Minimal failure probability for ceramic design via shape control

We consider the probability of failure for components made of brittle mate- rials under one time application of a load as introduced by Weibull and Batdorf - Crosse and more recently studied by NASA and the STAU cooperation as an objective functional in shape optimization and prove the existence of optimal shapes in the class of shapes with a uniform cone property. The corresponding integrand of the objective functional has convexity properties that allow to derive lower-semicontinuity according to Fujii (Opt. Th. Appl. 1988). These properties require less restrictive regularity assumptions for the boundaries and state functions compared to [arXiv:1210.4954]. Thereby, the weak formulation of linear elasticity can be kept for the abstract setting for shape optimization as presented in the book by Haslinger and Maekinen

A multigrid perspective on the parallel full approximation scheme in space and time

For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the "Parallel Full Approximation Scheme in Space and Time" (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type

Non-overlapping block smoothers for the Stokes equations

Overlapping block smoothers efficiently damp the error contributions from highly oscillatory components within multigrid methods for the Stokes equations but they are computationally expensive. This paper is concentrated on the development and analysis of new block smoothers for the Stokes equations that are discretized on staggered grids. These smoothers are non-overlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, the efficiency of multigrid methods for solving more difficult problems such as the Stokes equations lead to computationally more expensive smoothers, e.g., overlapping block smoothers. Non-overlapping smoothers are less expensive, but have been considered less efficient in the literature. In this paper, we develop new non-overlapping smoothers, the so-called triad-wise smoothers, and show their efficiency within multigrid methods to solve the Stokes equations. In addition, we compare overlapping and non-overlapping smoothers by measuring their computational costs and analyzing their behavior by the use of local Fourier analysis.Comment: 17 pages, 34 figure

PROFESSIONALIZATION OF ADULT EDUCATORS FOR A DIGITAL WORLD: AN EUROPEAN PERSPECTIVE

Digital media play an increasingly important role in all areas of society. As a result, media literacy is one of the key qualifications for our information society. It enables social participation and opens up opportunities for professional development. Media literacy is not a static construct though – due to technological progress it must be continually developed. For this reason, adult education has a central function in promoting media literacy. At the same time, for education too new opportunities for promoting learning are constantly opening up via digital media. The media education competencies of adult educators are therefore of central significance for assessing and utilising the opportunities and risks of current developments. In light of this, this article discusses the current situation with regards to standards and pathways of professionalization of adult educators in terms of media pedagogic competences in Europe.  Article visualizations

Multigrid methods for structured grids and their application in particle simulation

Mehrgitterverfahren sind optimale, d.h. linear skalierende, Verfahren zur Lösung einer großen Zahl von Problemen, z.B. von elliptischen partiellen Differentialgleichungen. Viele dieser Anwendungen, z.B. partielle Differentialgleichungen die auf strukturierten Gittern diskretisiert sind, besitzen viel Struktur, die eine effiziente Implementierung des Mehrgitterverfahrens auf seriellen und parallelen Computern erlaubt. Neben der Standard-Theorie für geometrische Mehrgitterverfahren wurde eine variationelle Theorie von einer Vielzahl von Autoren entwickelt, wobei die variationelle Theorie auf der Verwendung des Galerkin-Grobgitteroperators basiert. In dieser Arbeit werden Mehrgitterverfahren zur Lösung linearer Gleichungssysteme mit strukturierten Koeffizientenmatrizen analysiert. Ein modifiziertes Schema zur Lösung der Poissongleichung in unbeschränkten Gebieten wird vorgestellt und sein Fehlerverhalten im Detail analysiert. Diese Methode nutzt eine endliche Anzahl von hierarchischen Vergröberungen und Vergrößerungen des Diskretisierungsgitters und Einsetzen von speziellen Randbedingungen auf dem gröbsten Level. Ein FAS-artiges Mehrgitterverfahren eignet sich gut zur Lösung dieses Problems. Partielle Differentialgleichungen mit konstanten Koeffizienten und periodischen Randbedingungen die auf equidistanten Gittern diskretisiert sind führen auf zirkulante Koeffizientenmatrizen. Daher kann in diesem Fall die vorhandene Theorie für zirkulante Matrizen angewandt werden. Diese Theorie basiert auf der klassischen Theorie für algebraische Mehrgitterverfahren, die in diesem Zusammenhang um die Verwendung von nicht-Galerkin-Grobgitteroperatoren erweitert wird. Ein Parallelisierungsansatz für zirkulante Matrizen basierend auf Gebietszerlegung wird vorgestellt. Die entwickelten Verfahren werden in Partikelsimulationsmethoden, wie sie in vielen Feldern der Physik gebraucht werden, angewandt. Das Problem wird vorgestellt und konsistent in einer Art formuliert, die es erlaubt das Problem durch Lösen der Poissongleichung in unbeschränkten oder periodischen Gebieten und eine Nahfeldkorrektur zu behandeln. Motiviert durch Methoden wie P3M basiert diese Umformulierung auf dem Ersatz von Punktladungen durch Distributionen mit beschränkten Träger. Durch die Umformulierung werden keine weiteren Fehler eingeführt. Daher ist der Fehler des Gesamtverfahrens nur durch den Diskretisierungsfehler und durch den Interpolationsfehler der verwendeten numerischen Schemata verursacht. Es werden numerische Beispiele für das FAS-artige Mehrgitterverfahren für das hierarchisch vergröberten Gitter, für das Mehrgitterverfahren für zirkulante Matrizen mit Ersatz des Galerkin-Grobgitteroperators, für sein paralleles Skalierungsverhalten auf Blue Gene/L und Blue Gene/P und für die Partikelsimulationsmethode, die das Mehrgitterverfahren nutzt, vorgestellt.Multigrid methods are optimal, i.e. linearly scaling, methods for the solution of a broad range of problems, including elliptic partial differential equations. Many of these applications, e.g. partial differential equations discretized on structured grids, posses a lot of structure that allows an efficient implementation of multigrid methods on serial and parallel computers. Besides the standard geometric multigrid theory, a variational theory was developed by a number of authors, where the variational theory is based on the use of the Galerkin coarse grid operator. In this work multigrid methods for the solution of linear systems of equations with structured coefficient matrices are analyzed. A modified discretization scheme for the solution of the Poisson equation in unbounded domains is presented and its error behavior is analyzed in detail. The method involves hierarchically coarsening and extending the discretization grid finitely often and imposing special boundary conditions on the boundary on the coarsest level. FAS-type multigrid is well suited to solve this problem. Partial differential equations with constant coefficients and periodic boundary conditions that are discretized on equispaced grids lead to circulant coefficient matrices. Therefor the theory of multigrid methods for circulant matrices is applicable to that case. The theory is based on the classical algebraic multigrid theory that is extended in this context to include non-Galerkin coarse grid operators, as well. A parallelization strategy based on domain decomposition is presented for circulant matrices. The developed methods are applied in particle simulation methods, as they are needed in various fields of physics. The problem is introduced and consistently reformulated in a way that allows to treat the problem with the solution of the Poisson equation in either unbounded or periodic domains with an added near-field correction. Motivated by methods like P3M this reformulation is based on replacing the point charges by finitely supported distributions. No additional errors are introduced by the reformulation. So the error of the resulting method is caused by the discretization error and the interpolation error of the used numerical schemes, only. Numerical examples are presented for the FAS-type multigrid solver for the hierarchically coarsened grid, for the multigrid solver for circulant matrices including the replacement of the Galerkin coarse grid operator, for its parallel scaling behavior on Blue Gene/L and Blue Gene/P and for the particle simulation method that uses the multigrid solver

Integration of continuous-time dynamics in a spiking neural network simulator

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here supports the combination of the two for multi-scale modeling approaches, the quantitative validation of mean-field approaches by spiking network simulations, and an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most efficient spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. We further demonstrate the broad applicability of the framework by considering various examples from the literature ranging from random networks to neural field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation

Symbol Based Convergence Analysis in Block Multigrid Methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear system with a particular saddle-point structure. In particular, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function $\mathbf{f}$, which consequently is a matrix-valued function with dimension depending on the block of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes equation. We show the efficiency of the methods studying the hidden $9\times 9$ block structure of the obtained matrix sequence proposing an efficient algebraic multigrid method with convergence rate independent of the matrix size. Moreover, we present several numerical tests comparing the results with different known strategies