type of problem by using multigrid preconditioners compared to classical CG/ILU methods.
Finally, several examples of model validation and simulation of pathological conditions based on animal data will be presented.
Nowadays, reliable and efficient numerical methods are available for solving the incompressible Navier-Stokes equation in a fixed domain. However, often the fluid domain has free boundaries and can change with time. As an example of such a situation we consider the deformation of a droplet after impinging a horizontal surface. First, we describe the mathematical model which takes into account the influence of the surface tension and the contact angle. Then, interface tracking and capturing methods for handling the unknown part of the boundary are discussed. Second order discretization in space and time using discontinuous piecewise linear pressure approximations guarantee a good mass balance. A multiple discretization multilevel solver is applied for solving the linearized systems of equations in each time step efficiently. Finally, the performance and accuracy of the method is demonstrated by some numerical examples.
We are concerned with a degenerate system of nonlinear partial differential equations modelling the competition of species in the presence of chemotaxis. The existence of weak solutions is proved by using Schauder fixed-point theorem. We show that weak solutions are Hölder continuous by adapting the technique of intrinsic scaling.
During the last few years, non-local theories in solid mechanics that account for effects of long-range forces have become topical again in the engineering community. One of these is the so-called peridynamic model, introduced by Silling in 1998, in which the equation of motion is a partial integro-differential equation without any spatial derivative. It is, therefore, a promising approach for the simulation of problems in which discontinuities emerge such as fracture or cracking.
In this talk, we commence with an overview of the peridynamic modelling and the related results obtained so far. We then concentrate on the description of a linear microelastic material. Interpreting the resulting equation of motion as an evolutionary equation of second order, well-posedness is shown. Moreover, jump relations and a perturbation of the integral kernel are studied.
Finally, we consider energy conserving spatial approximations by means of a quadrature formula or finite element method and present some first results.
This is joint work with O. Weckner (MIT Department of Mechanical Engineering).
Linear and time-invariant control systems governed by coupled algebraic and ordinary differential equations are usually modelled by descriptor systems Ex'=Ax+Bu, y=Cx. On the other hand, many controlled partial differential equations (like e.g. heat equation with boundary action) can be rewritten as a system x'=Ax+Bu, y=Cx, where A, B and C are operators acting on some infinite dimensional spaces. Consequently, controlled systems consisting of partial differential equations coupled with differential-algebraic equations can modelled by descriptor systems, but now with an infinite dimensional state space. Although for finite dimensional descriptor systems as well as for ordinary infinite dimensional systems there is a lot of theory available, very few about their combination is known. We present some results about the analytical properties like solvability, index and consistent initialization of infinite dimensional descriptor systems and we consider systems theoretic aspects like controllability and observability. The results are applied to electrical circuits with transmission lines.
We will survey some methods to prove regularity of optimal shapes. We will recall classical results coming from the theory of minimal surfaces, like the breaking point of dimension 8 for regularity. Then, we will consider functional shapes involving state functions which are solutions of elliptic equations. In this context, existence results for optimal shapes are obtained via topological and functional analysis arguments which provide shapes without any a priori regularity, like open subsets of R^d or even only measurable sets. It is generally a hard task to reach a first level of regularity. One main step is to obtain regularity of the state function (like Lipschitz regularity). This step will be discussed. The case of optimal shapes for eigenvalues of the Laplace operator will be also considered.
We will discuss the question of global existence in time of solutions for reaction-diffusion systems for which L^1-bounds hold for the nonlinear terms. It turns out that, in the positive case, one always gets global weak supersolutions. They are even solutions for a wide class of systems with a priori bounds on the total mass. Strangely enough, L^2-estimates are also hidden in these systems. Coupling all these estimates provide for instance global solutions for all quadratic systems of this type. Many open questions are still open in this context.
Viele physikalischen Probleme werden mathematisch durch partielle Differentialgleichungen auf unbeschränkten Gebieten beschrieben. Meistens ist die Lösung eines Problems nur auf einem endlichen Teilgebiet Ω nötigt, das (eventuell unbeschränkte) Gesamtgebiet nimmt aber Einfluss auf die Lösung in diesem Teilgebiet. Zur numerischen Lösung der Differentialgleichung müssen an dem Rand des Teilgebietes neue Randbedingungen formuliert werden. Diese hei\ss en transparent, wenn die Lösung eines Anfangs-Randwertproblems auf Ω mit der auf Ω eingeschränkten Lösung des Ganzraumproblems übereinstimmt.
In diesem Vortrag werden inhomogene transparente Randbedingungen (TRB) für die zeitabhängige Schrödinger Gleichung in 2D für ein kompaktes 9-Punkt-Schema 4.Ordnung vorgestellt. Anwendungen der TRB für die Schrödinger Gleichung liegen z.B. in der Simulation von Quantenwellenleitern. Vorgestellt wird die Simulation eines Quantentransistors und ein Verfahren zur realistischeren Simulation von Quantenwellenleitern. Weiterhin wird kurz auf die numerische Lösung der Schrödinger Gleichung auf Kreisgebieten eingegangen, welche zur Zeit in Kooperation mit M.Ehrhardt (TU Berlin) entwickelt wird.
The computer simulation of electromagnetic fields plays an increasingly important role in many applications. The optimal design of electro-technical devices via mathematical optimization procedures usually requires fast numerical solution techniques for the direct field problem that is described by Maxwell's equations.
In the first part of the talk we propose a multigrid solver for nonlinear time-periodic eddy current problems in the frequency domain based on a multiharmonic approach. The simulation of a 3D shielding and a 3D welding problem shows the efficiency of our solver in comparison with the solution procedures in the time domain.
The second part of the talk is devoted to topology and shape optimization in magnetostatics. We consider a DC electromagnet which is used for measurements of magneto-optical effects on thin layers. The aim is to find an optimal geometrical design of the ferromagnetic yoke and poles such that the magnetostatic field is as constant as possible in the area of measurements. 3D topology optimization resulted in a single-component spherical geometry. We further optimized the shape of the pole heads using a multilevel shape optimization method. The resulted shape was manufactured and is used in several laboratories.
Finally, we discuss some fast iteration techniques for solving KKT systems which arise from the all-at-once approach to optimization problems with PDE constraints.
The results of the first part (simulation) were obtained in a joint work with F. Bachinger and J. Schöberl who were supported by the Austrian Science Fund "Fonds zur Förderung der wissenschaftlichen Forschung (FWF)" under the grants P 14953, SFB F013 and START Y192, whereas the results of the second and third part (optimization) were obtained in a joint work with D. Lukáš, R. Simon, R. Stainko and W. Zulehner within the Project F1309 of the SFB F013 "Numerical and Symbolic Scientific Computing" in Linz.
|Impressum||Christian Mehl 10.3.2006|