Bibliothek

Beschreibung: An edge of a perfect graph $G$ is critical if $G-e$ is imperfect. We would like to decide whether $G - e$ is still {\sl almost perfect} or already {\sl very imperfect}. Via relaxations of the stable set polytope of a graph, we define two superclasses of perfect graphs: rank-perfect and weakly rank-perfect graphs. Membership in those two classes indicates how far an imperfect graph is away from being perfect. We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from the complement of such a graph.

Beschreibung: Collection of abstracts of the first SIAM-EMS conference Applied Mathematics in our Changing World'' in Berlin, September 2-6, 2001.

Beschreibung: {\sc Zimpl} is a little language to translate the mathematical model of a problem into a linear or (mixed-)integer mathematical program expressed in {\tt lp} or {\tt mps} file format which can be read by a LP or MIP solver.

Beschreibung: Given a set of service requests (events), a set of guided servers (units), and a set of unguided service contractors (conts), the vehicle dispatching problem {\sl vdp} is the task to find an assignment of events to units and conts as well as tours for all units starting at their current positions and ending at their home positions (dispatch) such that the total cost of the dispatch is minimized. The cost of a dispatch is the sum of unit costs, cont costs, and event costs. Unit costs consist of driving costs, service costs and overtime costs; cont costs consist of a fixed cost per service; event costs consist of late costs linear in the late time, which occur whenever the service of the event starts later than its deadline. The program \textsf{ZIBDIP} based on dynamic column generation and set partitioning yields solutions on heavy-load real-world instances (215 events, 95 units) in less than a minute that are no worse than 1\% from optimum on state-of-the-art personal computers.

Beschreibung: Scenario tree models of stochastic programs arise naturally under standard nonanticipativity assumptions. We demonstrate how tree-sparse programs cover the general case, with \emph{arbitrary} information constraints. Detailed examples and intuitive interpretations illuminate the basic thoughts behind the abstract but elementary construction.

Beschreibung: In this paper we present an algorithm that accelerates 3D texture-based volume rendering of large and sparse data sets. A hierarchical data structure (known as AMR tree) consisting of nested uniform grids is employed in order to efficiently encode regions of interest. The hierarchies resulting from this kind of space partitioning yield a good balance between the amount of volume to render and the number of texture bricks -- a prerequisite for fast rendering. Comparing our approach to an octree based algorithm we show that our algorithm increases rendering performance significantly for sparse data. A further advantage is that less parameter tuning is necessary.

Beschreibung: Mathematical optimization techniques are on their way to becoming a standard tool in chemical process engineering. While such approaches are usually based on deterministic models, uncertainties such as external disturbances play a significant role in many real-life applications. The present article gives an introduction to practical issues of process operation and to basic mathematical concepts required for the explicit treatment of uncertainties by stochastic optimization.

Beschreibung: Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. UCMC aims at avoiding the typical metastable or trapping behavior of Monte Carlo techniques. From the viewpoint of Monte Carlo, a slowly converging long-time Markov chain is replaced by a limited number of rapidly mixing short-time ones. Therefore, the state space of the chain has to be hierarchically decomposed into its metastable conformations. This is done by means of combining the technique of conformation analysis as recently introduced by the authors, and appropriate annealing strategies. We present a detailed examination of the uncoupling-coupling procedure which uncovers its theoretical background, and illustrates the hierarchical algorithmic approach. Furthermore, application of the UCMC algorithm to the $n$-pentane molecule allows us to discuss the effect of its crucial steps in a typical molecular scenario.

Beschreibung: Standard model predictive control for real-time operation of industrial production processes may be inefficient in the presence of substantial uncertainties. To avoid overly conservative disturbance corrections while ensuring safe operation, random influences should be taken into account explicitly. We propose a multistage stochastic programming approach within the model predictive control framework and apply it to a distillation process with a feed tank buffering external sources. A preliminary comparison to a probabilistic constraints approach is given and first computational results for the distillation process are presented.

Beschreibung: Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations are theoretically analyzed from a control-theoretic viewpoint and compared to each other. Associated KKT solution algorithms with linear complexity are developed and comparisons to other interior approaches and related problem formulations are discussed.

Beschreibung: We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly enviroment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size~$B\in\mathbb{N}$. The packing process is subject to the constraint that at any moment in time at most $q\in\mathbb{N}$ bins may be partially filled. Moreover, bins may only be closed if they are filled completely. An online algorithm must pack each item must be packed without knowledge of any future items. We investigate the existence of competitive online algorithms for the online uniform binpacking problem. We show upper bounds for the bin coloring problem. We prove an upper bound of $3q$ - 1 and a lower bound of $2q$ for the competitive ratio of a natural greedy-type algorithm, and show that surprisingly a trivial algorithm which uses only one open bin has a strictly better competitive ratio of $2q$ - 1. Morever, we show that any deterministic algorithm has a competitive ratio $\Omega (q)$ and that randomization does not improve this lower bound even when the adversary is oblivious.

Beschreibung: In circuit switching networks call streams are characterized by their mean and peakedness (two-moment method). The $GI/M/C/0$ system is used to model a single link, where the $GI$-stream is determined by fitting moments appropriately. For the moments of the overflow traffic of a $GI/M/C/0$ system there are efficient numerical algorithms available. However, for the moments of the freed carried traffic, defined as the moments of a virtual link of infinite capacity to which the process of calls accepted by the link (carried arrival process) is virtually directed and where the virtual calls get fresh exponential i.i.d.\ holding times, only complex numerical algorithms are available. This is the reason why the concept of the freed carried traffic is not used rigorously. The main result of this paper is an efficient algorithm for computing the moments of the freed carried traffic, in particular an explicit formula for its peakedness. This result offers a unified handling of both overflow and carried traffics in networks. Furthermore, some refined characteristics for the overflow and freed carried streams are derived.

Beschreibung: In this work we concentrate on developing methods which determine good lower bounds for set partitioning problems (SPP) in an appropriate amount of time. We found out that it makes sense to use the Lagrangian relaxation method for this task. The Lagrangian relaxed problem of SPP has a simple structure, which leads to algorithms and heuristics, whose total complexity per iteration depends linearly on the number of non-zeros of the problem matrix of SPP. In contrast, other methods like simplex methods or interior point methods have a complexity of higher order. Because the problem matrices of our tested instances are sparse, the linear dependence becomes an advantage for the algorithms and heuristics mentioned above. As a reference for the state-of-the-art we have applied the dual simplex method and the barrier function method, implemented in CPLEX. The methods, which we have developed and compared with those of CPLEX, are SBM, CAM, CCBM, and CBM. SBM is a subgradient bundle method derived from the basic subgradient method, which is a global convergent method for determining the maximum of concave functions. CAM is a coordinate ascent method, where the convex coordinate bundle method CCBM and the coordinate bundle method CBM are derivatives from CAM. We observed that the basic subgradient and the coordinate ascent method are improved if bundling techniques can be used. But the motivation for bundling differs for both approaches. In the former case bundling helps to approximate a minimum norm subgradient, which provides a steepest ascent direction, in order to speed up the performance. In the latter case bundling enables proceeding along directions, which are not restricted on the coordinate directions. By this the performance is accelerated. Among all used techniques stabilization is worth mentioning. Stabilization improves the performance especially at the beginning by avoiding too big steps during the proceeding. This leads to a more stabilized progression. Stabilization was successfully applied to SBM, CAM, CCBM, and CBM. As an overall result we conclude the following: \begin{enumerate} \item CPLEX computes the optimal objective values, whereas SBM and CBM has on average a gap of under $1.5\%$. \item In comparison to CPLEX baropt, SBM, CAM, and CBM the algorithm CCBM has a slow convergence because of the convex combination of ascent coordinate directions. An alternative is to relax the convex combination to a simple sum of the corresponding directions. This idea is realized in CBM. \item If we focus on the running time rather than on optimality then CBM is on average the fastest algorithm. \end{enumerate} Note that methods like SBM or CBM are applied on static SPP instances in order to determine a good lower bound. For solving SPP we need dynamical methods. Due to the complex topic of dynamical methods we will not discuss them, but a certain technique is worth mentioning. It is called column generation. We have indicated that this technique needs good Lagrangian multipliers of the corresponding SPP instances in order to generate further columns (in our case duties), which are added to the current SPP instance. Those multipliers are by-products of methods like our six considered methods. Due to the large number of such generation steps the running time depends on the computation time of these methods. Therefore, CBM fits more to this technique than CPLEX baropt or SBM. To sum it up it can be said that applications such as a duty scheduling can be described as set partitioning problems, whose lower bound can be solved by simplex, interior points, subgradient, or coordinate ascent methods. It turns out that the interior points method CPLEX baropt and the heuristic CBM have good performances. Furthermore, good Lagrangian multipliers, which are by-products of these methods, can be used by techniques like column generation. For this particular technique it also turns out that among our tested algorithms CBM is the most efficient one. In general we can state that real-world applications, which have to solve a large number of Lagrangian relaxed SPP instances can improve their performance by using CBM.

Beschreibung: The success of large-scale multi-national projects like the forthcoming analysis of the LHC particle collision data at CERN relies to a great extent on the ability to efficiently utilize computing a management software (Datagrid, Globus, etc.), while the effective integration of computing nodes has been largely neglected up to now. This is the focus of our work. We present a framework for a high-performance cluster that can be used as a reliable computing node in the Grid. We outline the cluster architecture, the management of distributed data and the seamless intergration of the cluster into the Grid environment.

Beschreibung: A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.

Beschreibung: By computed tomography data (CT), the individual geometry of the mandible is quite well reproduced, also the separation between cortical and trabecular bone. Using anatomical knowledge about the architecture and the functional potential of the masticatory muscles, realistic situations were approximated. The solution of the underlying partial differential equations describing linear elastic material behaviour is provided by an adaptive finite element method. Estimations of the discretization error, local grid refinement, and multilevel techniques guarantee the reliability and efficiency of the method.

Beschreibung: Optimization is the task of finding an optimum solution to a given problem. When the decision variables are discrete we speak of a combinatorial optimization problem. Such a problem is online when decisions have to be made before all data of the problem are known. And we speak of a real-time online problem when online decisions have to be computed within very tight time bounds. This paper surveys the are of combinatorial online and real-time optimization, it discusses, in particular, the concepts with which online and real-time algorithms can be analyzed.

Beschreibung: This paper describes a new simulation tool for the prediction of aerosol formation and behavior in gas--liquid contact devices such as absorbers, scrubbers, quench coolers, and condensers as well as multistage gas cleaning processes, respectively. Aerosol formation can impact severely the separation efficiency of gas cleaning processes. Aerosol or fog formation can arise by spontaneous condensation or desublimation in supersaturated gas phases. The rigorous description of the mass and energy transfer between the gas phase, the liquid phase, and the growing aerosol droplets leads to a system of partial differential and algebraic equations. For the solution of these systems we have developed the plant simulation tool AerCoDe. This program bases upon the linearly--implicit Euler discretisation, which in combination with extrapolation permits an adaptive step size and order control. Typical simulation results of a multistage industrial flue gas scrubbing process are presented. It is shown, that experimental data can be confirmed if the number concentration of condensation nuclei as an input parameter is roughly known.

Beschreibung: This paper discusses online optimization of real-world transportation systems. We concentrate on transportation problems arising in production and manufacturing processes, in particular in company internal logistics. We describe basic techniques to design online optimization algorithms for such systems, but our main focus is decision support for the planner: which online algorithm is the most appropriate one in a particular setting? We show by means of several examples that traditional methods for the evaluation of online algorithms often do not suffice to judge the strengths and weaknesses of online algorithms. We present modifications of well-known evaluation techniques and some new methods, and we argue that the selection of an online algorithm to be employed in practice should be based on a sound combination of several theoretical and practical evaluation criteria, including simulation.

Beschreibung: Wireless communication networks employ radio frequencies to establish communication links. The available radio spectrum is very limited. To meet today's radio communication demand, this resource has to be administered and reused carefully in order to control mutual interference. The reuse can be organized via separation in space, time, or frequency, for example. The problem, therefore, arises to distribute frequencies to links in a ``reasonable manner''. This is the basic form of the frequency assignment problem. What ``reasonable'' means, how to quantify this measure of quality, which technical side constraints to consider cannot be answered in general. The exact specification of this task and its mathematical model depend heavily on the particular application considered. In this paper we discuss this issue with respect to the GSM standard for mobile communication.

Beschreibung: This paper is a summary of the Round Table: ``The Impact of Mathematical Research on Industry and Vice Versa'' held at 3ecm in Barcelona on July 11, 2000. The round table started with contributions of the three panelists. Irene Fonseca, the panel chair, opened the discussion by stating six questions addressing the main issues of the round table topic. She presented the panel's answers to these questions, drawing on many examples from her own academic experience. In the following additional presentations, the other two panel members added further points of view based on their personal involvement with industry. The round table ended with a lively discussion with members from the audience. This written summary of the oral presentations follows the structure of the round table indicated above.

Beschreibung: This article is about \emph{adaptive column generation techniques} for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of a large problem. Computational results for three European scenarios are reported.

Beschreibung: The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating odd-cycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.

Beschreibung: Many phenomena in nature and engineering happen simultaneously on rather diverse spatial and temporal scales, i.e.\ exhibit a multi-scale character. Therefore various hierarchical data structures and numerical schemes have been devised to represent quantitatively such phenomena. A special numerical multilevel technique, associated with a particular hierarchical data structure, is so-called Adaptive Mesh Refinement (AMR). This scheme achieves locally very high spatial and temporal resolutions. Due to its popularity, many scientists are in need of interactive visualization tools for AMR data. In this article we present a 3D texture-based volume rendering algorithm for AMR data, that directly utilizes the hierarchical structure. Thereby interactive rendering even for large data sets is achieved. In particular the problems of interpolation artifacts, opacity corrections, and texture memory limitations are addressed. The algorithm's value in practice is demonstrated with simulation and image data.

Beschreibung: By combining techniques of preparation, histology, confocal microscopy, data visualization and data processing, we have created and recently published a standard brain model for drosophila and honey bee brains. This report describes the algorithms and implementation of the corresponding software modules. At the same time it serves as a user's guide for scientist who want to reproduce the results for differerent species or mutants.

Beschreibung: Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are listed.

Beschreibung: Two traffic streams $\Phi_1$, $\Phi_2$ are offered a link. The calls of $\Phi_i$ require exponential holding times with parameter $\mu$ and are accepted if less than $C_i$ trunks are occupied. Approximating the $\Phi_i$ by appropriate renewal processes meeting their first two moments, defined as the moments of the numbers of calls in virtual links of infinite capacity to which the traffic streams as freed traffics are virtually directed and where the calls get fresh exponential i.i.d.\ holding times with parameter $\mu$, stable recursive algorithms of complexity $O(\max(C_1,C_2))$ are derived for the first two defined as above moments of the individual overflow and freed carried traffics. The results offer a unified handling of both overflow and carried traffics in circuit switching networks with trunk reservation, providing a basis for new two-moment network dimensioning algorithms.

Beschreibung: We study the performance of QCD simulations with dynamical Wilson fermions by combining the Hybrid Monte Carlo algorithm with parallel tempering on $10^4$ and $12^4$ lattices. In order to compare tempered with standard simulations, covariance matrices between sub-ensembles have to be formulated and evaluated using the general properties of autocorrelations of the parallel tempering algorithm. We find that rendering the hopping parameter $\kappa$ dynamical does not lead to an essential improvement. We point out possible reasons for this observation and discuss more suitable ways of applying parallel tempering to QCD.

Beschreibung: We report numerical results for SBmethod --- a publically available implementation of the spectral bundle method --- applied to the 7$^{th}$ DIMACS challenge test sets that are semidefinite relaxations of combinatorial optimization problems. The performance of the code is heavily influenced by parameters that control bundle update and eigenvalue computation. Unfortunately, no mathematically sound guidelines for setting them are known. Based on our experience with SBmethod, we propose heuristics for dynamically updating the parameters as well as a heuristc for improving the starting point. These are now the default settings of SBmethod Version 1.1. We compare their performance on the DIMACS instances to our previous best choices for Version 1.0. SBmethod Version 1.1 is also part of the independent DIMACS benchmark by H.~Mittelmann. Based on these results we try to analyze strengths and weaknesses of our approach in comparison to other codes for large scale semidefinite programming.

Beschreibung: Many {\cal NP}-hard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for diverse optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed $k$, linear time algorithms exist to solve the decision problem ``treewidth $\leq k$'', their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.

Beschreibung: The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that pre-processing can help in finding good triangulations for probabilistic networks, that is, triangulations with a minimal maximum clique size. We provide a set of rules for stepwise reducing a graph, without losing optimality. This reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some well-known real-life probabilistic networks can be triangulated optimally just by preprocessing; for other networks, huge reductions in their graph's size are obtained.

Beschreibung: {\begin{rawhtml} 〈a href="http://dx.doi.org/10.1007/s10479-007-0178-0"〉 Revised Version unter http://dx.doi.org/10.1007/s10479-007-0178-0〈/a〉 \end{rawhtml}} Wireless communication is used in many different situations such as mobile telephony, radio and TV broadcasting, satellite communication, and military operations. In each of these situations a frequency assignment problem arises with application specific characteristics. Researchers have developed different modelling ideas for each of the features of the problem, such as the handling of interference among radio signals, the availability of frequencies, and the optimization criterion. This survey gives an overview of the models and methods that the literature provides on the topic. We present a broad description of the practical settings in which frequency assignment is applied. We also present a classification of the different models and formulations described in the literature, such that the common features of the models are emphasized. The solution methods are divided in two parts. Optimization and lower bounding techniques on the one hand, and heuristic search techniques on the other hand. The literature is classified according to the used methods. Again, we emphasize the common features, used in the different papers. The quality of the solution methods is compared, whenever possible, on publicly available benchmark instances.

Beschreibung: The pole condition is a general concept for the theoretical analysis and the numerical solution of a variety of wave propagation problems. It says that the Laplace transform of the physical solution in radial direction has no poles in the lower complex half-plane. In the present paper we show that for the Helmholtz equation with a radially symmetric potential the pole condition is equivalent to Sommerfeld's radiation condition. Moreover, a new representation formula based on the pole condition is derived and used to prove existence, uniqueness and asymptotic properties of solutions. This lays the foundations of a promising new algorithm to solve time-harmonic scattering problems numerically and provides a new approach for analyzing existing algorithms such as the Perfectly Matched Layer (PML) method and the Bayliss-Gunzburger-Turkel (BGT) algorithm.

Beschreibung: In this paper we study the PML method for Helmholtz-type scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a \textbf{P}erfectly \textbf{M}atched sponge \textbf{L}ayer. We prove that the approximate solution obtained by the PML method converges exponentially fast to the true solution in the computational domain as the thickness of the sponge layer tends to infinity. This is a generalization of results by Lassas and Somersalo based on boundary integral eqaution techniques. Here we use techniques based on the pole condition instead. This makes it possible to treat problems without an explicitly known fundamental solution.

Beschreibung: The present paper contains a generalization of a refinement of the Newton- Mysovskii theorem, recently obtained by the authors, to the case of Gauss-Newton procedures for solving nonlinear least-squares problems with full Jacobians. Invariant sufficient conditions are given that ensure the convergence of the Gauss-Newton iterates towards a solution of the problem, as well as the uniqueness of that solution in an explicitely defined neighborhood. It is shown by a counter- example that the results do not carry over to the rank deficient case.

Beschreibung: The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm.

Beschreibung: Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.

Beschreibung: Most nonlinear computations require the evaluation of first and higher derivatives of vector functions defined by computer programs. It is shown here how vectors of such partial derivatives can be obtained automatically and efficiently if the computer language allows overloading (as is or will be the case for C++, PASCAL-XSC, FORTRAN90, and other modern languages). Here, overloading facilitates the extension of arithmetic operations and univariate functions from real or complex arguments to truncated Taylor-series (or other user- defined types), and it generates instructions for the subsequent evaluation of adjoints. Similar effects can be achieved by precompilation of FORTRAN77 programs. The proposed differentiation algorithm yields gradients and higher derivatives at a small multiple of the run-time and RAM requirement of the original function evaluation program. {\bf Keywords:} Automatic Differentiation, Chain Rule, Overloading, Taylor Coefficients, Gradients, Hessians, Reverse Accumulation, Adjoint Equations. {\bf Abbreviated title:} Automatic Differentiation by Overloading.

Beschreibung: The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency.

Beschreibung: The numerical solution of optimal control problems by indirect methods (such as multiple shooting or collocation) requires a considerable amount of analytic calculation to establish a numerically tractable system. These analytic calculations, though being rather tedious in realistic examples, are nowadays mostly still done by hand-and thus prone to calculation errors. The paper aims at automating this analytic processing to a reasonable extent by means of a modern symbolic manipulation language (here: REDUCE). In its present stage of development the package OCCAL (mnemotechnically for \underline{O}ptimal \underline{C}ontrol \underline{CAL}culator) permits an interactive use, covering tasks like automatic determination of control and, in case of a singular control, of its order. In simpler problems, the present version of OCCAL automatically produces the full subroutine input for a MULtiple shooting code (MULCON) with adaptive numerical CONtinuation. In more complicate problems where singular sub-arcs may occur or where the sequence of sub-arcs of the optimal trajectory is unclear OCCAL is a significant help in reducing analytic pre-processing. Numerical examples illustrate the performance of OCCAL/MULCON.

Beschreibung: If $B$ is an arrangement of linear complex Hyperplanes in $C^d$, then the following can be constructed from knowledge of its intersection lattice: (a) the cohomology groups of the complement [Br], (b) the cohomology algebra of the complement [OS], (c) the fundamental group of the complement, if $d\le2$, (d) the singularity link up to homeomorphism, if $d\le3$, (e) the singularity link up to homotopy type [ZZ]. If $B'$ is, more generally, a 2-arrangement in $ R^{2d}$ (an arrangement of real subspaces of codimension 2 with even-dimensional intersections), then the intersection lattice still determines (a) the cohomology groups of the complement [GM] and (e) the homotopy type of the singularity link [ZZ]. We show, however, that for 2-arrangements the data (b), (c) and (d) are not determined by the intersection lattice. They require the knowledge of extra information on sign patterns, which can be computed as determinants of linear relations, or (equivalently) as linking coefficients in the sense of knot theory.

Beschreibung: We prove combinatorial formulas for the homotopy type of the union of the subspaces in an (affine, compactified affine, spherical or projective) subspace arrangement. From these formulas we derive results of Goresky & MacPherson on the homology of the arrangement and the cohomology of its complement. The union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset. A closely related space is obtained by taking the homotopy direct limit of this diagram. Our method consists in constructing a combinatorial model diagram over the same poset, whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces.

Beschreibung: A new adaptive multilevel approach for linear partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A Multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy.

Beschreibung: NUMSIM 91 fand vom 6.-08. Mai 1991 im Wissenschaftlichen Kommunikations- & Konferenz- Zentrum der Humboldt-Universität in Gosen bei Berlin statt. Im Mittelpunkt dieses Seminars standen Probleme der numerischen Simulation von Ladungstransport- und Technologieprozessen der Mikro- und Optoelektronik. Es führte Spezialisten der physikalischen Modellierung, der numerischen Mathematik und mathematischen Analysis sowie Nutzer von Simulationsprogrammen aus dem deutschsprachigen Raum zusammen. Ziel war die Vermittlung und der Austausch von Erfahrungen, die Diskussion gemeinsamer Aufgabenstellungen und Projekte und - in Anbetracht der deutschen Vereinigung nicht zuletzt - das gegenseitige Kennen- und Verstehenlernen.

Beschreibung: In this paper we introduce a discontinuous finite element method. In our approach, it is possible to combine the advantages of finite element and finite difference methods. The main ingredients are numerical flux approximation and local orthogonal basis functions. The scheme is defined on arbitrary triangulations and can be easily extended to nonlinear problems. Two different error indicators are derived. Especially the second one is closely connected to our approach and able to handle arbitrary variing flow directions. Numerical results are given for boundary value problems in two dimensions. They demonstrate the performance of the scheme, combined with the two error indicators. {\bf Key words:} neutron transport equation, discontinuous finite element, adaptive grid refinement. {\bf Subject classifications:} AMS(MOS) 65N30, 65M15.

Beschreibung: In this paper it is shown that for highly nonuniformly refined triangulations the condition number of the BPX preconditioner for elliptic finite element problems grows at most linearly in the depth of refinement. This is achieved by viewing the computational available version of the BPX preconditioner as an abstract additive Schwarz method with exact solvers. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65N20, 65N30.

Beschreibung: We study the space of all extensions of a real hyperplane arrangement by a new pseudo- hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kapranov & Sturmfels, via the Bohne-Dress Theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not even known whether the extension space is connected. We show that the subspace of realizable extensions is always connected but not necessarily spherical.

Beschreibung: We consider such variational inequalities which either describe obstacle problems or result from an implicit time discretization of moving boundary problems of two phase Stefan type. Based on a discretization in space by means of continuous, piecewise linear finite elements with respect to a nested hierarchy of triangulations, in both cases we use iterative processes consisting of inner and outer iterations. The outer iterations are either active set strategies or generalized Newton methods while the inner iterations are preconditioned cg- iterations with multilevel preconditioners.

Beschreibung: One-step discretizations of order $p$ and step size $\varepsilon$ of ordinary differential equations can be viewed as time-$\varepsilon$ maps of \begin{displaymath} \dot{x} (t) = f(\lambda ,x(t)) + \varepsilon^p g(\varepsilon,\lambda,t/\varepsilon,x(t)), x \in R^N,\lambda \in R, \end{displaymath} where $g$ has period $\varepsilon$ in $t$. This is a rapidly forced nonautonomous system. We study the behavior of a homoclinit orbit $\Gamma$ for $\varepsilon = 0, \lambda =0$, under discretization. Under generic assumptions we show that $\Gamma$ becomes transverse for positive $\in$. The transversality effects are estimated from above to be exponentially small in $\in$. For example, the length $l(\varepsilon$) of the parameter interval of $\lambda$ for which $\Gamma$ persists can be estimated by \begin{displaymath} l(\varepsilon)\le Cexp(-2\pi\eta/\varepsilon), \end{displaymath} where $C,\eta$ are positive constants. The coefficient $\eta$ is related to the minimal distance from the real axis of the poles of $\Gamma(t)$ in the complex time domain. Likewise, the region where complicated, "chaotic" dynamics prevail is estimated to be exponentially small, provided $x \in R^2$ and the saddle quantity of the associated equilibrium is nonzero. Our results are visualized by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality becomes pratically invisible under normal circumstances, already for only moderately small step size. {\bf Keywords:} Homoclinic orbit, ordinary differential equations, discretization, transversality, averaging, exponential smallness, chaos. {\bf Subject Classifications:} (AMS): 34C15, 34C35, 58F14, 65L60

Beschreibung: We consider the discretization of obstacle problems for the Laplacian by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned c-g iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semi-local and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.

Beschreibung: Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and statistics. They can be considered as ordinary differential equations in sequence spaces. In this work, a fully adaptive algorithm for the computational treatment of such systems is developed. The method is based on a time discretization of an abstract Cauchy problem in Hilbert space and a discrete Galerkin approach for the discretization of the arising stationary subproblems. The Galerkin method uses orthogonal functions of a discrete variable, which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations are solved adaptively either by use of analytical properties of the orthogonal functions or by an appropriate numerical summation. The resulting algorithm CODEX is applied to examples of technological interest, in particular from polymer chemistry.

Beschreibung: The deformation of a hexagonal lattice dome under an external load is an example of a parameter dependent system which is equivariant under the symmetry group of a regular hexagon. In this paper the mixed symbolic-numerical algorithm SYMCON is applied to analyze its steady state solutions automatically showing their different symmetry and stability properties.

Beschreibung: We study the higher Bruhat orders $B(n,k)$ of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets $U(C^{n+1,r},n+1)$ of uniform extensions of the alternating oriented matroids $C^{n,r}$ for $r:=n-k$ (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), - show that $B(n,k)$ is a lattice for $k =1$ and for $r\le 3$, but not in general, - show that $B(n,k)$ is ordered by inclusion of inversion sets for $k=1$ and for $r\le 4$. However, $B(8,3)$ is not ordered by inclusion. This implies that the partial order $B_\subseteq (n,k)$ defined by inclusion of inversion sets differs from $B(n,k)$ in general. We show that the proper part of $B_\subseteq (n,k)$ is homotopy equivalent to $S^{r-2}$. Consequently, - $B(n,k)\simeq S^{r-2}$ for $k=1$ and for $r\le 4$. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for $r=3$, and that the proper part is not always homotopy equivalent to $S^{r(M)-2}$.

Beschreibung: The C-implementation of KASKADE, an adaptive solver for linear elliptic differential equations in 2D, is object of a set of numerical experiments to analyze the use of resources (time and memory) with respect to numerical accuracy. We study the dependency of the reliability, robustness, and efficiency of the program from the parameters controlling the algorithm.

Beschreibung: A high resolution Godunov-type method is developed and applied to a two-dimensional version of the Burgers equation. On one-dimensional testproblems the scheme is second order accurate for smooth flow and resolves discontinuities sharply. On two-dimensional problems, however, large numerical oscillations are introduced when discontinuities traveling obliquely to the grid directions are present. These oscillations are caused by a Strang- type operator splitting, and cannot be eliminated using flux limiters or small timesteps. {\bf Keywords:} High resolution methods, Godunovs method, operator splitting, complex Burgers equation. {\bf AMS:} 35: A40, L45, L60, L65, L, 67. 65: M05, M10.

Beschreibung: For adaptive solution of convection- difussion problems with the streamline-diffusion finite element method, an error estimator based on interpolation techniques is developed. It can be shown that for correctness of this error estimator a restriction of the maximum angle is to be sufficient. Compared to usual methods, the adaptive process leads to more accurate solutions at much less computational cost. Numerical tests are enclosed. {\bf Keywords: } Adaptive finite elements, convection- diffusion equation, internal and boundary layers, streamline-diffusion. {\bf Subject Classifications:} AMS(MOS): 65N15, 65N30

Beschreibung: The success of the German Reunification requires a unique migration from the traditional Eastern Block system to a modern free society. The successful migration must include a shift from a low state of the art in technology to the high level of science and technology present in the Western Community. The role of supercomputing in this transition will include leading the process of organizational change in research and development. The paper contains a short overview on the current state of this evolution, on the currently used political and administrative tools to keep this evolution on track, and on the influence on the Konrad-Zuse-Zentrum, which is, as a Berlin institution, in the geographical and cultural center of this transformation.

Beschreibung: The following report intends to provide a survey over the computational chemistry molecular structure software installed on CRAY X-MP/24 at ZIB. It shows what kind of problems can be tackled with the existing chemistry software, which covers a wide range of ab initio, semiempirical, molecular mechanics, and dynamics applications.

Beschreibung: An adaptive approach to the numerical solution of the wave propagation in integrated optics devices with 1D cross sections is described. First, Fresnel's approximation of the exact wave equation resulting from Maxwell's equations is considered. A criterion to estimate the validity of this approximation is derived. Fresnel's wave equation being formally equivalent to Schroedinger's equation uniquely defines an initial-boundary-value problem, which is solved numerically by a stepwise calculation of the propagating field. Discretization in longitudinal direction first with stepsize control leads to a stationary subproblem for the transversal field distribution, which is then handled by an adaptive finite element method. Thus full adaptivity of the algorithm is realized. The numerical examples are concentrated on taper structures playing an essential role in integrated optics devices for telecommunication systems.