Library

Description: This article is about the optimal track allocation problem (OPTRA) to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations: a standard formulation that models block conflicts in terms of packing constraints, and a new extended formulation that is based on additional configuration' variables. We show that the packing constraints in the standard formulation stem from an interval graph, and that they can be separated in polynomial time. It follows that the LP relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the extended formulation produces the same LP bound, and that it can also be computed with this model in polynomial time. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view, because it features a constant number of rows and is therefore amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hannover-Fulda-Kassel region of the German long distance railway network are reported.

Description: We study barrier methods for state constrained optimal control problems with PDEs. In the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented path-following algorithms. We establish results on existence, continuity and convergence of this path. Moreover, we consider the structure of barrier subdifferentials, which play the role of dual variables.

Description: For the treatment of equilibrated molecular systems in a heat bath we propose a transition state theory that is based on conformation dynamics. In general, a set-based discretization of a Markov operator ${\cal P}^\tau$ does not preserve the Markov property. In this article, we propose a discretization method which is based on a Galerkin approach. This discretization method preserves the Markov property of the operator and can be interpreted as a decomposition of the state space into (fuzzy) sets. The conformation-based transition state theory presented here can be seen as a first step in conformation dynamics towards the computation of essential dynamical properties of molecular systems without time-consuming molecular dynamics simulations.

Description: We present a middleware to store multidimensional data sets on Internet-scale distributed systems and to efficiently perform range queries on them. Our structured overlay network \emph{SONAR (Structured Overlay Network with Arbitrary Range queries)} puts keys which are adjacent in the key space on logically adjacent nodes in the overlay and is thereby able to process multidimensional range queries with a single logarithmic data lookup and local forwarding. The specified ranges may have arbitrary shapes like rectangles, circles, spheres or polygons. Empirical results demonstrate the routing performance of SONAR on several data sets, ranging from real-world data to artificially constructed worst case distributions. We study the quality of SONAR's routing information which is based on local knowledge only and measure the indegree of the overlay nodes to find potential hot spots in the routing process. We show that SONAR's routing table is self-adjusting, even under extreme situations, keeping always a maximum of $\lceil \log N \rceil$ routing entries.

Description: \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract}

Description: Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.

Description: In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicola{\"i}des (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered to the computation of incompressible variable density flow problems.

Description: We provide information on the Survivable Network Design Library (SNDlib), a data library for fixed telecommunication network design that can be accessed at http://sndlib.zib.de. In version 1.0, the library contains data related to 22 networks which, combined with a set of selected planning parameters, leads to 830 network planning problem instances. In this paper, we provide a mathematical model for each planning problem considered in the library and describe the data concepts of the SNDlib. Furthermore, we provide statistical information and details about the origin of the data sets.

Description: The \emph{optimal track allocation problem} (\textsc{OPTRA}), also known as the train routing problem or the train timetabling problem, is to find, in a given railway network, a conflict-free set of train routes of maximum value. We propose a novel integer programming formulation for this problem that is based on additional configuration' variables. Its LP-relaxation can be solved in polynomial time. These results are the theoretical basis for a column generation algorithm to solve large-scale track allocation problems. Computational results for the Hanover-Kassel-Fulda area of the German long distance railway network involving up to 570 trains are reported.

Description: We consider a multicommodity routing problem, where demands are released \emph{online} and have to be routed in a network during specified time windows. The objective is to minimize a time and load dependent convex cost function of the aggregate arc flow. First, we study the fractional routing variant. We present two online algorithms, called Seq and Seq$^2$. Our first main result states that, for cost functions defined by polynomial price functions with nonnegative coefficients and maximum degree~$d$, the competitive ratio of Seq and Seq$^2$ is at most $(d+1)^{d+1}$, which is tight. We also present lower bounds of $(0.265\,(d+1))^{d+1}$ for any online algorithm. In the case of a network with two nodes and parallel arcs, we prove a lower bound of $(2-\frac{1}{2} \sqrt{3})$ on the competitive ratio for Seq and Seq$^2$, even for affine linear price functions. Furthermore, we study resource augmentation, where the online algorithm has to route less demand than the offline adversary. Second, we consider unsplittable routings. For this setting, we present two online algorithms, called U-Seq and U-Seq$^2$. We prove that for polynomial price functions with nonnegative coefficients and maximum degree~$d$, the competitive ratio of U-Seq and U-Seq$^2$ is bounded by $O{1.77^d\,d^{d+1}}$. We present lower bounds of $(0.5307\,(d+1))^{d+1}$ for any online algorithm and $(d+1)^{d+1}$ for our algorithms. Third, we consider a special case of our framework: online load balancing in the $\ell_p$-norm. For the fractional and unsplittable variant of this problem, we show that our online algorithms are $p$ and $O{p}$ competitive, respectively. Such results where previously known only for scheduling jobs on restricted (un)related parallel machines.

Description: In this paper we study capacitated network design problems, differentiating directed, bidirected and undirected link capacity models. We complement existing polyhedral results for the three variants by new classes of facet-defining valid inequalities and unified lifting results. For this, we study the restriction of the problems to a cut of the network. First, we show that facets of the resulting cutset polyhedra translate into facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Second, we provide an analysis of the facial structure of cutset polyhedra, elaborating the differences caused by the three different types of capacity constraints. We present flow-cutset inequalities for all three models and show under which conditions these are facet-defining. We also state a new class of facets for the bidirected and undirected case and it is shown how to handle multiple capacity modules by Mixed Integer Rounding (MIR).

Description: In this paper we study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is 4K2 (1+K)2 -competitive, where K is the number of commodities. For the single-source single-destination case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.

Description: In this paper, we empirically investigate the NP-hard problem of finding sparse solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a branch-and-cut approach via the maximum feasible subsystem problem to compute optimal solutions for small instances and investigate the uniqueness of the optimal solutions. We furthermore discuss five (modifications of) heuristics for this problem that appear in different parts of the literature. For small instances, the exact optimal solutions allow us to evaluate the quality of the heuristics, while for larger instances we compare their relative performance. One outcome is that the basis pursuit heuristic performs worse, compared to the other methods. Among the best heuristics are a method due to Mangasarian and a bilinear approach.

Description: The performance evaluation of W-CDMA networks is intricate as cells are strongly coupled through interference. Pole equations have been developed as a simple tool to analyze cell capacity. Numerous scientific contributions have been made on their basis. In the established forms, the pole equations rely on strong assumptions such as homogeneous traffic, uniform users, and constant downlink orthogonality factor. These assumptions are not met in realistic scenarios. Hence, the pole equations are typically used during initial network dimensioning only. Actual network (fine-) planning requires a more faithful analysis of each individual cell's capacity. Complex analytical analysis or Monte-Carlo simulations are used for this purposes. In this paper, we generalize the pole equations to include inhomogeneous data. We show how the equations can be parametrized in a cell-specific way provided the transmit powers are known. This allows to carry over prior results to realistic settings. This is illustrated with an example: Based on the pole equation, we investigate the accuracy of average snapshot'' approximations for downlink transmit powers used in state-of-the-art network optimization schemes. We confirm that the analytical insights apply to practice-relevant settings on the basis of results from detailed Monte-Carlo simulation on realistic datasets.

Description: This paper deals with directed, bidirected, and undirected capacitated network design problems. Using mixed integer rounding (MIR), we generalize flow-cutset inequalities to these three link types and to an arbitrary modular link capacity structure, and propose a generic separation algorithm. In an extensive computational study on 54 instances from the Survivable Network Design Library (SNDlib), we show that the performance of cplex can significantly be enhanced by this class of cutting planes. The computations reveal the particular importance of the subclass of cutset-inequalities.

Description: Wigner transformation provides a one-to-one correspondence between functions on position space (wave functions) and functions on phase space (Wigner functions). Weighted integrals of Wigner functions yield quadratic quantities of wave functions like position and momentum densities or expectation values. For molecular quantum systems, suitably modified classical transport of Wigner functions provides an asymptotic approximation of the dynamics in the high energy regime. The article addresses the computation of Wigner functions by Monte Carlo quadrature. An ad aption of the Metropolis algorithm for the approximation of signed measures with disconnected support is systematically tested in combination with a surface hopping algorithm for non-adiabatic quantum dynamics. The numerical experiments give expectation values and level populations with an error of two to three percent, which agrees with the theoretically expected accuracy.

Description: We study a planning problem arising in SDH/WDM multi-layer telecommunication network design. The goal is to find a minimum cost installation of link and node hardware of both network layers such that traffic demands can be realized via grooming and a survivable routing. We present a mixed-integer programming formulation that takes many practical side constraints into account, including node hardware, several bitrates, and survivability against single physical node or link failures. This model is solved using a branch-and-cut approach with problem-specific preprocessing and cutting planes based on either of the two layers. On several realistic two-layer planning scenarios, we show that these cutting planes are still useful in the multi-layer context, helping to increase the dual bound and to reduce the optimality gaps.

Description: In \emph{classical optimization} it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so nice'' to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of \emph{online optimization} which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not the result of bad data handling but an inevitable phenomenon.

Description: A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.

Description: We present a unified approach for consistent remeshing of arbitrary non-manifold triangle meshes with additional user-defined feature lines, which together form a feature skeleton. Our method is based on local operations only and produces meshes of high regularity and triangle quality while preserving the geometry as well as topology of the feature skeleton and the input mesh.

Description: 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.

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

Description: {\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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: {\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.

Description: 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.

Description: 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.