@misc{GroetschelMonmaStoer1992, author = {Gr{\"o}tschel, Martin and Monma, Clyde and Stoer, Mechthild}, title = {A Polyhedral Approach to Network Connectivity Problems (Extended Abstract).}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-924}, number = {SC-92-22}, year = {1992}, abstract = {We present a polyhedral approach for the general problem of designing a minimum-cost network with specified connectivity requir}, language = {en} } @misc{Chu1992, author = {Chu, Nguyen Ngoc}, title = {An Algorithm for Optimizing a Linear Function over the Integer Efficient Set.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-930}, number = {SC-92-23}, year = {1992}, abstract = {Whereas optimization of a linear function over an efficient set is a favourite topic for theoretical studies, the problem (\$P^I\$) of finding a maximal value of a linear function \$dx\$ over an integer efficient set is still open. The problem (\$P^I\$) is NP-hard and it is very unlikely that the maximal objective value of the integer problem (\$P^I\$) in many cases is greater than the maximal objective value of it's corresponding continuous problem (\$P\$). In this paper we pay atention to the study of the problem (\$P^I\$) and some related properties of the problem (\$P\$). In particular, we establish conditions determining whether or not an optimal solution to the problem (\$P\$) is an optimal solution to the it's corresponding linear program. For the problem (\$P^I\$) we find an upper bound for it's optimal objective value and present an algorithm which gives a global optimal solution after a finite number of steps. We also study two particular classes of problems (\$P^I\$) : the bicriteria case and the case when \$d\$ is a nonnegative linear combination of the vectors-criteria defining the efficient set. Key words: Multiple objective linear programming, integer efficient set, efficient cone, cutting plane.}, language = {en} } @misc{GroetschelLueggerSperber1992, author = {Gr{\"o}tschel, Martin and L{\"u}gger, Joachim and Sperber, Wolfram}, title = {Elektronische Fachinformation im Bereich der Mathematik an Hochschulen und Statusbericht zum DMV-Projekt. "Fachinformation".}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4910}, number = {TR-92-06}, year = {1992}, abstract = {Im Bereich der Mathematik entwickelte sich in Deutschland die elektronische Fachinformation in Form der Online-Datenbank MATH und der CD-ROM CompactMATH auf der Grundlage der Arbeiten des Referateorgans "Zentralblatt f{\"u}r Mathematik und Ihre Grenzgebiete". Das "Zentralblatt" fand weithin Anerkennung und Verbreitung. Die Rezeption elektronischer Fachinformation ging jedoch - nicht nur im Bereich der Mathematik - nur z{\"o}gerlich vonstatten. Die Gr{\"u}nde daf{\"u}r werden in einer vom Bundesminister f{\"u}r Forschung und Technologie (BMFT) initiierten Studie analysiert. Ein k{\"u}rzlich von der Deutschen Mathematiker - Vereinigung begonnenes und vom BMFT gef{\"o}rdertes Vorhaben hat die "Verbesserung des benutzerorientierten Zugriffs auf fachspezifische Online-Datenbanken und CD-ROM f{\"u}r Mathematische Institute in der Bundesrepublik Deutschland" zum Ziel. Insgesamt 51 mathematische Fachbereiche und Institute nehmen daran teil. Elektronische Fachinformation soll zu einem festen Bestandteil des methodischen Instrumentariums der wissenschaftlichen Arbeit werden. Diese Schrift soll die Ausgangslage in den Fachbereichen darstellen und die organisatorischen und technischen Infrastrukturmaßnahmen erl{\"a}utern, mit denen den spezifischen Nutzungshemmnissen begegnet werden soll.}, language = {de} } @misc{Nowak1992, author = {Nowak, Ulrich}, title = {Dynamic Sparsing in Stiff Extrapolation Methods.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-911}, number = {SC-92-21}, year = {1992}, abstract = {Based on a simple stability analysis for the semi--implicit Euler discretization a new dynamic sparsing procedure is derived. This procedure automatically eliminates ``small'' elements of the Jacobian matrix. As a consequence, the amount of work needed to handle the linear algebra within a semi--implicit extrapolation integrator can be reduced drastically. Within the course of integration the sparsing criterion, which decides what ``small'' means, is dynamically adapted to ensure stability of the discretization scheme. Thus, stepsize restrictions due to instability can be avoided. Numerical experiments for quite different problems show robustness and efficiency of this dynamic sparsing technique. The techniques developed here in the context of stiff extrapolation integrators can, in principle, be applied to W--methods, where exact Jacobians may be replaced by ``sufficiently good'' approximations. {\bf Keywords:} Large scale integration, extrapolation methods, stiff ODEs, W--methods, sparse matrix techniques.}, language = {en} } @misc{Yang1992, author = {Yang, Shuli}, title = {Numerical Riemann solutions in multi-pieces for 2-D gas dynamics systems I. Contact discontinuities.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-903}, number = {SC-92-20}, year = {1992}, abstract = {The numerical solutions of Riemann problems in three, four, five and six pieces, which only contain contact discontinuities, are presented by using Taylor FVM MmB schemes on regular triangular meshes for 2-D gas dynamics systems. The 2-D Riemann initial data are as defined in [1], under the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves , or contact discontinuities. The main ends of the paper are that spirals will be shown for some configurations and the relations of the solutions between different distibutions of Riemann initial data are explained by the numerical solutions of modified Riemann problems. Key words and phrases: Riemann problem, gas dynamics systems, spiral, MmB schemes.}, language = {en} } @misc{Ziegler1992, author = {Ziegler, G{\"u}nter M.}, title = {Shellability of Chessboard Complexes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-890}, number = {SC-92-19}, year = {1992}, abstract = {The matchings in a complete bipartite graph form a simplicial complex, which in many cases has strong structural properties. We use an equivalent description as chessboard complexes: the complexes of all non-taking rook positions on chessboards of various shapes. In this paper we construct `certificate \$k\$-shapes' \$\Sigma(m,n,k)\$ such that if the shape \$A\$ contains some \$\Sigma(m,n,k)\$, then the \$(k{-}1)\$-skeleton of the chessboard complex \$\Delta(A)\$ is vertex decomposable in the sense of Provan \& Billera. This covers, in particular, the case of rectangular chessboards \$A=[m]{\times}[n]\$, for which \$\Delta(A)\$ is vertex decomposable if \$n\ge 2m{-}1\$, and the \$(\lfloor{m+n+1\over3}\rfloor{-}1)\$-skeleton is vertex decomposable in general. The notion of vertex decomposability is a very convenient tool to prove shellability of such combinatorially defined simplicial complexes. We establish a relation between vertex decomposability and the CL-shellability technique (for posets) of Bj{\"o}rner \& Wachs.}, language = {en} } @misc{HohmannWulff1992, author = {Hohmann, Andreas and Wulff, Claudia}, title = {Modular Design of Extrapolation Codes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4909}, number = {TR-92-05}, year = {1992}, abstract = {In this paper we describe a modular implementation of the well-established extrapolation codes EULEX, EULSIM and DIFEX for initial value problems of ordinary differential equations. The basic module embodies an abstract extrapolation method with order and stepsize control. Based on this module the particular integration codes only have to provide the underlying discretization schemes.}, language = {en} } @misc{Bornemann1992, author = {Bornemann, Folkmar A.}, title = {Adaptive Solution of One-Dimensional Scalar Conservation Laws with Convex Flux.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-880}, number = {SC-92-18}, year = {1992}, abstract = {A new adaptive approach for one-dimensional scalar conservation laws with convex flux is proposed. The initial data are approximated on an adaptive grid by a problem dependent, monotone interpolation procedure in such a way, that the multivalued problem of characteristic transport can be easily and explicitly solved. The unique entropy solution is chosen by means of a selection criterion due to LAX. For arbitrary times, the solutions is represented by an adaptive monotone spline interpolation. The spatial approximation is controlled by local \$L^1\$-error estimated. As a distinctive feature of the approach, there is no discretization in time. The method is monotone on fixed grids. Numerical examples are included, to demonstrate the predicted behavior. {\bf Key words.} method of characteristics, adaptive grids, monotone interpolation, \$L^1\$-error estimates {\bf AMS(MOS) subject classification.} 65M15, 65M25, 65M50.}, language = {en} } @misc{Yang1992, author = {Yang, Shuli}, title = {MmB schemes on regular triangular meshes for 2-D conservation laws.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-873}, number = {SC-92-17}, year = {1992}, abstract = {In this paper, two classes of second order accurate high resolution schemes are presented on regular triangular meshes for initial value problem of two dimensional conservation laws. The first class are called Runge-Kutta-FVM MmB (locally Maximum- minimum Bounds preserving) schemes, which are first discretized by (FVM) finite volume method in space direction and modifying numerical fluxes, and then by Runge-Kutta methods in time direction; The second class, constructed by Taylor expansion in time, and then by FVM methods and making modifications to fluxes, are called Taylor- FVM MmB schemes. MmB properties of both schemes are proved for 2-D scalar conservation law. Numerical results are given for Riemann problems of 2-D scalar conservation law and 2-D gas dynamics systems and some comparisons are made between the two classes of the schemes. Key words and phrases: MmB schemes, 2-D, conservation laws, gas dynamics systems, Runge-Kutta-FVM, Taylor-FVM.}, language = {en} } @phdthesis{Weismantel1992, author = {Weismantel, Robert}, title = {Plazieren von Zellen: Theorie und L{\"o}sung eines quadratischen 0/1- Optimierungsproblems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4883}, number = {TR-92-03}, year = {1992}, abstract = {Die vorliegende Arbeit besch{\"a}ftigt sich mit dem Plazierungsproblem, welches beim Entwurf elektronischer Schaltungen auftritt. Das Plazierungsproblem modellieren wir als ein quadratisches 0/1 Optimierungsproblem unter linearen Nebenbedingungen und untersuchen das Modell komplexit{\"a}tstheoretisch. Der zweite Aspekt der Arbeit bezieht sich auf die L{\"o}sung praktischer Problembeispiele im sogenannten Sea of cells"-Entwurfsstil. Zur L{\"o}sung dieser Beispiele wurde ein Prototyp implementiert und mit state of the art"-Plazierungsverfahren verglichen. Schlie\ss lich werden wir uns mit dem Clusteringproblem, das eine Variante des Mehrfachschnitt-Problems darstellt, besch{\"a}ftigen. Dabei steht einerseits im Vordergrund, wie diese Probleme heuristisch gel{\"o}st werden k{\"o}nnen und wie die Integration des Ansatzes in das Plazierungsprogramm erfolgt. Andererseits soll das Clusteringproblem polyedrisch untersucht werden.}, language = {de} } @phdthesis{Martin1992, author = {Martin, Alexander}, title = {Packen von Steinerb{\"a}umen: Polyedrische Studien und Anwendung.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4894}, number = {TR-92-04}, year = {1992}, abstract = {Gegeben sei ein Graph \$G=(V,E)\$ mit positiven Kantenkapazit{\"a}ten \$c_e\$ und Knotenmengen \$T_1,\ldots,T_N\$. Das Steinerbaumpackungs-Problem besteht darin, Kantenmengen \$S_1,\ldots,S_N\$ zu finden, so da\ss\ jedes \$S_k\$ die Knoten aus \$T_k\$ verbindet und jede Kante \$e\$ in h{\"o}chstens \$c_e\$ Kantenmengen aus \$S_1,\ldots,S_N\$ vorkommt. Eine zul{\"a}ssige L{\"o}sung dieses Problems nennen wir eine Steinerbaumpackung. Ist zus{\"a}tzlich eine Gewichtung der Kanten gegeben und nach einer bez{\"u}glich dieser Gewichtung minimalen Steinerbaumpackung gesucht, so sprechen wir vom gewichteten Steinerbaumpackungs-Problem. Die Motivation zum Studium dieses Problems kommt aus dem Entwurf elektronischer Schaltungen. Ein dort auftretendes Teilproblem ist das sogenannte Verdrahtungsproblem, das im wesentlichen darin besteht, gegebene Punktmengen unter bestimmten Nebenbedingungen und Optimalit{\"a}tskriterien auf einer Grundfl{\"a}che zu verbinden. Wir studieren das Steinerbaumpackungs-Problem aus polyedrischer Sicht und definieren ein Polyeder, dessen Ecken genau den Steinerbaumpackungen entsprechen. Anschlie\ss end versuchen wir, dieses Polyeder durch gute'' beziehungsweise facetten-definierenden Ungleichungen zu beschreiben. Basierend auf diesen Ungleichungen entwickeln wir ein Schnittebenenverfahren. Die L{\"o}sung des Schnittebenenverfahrens liefert eine untere Schranke f{\"u}r die Optimall{\"o}sung und dient als Grundlage f{\"u}r die Entwicklung guter Primalheuristiken. Wir haben das von uns implementierte Schnittebenenverfahren an einem Spezialfall des Verdrahtungsproblems, dem sogenannten Switchbox-Verdrahtungsproblem, getestet und vielversprechende Ergebnisse erzielt.}, language = {de} } @misc{LeimkuhlerReich1992, author = {Leimkuhler, Benedict and Reich, Sebastian}, title = {The Numerical Solution of Constrained Hamiltonian Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-868}, number = {SC-92-16}, year = {1992}, abstract = {A Hamiltonian system subject to smooth constraints can typically be viewed as a Hamiltonian system on a manifold. Numerical computations, however, must be performed in \$ R^n\$. In this paper, canonical transformations from ``Hamiltonian differential--algebraic equations'' to ODEs in Euclidean space are considered. In \S2, canonical parameterizations or local charts are developed and it is shown how these can be computed in a practical framework. In \S3 we consider the construction of unconstrained Hamiltonian ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an integral invariant and whose flow reduces to the flow of the constrained system along the manifold. It is shown that certain of these unconstrained Hamiltonian systems force Lyapunov stability of the constraint--invariants, while others lead to an unstable invariant. In \S4, we compare various projection techniques which might be incorporated to better insure preservation of the constraint--invariants in the context of numerical discretization. Numerical experiments illustrate the degree to which the constraint and symplectic invariants are maintained under discretization of various formulations. {\bf Keywords:} differential--algebraic equations, Hamiltonian systems, canonical discretization schemes. {\bf AMS(MOS):} subject classification 65L05.}, language = {en} } @misc{Moeller1992, author = {M{\"o}ller, H. Michael}, title = {On decomposing systems of polynomial equations with finitely many solutions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-851}, number = {SC-92-15}, year = {1992}, abstract = {This paper deals with systems of \$m\$ polynomial equations in \$n\$ unknown, which have only finitely many solutions. A method is presented which decomposes the solution set into finitely many subsets, each of them given by a system of type \begin{displaymath} f_1(x_1)=0, f_2(x_1,x_2)=0,...,f_n(x_1,...,x_n)=0. \end{displaymath} The main tools for the decomposition are from ideal theory and use symbolical manipulations. For the ideal generated by the polynomials which describe the solution set, a lexicographical Gr{\"o}bner basis is required. A particular element of this basis allows the decomposition of the solution set. A recursive application of these decomposition techniques gives finally the triangular subsystems. The algorithm gives even for non-finite solution sets often also usable decompositions. {\bf Keywords:} Algebraic variety decomposition, Gr{\"o}bner bases, systems of nonlinear equations.}, language = {en} } @misc{BornemannErdmannKornhuber1992, author = {Bornemann, Folkmar A. and Erdmann, Bodo and Kornhuber, Ralf}, title = {Adaptive Multilevel-Methods in 3-Space Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-843}, number = {SC-92-14}, year = {1992}, abstract = {We consider the approximate solution of selfadjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. as the performance of hierarchical bases preconditioners deteriorate in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a-posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.}, language = {en} } @misc{Gatermann1992, author = {Gatermann, Karin}, title = {Computation of Bifurcation Graphs.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-837}, number = {SC-92-13}, year = {1992}, abstract = {The numerical treatment of Equivariant parameter-dependent onlinear equation systems, and even more its automation requires the intensive use of group theory. This paper illustrates the group theoretic computations which are done in the preparation of the numerical computations. The bifurcation graph which gives the bifurcation subgroups is determined from the interrelationship of the irreducible representations of a group and its subgroups. The Jacobian is transformed to block diagonal structure using a modification of the transformation which transforms to block diagonal structure with respect to a supergroup. The principle of conjugacy is used everywhere to make symbolic and numerical computations even more efficient. Finally, when the symmetry reduced problems and blocks of Jacobian matrices are evaluated numerically, the fact that the given representation is a quasi-permutation representation is exploited automatically.}, language = {en} } @misc{OPUS4-982, title = {Jahresbericht 1991}, issn = {0934-5892}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9825}, number = {1991}, year = {1992}, language = {de} } @misc{LubichNowakPoehleetal.1992, author = {Lubich, Christian and Nowak, Ulrich and P{\"o}hle, Uwe and Engstler, Ch.}, title = {MEXX - Numerical Software for the Integration of Constrained Mechanical Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-823}, number = {SC-92-12}, year = {1992}, abstract = {MEXX (short for MEXanical systems eXtrapolation integrator) is a Fortran code for time integration of constrained mechanical systems. MEXX is suited for direct integration of the equations of motion in descriptor form. It is based on extrapolation of a time stepping method that is explicit in the differential equations and linearly implicit in the nonlinear constraints. It only requires the solution of well--structured systems of linear equations which can be solved with a computational work growing linearly with the number of bodies, in the case of multibody systems with few closed kinematic loops. Position and velocity constraints are enforced throughout the integration interval, whereas acceleration constraints need not be formulated. MEXX has options for time--continuous solution representation (useful for graphics) and for the location of events such as impacts. The present article describes MEXX and its underlying concepts.}, language = {en} } @misc{Bixby1992, author = {Bixby, Robert E.}, title = {Das Implementieren des Simplex- Verfahrens: Die Startbasis.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-816}, number = {SC-92-11}, year = {1992}, abstract = {Dieses Paper enth{\"a}lt die ersten beiden Teile einer geplanten Serie von Aufs{\"a}tzen {\"u}ber die \$CPLEX^2\$-Implementierung des Simplex- Verfahrens. Der erste Teil ist eine Einf{\"u}hrung: er liefert eine kurze Beschreibung des Verfahrens f{\"u}r Probleme mit beschr{\"a}nkten Variablen, zusammen mit einer relativ ausf{\"u}hrlichen Diskussion der numerischen Eigenschaften der Netlib-Probleme. Diese Probleme bilden auch das Fundament der rechnerischen Untersuchungen in den folgenden Teilen. Der zweite Teil enth{\"a}lt die Hauptergebnisse dieses Papers, eine Beschreibung der Methode, die von CPLEX verwendet wird, um eine Startbasis zu konstruieren.}, language = {de} } @misc{GroetschelMartinWeismantel1992, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Packing Steiner Trees: Polyhedral Investigations.}, doi = {10.1007/BF02592085}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-786}, number = {SC-92-08}, year = {1992}, abstract = {Let \$G=(V,E)\$ be a graph and \$T\subseteq V\$ be a node set. We call an edge set \$S\$ a Steiner tree with respect to \$T\$ if \$S\$ connects all pairs of nodes in \$T\$. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graph \$G=(V,E)\$ with edge weights \$w_e\$, edge capacities \$c_e, e \in E,\$ and node sets \$T_1,\ldots,T_N\$, find edge sets \$S_1,\ldots,S_N\$ such that each \$S_k\$ is a Steiner tree with respect to \$T_k\$, at most \$c_e\$ of these edge sets use edge \$e\$ for each \$e\in E\$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing Problem from a polyhedral point of view and define an appropriate polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch \& cut algorithm. This algorithm is described in our companion paper SC 92-09.}, language = {en} } @misc{GroetschelMartinWeismantel1992, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Packing Steiners Trees: A Cutting Plane Algorithm and Computational Results.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-792}, number = {SC-92-09}, year = {1992}, abstract = {In this paper we describe a cutting plane algorithm for the Steiner tree packing problem. We use our algorithm to solve some switchbox routing problems of VLSI-design and report on our computational experience. This includes a brief discussion of separation algorithms, a new LP-based primal heuristic and implementation details. The paper is based on the polyhedral theory for the Steiner tree packing polyhedron developed in our companion paper SC 92-8 and meant to turn this theory into an algorithmic tool for the solution of practical problems.}, language = {en} } @misc{JuengerMartinReineltetal.1992, author = {J{\"u}nger, Michael and Martin, Alexander and Reinelt, Gerhard and Weismantel, Robert}, title = {Quadratic 0/1 Optimization and a Decomposition Approach for the Placement of Electronic Circuits.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-803}, number = {SC-92-10}, year = {1992}, abstract = {The placement in the layout design of electronic circiuts consists of finding a non- overlapping assignment of rectangular cells to positions on the chip so what wireability is guaranteed and certain technical constraints are met.This problem can be modelled as a quadratic 0/1- program subject to linear constraints. We will present a decomposition approach to the placement problem and give results about \$NP\$-hardness and the existence of \$\varepsilon\$-approximative algorithms for the involved optimization problems. A graphtheoretic formulation of these problems will enable us to develop approximative algorithms. Finally we will present details of the implementation of our approach and compare it to industrial state of the art placement routines. {\bf Keywords:} Quadratic 0/1 optimization, Computational Complexity, VLSI-Design.}, language = {en} } @misc{SchuetteWulkow1992, author = {Sch{\"u}tte, Christof and Wulkow, Michael}, title = {Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-778}, number = {SC-92-07}, year = {1992}, abstract = {Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.}, language = {en} } @misc{KornhuberWittum1992, author = {Kornhuber, Ralf and Wittum, Gabriel}, title = {Discretization and Iterative Solution of Convection Diffusion Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-762}, number = {SC-92-06}, year = {1992}, abstract = {We propose an extended box method which turns out to be a variant of standard finite element methods in the case of pure diffusion and an extension of backward differencing to irregular grids if only convective transport is present. Together with the adaptive orientation proposed in a recent paper and a streamline ordering of the unknowns, this discretization leads to a highly efficient adaptive method for the approximation of internal layers in the case of large local Peclet numbers.}, language = {en} } @misc{LangWalter1992, author = {Lang, Jens and Walter, Artur}, title = {A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-755}, number = {SC-92-05}, year = {1992}, abstract = {Large scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction- diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means, we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods whereas for the space discretization a finite element method is used. The different behaviour of the proposed approaches are demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics and combustion theory. {\bf Keywords:} initial boundary value problem, Rothe- method, adaptive Runge-Kutta method, finite elements, mesh refinement. {\bf AMS CLASSIFICATION:} 65J15, 65M30, 65M50.}, language = {en} } @misc{Groetschel1992, author = {Gr{\"o}tschel, Martin}, title = {Discrete Mathematics in Manufacturing.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-738}, number = {SC-92-03}, year = {1992}, abstract = {Manufacturing is a topic that provides rich opportunities for important mathematical contributions to real-world problems. The purpose of this paper is to show, by means of several examples, where and how mathematical problems of a discrete nature arise in manufacturing and to demonstrate the savings and improvements that can be achieved by employing the techniques of combinatorial optimization. The topics covered range from the design phase of a product (e. g.,routing, placement and via minimization in VLSI design), the control of CNC machines (e. g., drilling and plotting), to the management of assembly lines, storage systems and whole factories. We also point out difficulties in the modelling of complex situations and outline the algorithmic methods that are used for the solution of the mathematical problems arising in manufacturing. {\bf Key words:} discrete mathematics , combinatorial optimization, applications to manufacturing.}, language = {en} } @misc{Wakabayashi1992, author = {Wakabayashi, Yoshiko}, title = {Medians of Binary Relations: Computational Complexity.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-746}, number = {SC-92-04}, year = {1992}, abstract = {Let \$\Re\$ be the set of all binary relations on a finite set \$N\$ and \$d\$ be the symmetric difference distance defined on \$\Re\$. For a given profile \$\Pi = (R_1,...,R_m) \in R^m\$, a relation \$R* \in \Re \$ that minimizes the function \$\sum^m_{k=1} d(R_k,R) \$ is called a median relation of \$\Pi\$. A number of problems occuring in the social sciences, in qualitative data analysis and in multicriteria decision making can be modelled as problems of finding medians of a profile of binary relations. In these contexts the profile \$\Pi\$ represents collected data (preferences, similarities, games) and the objective is that of finding a median relation of \$\Pi\$ with some special feature (representing e. g., consensus of preferences, clustering of similar objects, ranking of teams, etc.). In this paper we analyse the computational complexity of all such problems in which the median is required to satisfy one or more of the properties: reflexitivity, symmetry, antisymmetry, transitivity and completeness. We prove that whenever transitivity is required (except when symmetry and completeness are also simultaneously required) then the corresponding median problem is \$NP\$-hard. In some cases we prove that they remain \$NP\$-hard when the profile \$\Pi\$ has a fixed number of binary relations.}, language = {en} } @misc{Schoeffel1992, author = {Sch{\"o}ffel, Klaus}, title = {Computational Chemistry Software at ZIB.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4879}, number = {TR-92-02}, year = {1992}, abstract = {The following report intends to provide a survey over the computational chemistry molecular structure software installed on the supercomputers CRAY X-MP/216 and CRAY Y-MP2E/164 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.}, language = {en} } @misc{AckermannHELFRICH1992, author = {Ackermann, J{\"o}rg and HELFRICH, K.}, title = {Radius of Convergence of the 1/Z-Expansion for Diatomic Molecules: The Ground State of the Isoelectronic H2Sequence.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-721}, number = {SC-92-02}, year = {1992}, abstract = {Using the perturbational-variational Rayleigh-Ritz matrix formalism, the 1/Z-expansion for the ground state of the isoelectronic \$H_2\$ sequence in the range of the internuclear distance \$0.2\le R \le 9.0\$ is calculated. Also lower bounds of the radius of convergence, based on Kato's theory of linear operators, are given. The numerical results of the 1/Z-expansion can be compared with the exact results and do not converge in the whole R-range. This behavior is in qualitative agreement with the lower bounds for the radius of convergence and enlights some still open properties of 1/Z- expansions for this sequence in the literature. {\bf PACS:} 31.15 + q; 31.20 Di; 31.20 Tz.}, language = {en} } @misc{BornemannYserentant1992, author = {Bornemann, Folkmar A. and Yserentant, Harry}, title = {A Basic Norm Equivalence for the Theory of Multilevel Methods.}, doi = {10.1007/BF01388699}, number = {SC-92-01}, year = {1992}, abstract = {Subspace decompositions of finite element spaces based on \$L2\$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the \$H^1\$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of \$ K\$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50.}, language = {en} } @misc{NowakWeimann1992, author = {Nowak, Ulrich and Weimann, Lutz}, title = {A Family of Newton Codes for Systems of Highly Nonlinear Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4855}, number = {TR-91-10}, year = {1992}, abstract = {This report presents new codes for the numerical solution of highly nonlinear systems. They realize the most recent variants of affine invariant Newton Techniques due to Deuflhard. The standard method is implemented in the code NLEQ1, whereas the code NLEQ2 contains a rank reduction device additionally. The code NLEQ1S is the sparse version of NLEQ1, i.e. the arising linear systems are solved with sparse matrix techniques. Within the new implementations a common design of the software in view of user interface and internal modularization is realized. Numerical experiments for some rather challenging examples illustrate robustness and efficiency of algorithm and software.}, language = {en} } @misc{Schoeffel1992, author = {Sch{\"o}ffel, Klaus}, title = {Ab initio Quantum Chemical Calculations with GAMESS-UK and GAUSSIAN90 Program Packeges - A Comparison -.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4867}, number = {TR-92-01}, year = {1992}, abstract = {Two commercially available molecular electronic structure software packages GAUSSIAN90 and GAMESS-UK are compared. Basis for this comparison is a benchmark suite which is designed to highlight the typical range of calculations commonly performed by the ab initio computational chemist.}, language = {en} } @misc{Hohmann1992, author = {Hohmann, Andreas}, title = {An Adaptive Continuation Method for Implicitly Defined Surfaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-705}, number = {SC-91-20}, year = {1992}, abstract = {A new method for the numerical aproximation of an implicitly defined surface is presented. It is a generalization of the Euler- Gauss-Newton method for implicitly defined (one- parameter) curves to the case of (two-parameter) surfaces. The basic task in the more general case is an efficient combination of modern CAGD techniques (such as triangular Bernstein-Bzier patches and the nine parameter Hermite interpolant) and the rank deficient Gauss-Newton method.}, language = {en} } @misc{GodsilGroetschelWelsh1992, author = {Godsil, C. D. and Gr{\"o}tschel, Martin and Welsh, D. J. A.}, title = {Combinatorics in Statistical Physics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-694}, number = {SC-91-19}, year = {1992}, language = {en} } @misc{Wulkow1991, author = {Wulkow, Michael}, title = {Adaptive Treatment of Polyreactions in Weighted Sequence Spaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-673}, number = {SC-91-17}, year = {1991}, abstract = {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.}, language = {en} } @misc{AckermannWulkow1991, author = {Ackermann, J{\"o}rg and Wulkow, Michael}, title = {The Treatment of Macromolecular Processes with Chain-Length-Dependent Reaction Coefficients - An Examplefrom Soot Formation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-682}, number = {SC-91-18}, year = {1991}, abstract = {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.}, language = {en} } @misc{SchoepfDeuflhard1991, author = {Sch{\"o}pf, R. and Deuflhard, Peter}, title = {OCCAL: A mixed symbolic- numeric Optimal Control CALculator.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-638}, number = {SC-91-13}, year = {1991}, abstract = {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.}, language = {en} } @misc{HoppeKornhuber1991, author = {Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Adaptive Multilevel - Methods for Obstacle Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-667}, number = {SC-91-16}, year = {1991}, abstract = {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.}, language = {en} } @misc{GatermannHohmann1991, author = {Gatermann, Karin and Hohmann, Andreas}, title = {Hexagonal Lattice Dome - Illustration of a Nontrivial Bifurcation Problem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-587}, number = {SC-91-08}, year = {1991}, abstract = {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.}, language = {en} } @misc{Ziegler1991, author = {Ziegler, G{\"u}nter M.}, title = {On the Difference Between Real and Complex Arrangements.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-648}, number = {SC-91-14}, year = {1991}, abstract = {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.}, language = {en} } @misc{ZieglerZivaljevic1991, author = {Ziegler, G{\"u}nter M. and Zivaljevic, R. T.}, title = {Homotopy Types of Subspace Arrangements via Diagrams of Spaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-658}, number = {SC-91-15}, year = {1991}, abstract = {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.}, language = {en} } @misc{Schmidt1991, author = {Schmidt, Frank}, title = {An Adaptive Approach to the Numerical Solution of Fresnel's Wave Equation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-628}, number = {SC-91-12}, year = {1991}, abstract = {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.}, language = {en} } @misc{SturmfelsZiegler1991, author = {Sturmfels, Bernd and Ziegler, G{\"u}nter M.}, title = {Extension Spaces of Oriented Matriods.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-618}, number = {SC-91-11}, year = {1991}, abstract = {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.}, language = {en} } @misc{Ziegler1991, author = {Ziegler, G{\"u}nter M.}, title = {Higher Bruhat Orders and Cyclic Hyperplane Arrangements.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-607}, number = {SC-91-10}, year = {1991}, abstract = {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}\$.}, language = {en} } @misc{DeuflhardNowakPoehleetal.1991, author = {Deuflhard, Peter and Nowak, Ulrich and P{\"o}hle, Uwe and Schmidt, B. Ch. and Weyer, J.}, title = {Die Ausbreitung von HIV/AIDS in Ballungsgebieten.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4844}, number = {TR-91-09}, year = {1991}, abstract = {The rapidly increasing number of AIDS cases requires a realistic estimation of the future development of the HIV/AIDS disease. For that purpose we develop a large system of coupled nonlinear differential equations describing simultaneously the dynamics of the development of the disease, the population size, the gender and age structure. A set of 1650 coupled equations are linked by balanced parameters. The balancing procedure is described by a set of (formally) 2,178,000 additional algebraic conditions. As the considered system is stiff, it requires new special extrapolation methods combined with techniquees of dynamical sparsing for the solution of sparsely filled systems. According to our simulations we expect 19,0,000 deaths caused by AIDS in the Federal Republic of Germany (former territories) in the year 2000. Such an epidemical spread would tie up about 4-7 percent of the actual health care workers.}, language = {de} } @misc{Bornemann1991, author = {Bornemann, Folkmar A.}, title = {A Sharpened Condition Number Estimate for the BPX Preconditioner of Elliptic Finite Element Problems on Highly Nonuniform Triangulations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-596}, number = {SC-91-09}, year = {1991}, abstract = {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.}, language = {en} } @misc{DeuflhardPotra1991, author = {Deuflhard, Peter and Potra, Florian}, title = {A Refined Gauss-Newton- Mysovskii Theorem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-541}, number = {SC-91-04}, year = {1991}, abstract = {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.}, language = {en} } @misc{HoppeKornhuber1991, author = {Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Multilevel Preconditioned CG-Iterations for Variational Inequalities.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-561}, number = {SC-91-06}, year = {1991}, abstract = {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.}, language = {en} } @misc{GajewskiEdDeuflhardMarkowiched1991, author = {Gajewski(Ed.), Herbert and Deuflhard, Peter and Markowich(ed.), A.}, title = {Tagung NUMSIM '91 5.-08. Mai 1991. Collected Abstracts and Papers.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4838}, number = {TR-91-08}, year = {1991}, abstract = {NUMSIM 91 fand vom 6.-08. Mai 1991 im Wissenschaftlichen Kommunikations- \& Konferenz- Zentrum der Humboldt-Universit{\"a}t in Gosen bei Berlin statt. Im Mittelpunkt dieses Seminars standen Probleme der numerischen Simulation von Ladungstransport- und Technologieprozessen der Mikro- und Optoelektronik. Es f{\"u}hrte 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.}, language = {en} } @misc{Bornemann1991, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4821}, number = {TR-91-07}, year = {1991}, abstract = {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.}, language = {en} } @misc{FiedlerScheurle1991, author = {Fiedler, Bernold and Scheurle, J.}, title = {Discretization of Homoclinic Orbits, Rapid Forcing and "Invisible" Chaos.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-552}, number = {SC-91-05}, year = {1991}, abstract = {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}, language = {en} } @misc{LangWalter1991, author = {Lang, Jens and Walter, Artur}, title = {An Adaptive Discontinuous Finite Element Method for the Transport Equation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-579}, number = {SC-91-07}, year = {1991}, abstract = {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.}, language = {en} } @misc{Schoeffel1991, author = {Sch{\"o}ffel, Klaus}, title = {Computational Chemistry Software for CRAY X-MP/2.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4816}, number = {TR-91-06}, year = {1991}, abstract = {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.}, language = {en} } @misc{Griewank1991, author = {Griewank, Andreas}, title = {Sequential Evaluation of Adjoints and Higher Derivative Vectors by Overloading and Reverse Accumulation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-539}, number = {SC-91-03}, year = {1991}, abstract = {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.}, language = {en} } @misc{OPUS4-983, title = {Jahresbericht 1990}, issn = {0934-5892}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9831}, number = {1990}, year = {1991}, language = {de} } @misc{Gottschewski1991, author = {Gottschewski, J{\"u}rgen}, title = {Supercomputing During the German Reunification.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4803}, number = {TR-91-05}, year = {1991}, abstract = {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.}, language = {en} } @misc{KornhuberRoitzsch1991, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {Self Adaptive Computation of the Breakdown Voltage of Planar pn-Junctions with Multistep Field Plates.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-525}, number = {SC-91-02}, year = {1991}, abstract = {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.}, language = {en} } @misc{Lang1991, author = {Lang, Jens}, title = {An Adaptive Finite Element Method for Convection-Diffusion Problems by Interpolation Techniques.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4795}, number = {TR-91-04}, year = {1991}, abstract = {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}, language = {en} } @misc{Noelle1991, author = {Noelle, Sebastian W. C.}, title = {On the Limits of Operator Splitting: Numerical Experiments for the Complex Burgers Equation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4789}, number = {TR-91-03}, year = {1991}, abstract = {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.}, language = {en} } @misc{LueggerDalitz1991, author = {L{\"u}gger, Joachim and Dalitz, Wolfgang}, title = {Verteilung mathematischer Software mittels elektronischer Netze: Die elektronische Softwarebibliothek eLib.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4774}, number = {TR-91-02}, year = {1991}, abstract = {Mit zunehmender Dezentralisierung im Bereich des Wissenschaftlichen Rechnens steigt auch der Bedarf an Softwarequellen qualitativ hochwertiger mathematischer Algorithmen. F{\"u}r den Software- und Informationsaustausch wird die Einrichtung {\"o}ffentlicher, elektronischer Softwarebibliotheken vorgeschlagen. Am Beispiel der eLib des ZIB werden die notwendigen technischen Konzepte, die u.a. auch in den heutigen Gegebenheiten der technischen Kommunikation begr{\"u}ndet sind, diskutiert. Die eLib ist fenster- und men{\"u}orientiert aufgebaut und folglich im Dialog sehr einfach zu benutzen. Sie kann aber auch mit ihrer Kommandosprache allein auf dem Wege {\"u}ber elektronische Post abgefragt werden. {\"U}ber die eLib sind, nach dem Verbund mit den Netzbibliotheken NetLib und Reduce-NetLib jetzt etwa 6000 mathematische Programme und allgemeine Dokumente verf{\"u}gbar. Das Dienstleistungsangebot der eLib setzt Beitr{\"a}ge aus einer aktiven Benutzergemeinschaft voraus. Es besteht die Hoffnung, daß durch den {\"o}ffentlichen Softwareaustausch sich Entwickler und Anwender wechselseitig in ihrer Arbeit befruchten.}, language = {de} } @misc{BornemannErdmannRoitzsch1991, author = {Bornemann, Folkmar A. and Erdmann, Bodo and Roitzsch, Rainer}, title = {KASKADE - Numerical Experiments.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4762}, number = {TR-91-01}, year = {1991}, abstract = {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.}, language = {en} } @misc{Bornemann1991, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations III.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-519}, number = {SC-91-01}, year = {1991}, abstract = {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.}, language = {en} } @misc{Walter1990, author = {Walter, Artur}, title = {Sparse Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-483}, number = {SC-90-20}, year = {1990}, abstract = {A variety of secant methods has been revisited in view of the construction of iterative solvers for large nonsymmetric linear systems \$ Ax = b \$ stemming from the discretization of convection diffusion equations. In the first section, we tried to approximate \$ A ^{-1} \$ directly. Since the sparsity structure of A- is not known, additional storage vectors are needed during the iteration. In the next section, an incomplete factorization \$ LU \$ of \$ A \$ is the starting point and we tried to improve this easy invertible approximation of \$ A \$. The update is constructed in such a way that the sparsity structure of \$ L \$ and \$ U \$ is maintained. Two different sparsity preserving updates are investigated from theoretical and practical point of view. Numerical experiments on discretized PDEs of convection diffusion type in 2- D with internal layers and on "arbitrary" matrices with symmetric sparsity structure are given. {\bf Key words:} nonsymmetric linear system, sparse secant method, Broyden's method, incomplete factorization.}, language = {en} } @misc{WulkowDeuflhard1990, author = {Wulkow, Michael and Deuflhard, Peter}, title = {Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-292}, number = {SC-90-01}, year = {1990}, abstract = {The discrete Galerkin method developed by the authors has turned out to be an efficient tool for the computational treatment of very large scale ODE systems arising in polyreaction kinetics. Up to now, this approach has been worked out in detail for homogeneous polymer reactions. The present paper deals with one line of possible extensions of the method to the case of so-called heterogeneous processes, which may appear e. g. in smog reactions. The associated mathematical models involve reaction coefficients depending on the chain length of the reacting polymer. The herein suggested extension is worked out in some detail on the basis of the earlier paper. In addition, a numerical example describing polymer degradation is included.}, language = {en} } @misc{Giloi1990, author = {Giloi, Wolfgang K.}, title = {Konrad Zuses Plankalk{\"u}l als Vorl{\"a}ufer moderner Programmier-modelle. (Vortrag)}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4759}, number = {TR-90-13}, year = {1990}, language = {de} } @misc{NowakWeimann1990, author = {Nowak, Ulrich and Weimann, Lutz}, title = {GIANT - A Software Package for the Numerical Solution of Very Large Systems of Highly Nonlinear Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4746}, number = {TR-90-11}, year = {1990}, abstract = {This report presents the final realization and implementation of a global inexact Newton method proposed by Deuflhard. In order to create a complete piece of software, a recently developed iterative solver (program GBIT) due to Deuflhard, Freund, Walter is adapted and serves as the standard iterative linear solver. Alternative linear iterative solvers may be adapted as well, e.g. the widely distributed code GMRES. The new software package GIANT (Global Inexact Affine Invariant Newton Techniques) allows an efficient and robust numerical solution of very large scale highly nonlinear systems. Due to the user friendly interface and its modular design, the software package is open for an easy adaptation to specific problems. Numerical experiments for some selected problems illustrate performance and usage of the package.}, language = {en} } @misc{CaprasseDemaretGatermannetal.1990, author = {Caprasse, H. and Demaret, J. and Gatermann, Karin and Melenk, Herbert}, title = {Power-Law Type Solutions of Fourth-Order Gravity}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-466}, number = {SC-90-18}, year = {1990}, abstract = {We study the power-law type solutions of the fourth order field equations derived from a generic quadratic Lagrangian density in the case of multidimensional Bianchi I cosmological models. All the solutions of the system of algebraic equations have been found, using computer algebra, from a search of the Groebner bases associated to it. While, in space dimension \$ d = 3 \$ , the Einsteinian Kasner metric is still the most general power-law type solution, for \$ d > 3 \$ , no solution, other than the Minkowski space-time, is common to the three systems of equations associated with the three contributions to the Lagrangian density. In the case of a pure Riemann-squared contribution (suggested by a recent calculation of the effective action for the heterotic string), the possibility exists to realize a splitting of the \$ d \$-dimensional space into a ( \$ d - 3 \$)-dimensional internal space and a physical 3- dimensional space, the latter expanding in time as a power bigger than 2 (about 4.5 when \$ d = 9 \$).}, language = {en} } @misc{Wang1990, author = {Wang, Daoliu}, title = {Symplectic Difference Schemes for Perturbed Hamiltonian Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-456}, number = {SC-90-17}, year = {1990}, abstract = {In this paper we consider symplectic difference schemes for perturbed Hamiltonian systems of integrable ones, which can cover many important problems. Symplectic difference schemes for general Hamiltonian systems can also be used to these problems. But the perturbation property has not been paid proper attention to, which is important in the method proposed here. Numerical simulation shows that, for this method the time step size can be taken quite large and the qualitative property , such as preserving invariant tori, is also better than usual symplectic difference schemes.}, language = {en} } @misc{HegeStueben1990, author = {Hege, Hans-Christian and St{\"u}ben, Hinnerk}, title = {Vectorization and Parallelization of Irregular Problems via Graph coloring.}, doi = {/10.1145/109025.109042}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-443}, number = {SC-90-16}, year = {1990}, abstract = {Efficient implementations of irregular problems on vector and parallel architectures are generally hard to realize. An important class of problems are Gauß-Seidel iteration schemes applied to irregular data sets. The unstructured data dependences arising there prevent restructuring compilers from generating efficient code for vector or parallel machines. It is shown, how to structure the data dependences by decomposing the underlying data set using graph coloring techniques and by specifying a particular execution order already on the algorithm level. Methods to master the irregularities originating from different types of tasks are proposed. An application is given and some open issues and future developments are discussed.}, language = {en} } @misc{RoitzschKornhuber1990, author = {Roitzsch, Rainer and Kornhuber, Ralf}, title = {BOXES - a Program to Generate Triangulations from a Rectangular Domain Description.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4738}, number = {TR-90-09}, year = {1990}, abstract = {BOXES computes a triangulation from a 2D domain description which consists of an arbitrary set of rectangles. Each rectangle may have attributes to control the triangulating process, define subdomain classes, or specify boundary conditions. The output of the program can be used as a coarse grid for KASKADE or one of its variants. Additional features are extensive checking of the user input, graphical display, and simple editing.}, language = {en} } @misc{AckermannWulkow1990, author = {Ackermann, J{\"o}rg and Wulkow, Michael}, title = {MACRON - A Program Package for Macromalecular Reaction Kinetics.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-423}, number = {SC-90-14}, year = {1990}, abstract = {This paper presents the new program package MACRON for the simulation of macromolecular kinetics including standard chemical reactions. Such problems lead to countable (possibly) infinite systems of ordinary differential equations (CODE's), which are numerically treated by the so-called discrete Galerkin method here. By a chemical compiler the required analytical preprocessing is performed, such that the complete reaction system, standard kinetics as well as macromolecular reactions, can be entered in the chemical formalism. Typical macromolecular reaction steps are chain addition, termination, chain transfer and degradation (cracking). In order to ensure efficiency and reliability, high sophisticated numerical routines are built within the package. MACRON can be used without a detailed knowledge of the used numerical methods. As an illustration the application of MACRON to some realistic problems is presented.}, language = {en} } @misc{WulkowAckermann1990, author = {Wulkow, Michael and Ackermann, J{\"o}rg}, title = {Numerical Treatment of Polyreactions - Recent Developments.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-439}, number = {SC-90-15}, year = {1990}, abstract = {The mathematical modeling of macromolecular reactions leads to countable (possibly infinite) systems of ordinary differential equations (CODE's). This paper reviews two recent developments of the so-called discrete Galerkin method, which has been developed for the numerical treatment of countable systems, which arise e.g. in polymer chemistry. The first approach can be considered as a method of lines with moving basis functions and has been implemented recently in the program package MACRON. The second type of the Galerkin method is characterized by a so-called outer time discretization of the complete problem and an appropriate and efficient solution of the arising subproblems. This method is realized in the research code CODEX.}, language = {en} } @misc{Wulkow1990, author = {Wulkow, Michael}, title = {Numerical Treatment of Countable Systems of Ordinary Differential Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4728}, number = {TR-90-08}, year = {1990}, abstract = {Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and medicine. They can be considered as ordinary differential equations in sequence spaces. In this work, a full adaptive algorithm for the computational treatment of such systems is developed. The method combines time discretization with extrapolation in Hilbert spaces with a discrete Galerkin approach as discretization of the stationary subproblems. The Galerkin method is based on 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 can be assembled either by use of analytical properties of the orthogonal functions or numerically by a multilevel summation algorithm. The resulting algorithm CODEX is applied to many examples of technological interest, in particular from polymer chemistry.}, language = {en} } @misc{Bornemann1990, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations II.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-410}, number = {SC-90-13}, year = {1990}, abstract = {In continuation of part I this paper develops a variable-order time discretization in Hilbert space based on a multiplicative error correction. Matching of time and space errors as explained in part I allows to construct an adaptive multilevel discretization of the parabolic problem. In contrast to the extrapolation method in time, which has been used in part I, the new time discretization allows to separate space and time errors and further to solve fewer elliptic subproblems with less effort, which is essential in view of the application to space dimension greater than one. Numerical examples for space dimension one are included which clearly indicate the improvement.}, language = {en} } @misc{Walter1990, author = {Walter, Artur}, title = {Improvement of Incomplete Factorizations by a Sparse Secant Method.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-401}, number = {SC-90-12}, year = {1990}, abstract = {In the present paper, the improvement of an incomplete factorization of a non-symmetric matrix A is discussed. Starting from the ideas of sparsity preserving quasi-Newton methods, an algorithm is developed which improves the approximation of A by the incomplete factorization maintaining the sparsity structure of the matrices. No renumbering of the unknowns or the admittance of additional fill-in is necessary. The linear convergence of the algorithm is proved under the assumption, that \$ L \$ and \$ U \$* have the same sparsity structure and an incomplete factorization with some reasonable approximation property exits. In combination with this algorithm, the method of incomplete factorization and its several modifications are applicable to a wider class of problems with improved convergence qualities. This is shown by a numerical example. {\bf Key Words:} non-symmetric linear system, sparse secant method, incomplete factorization. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20, 65N30.}, language = {en} } @misc{KornhuberRoitzsch1990, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {Self Adaptive FEM Simulation of Reverse Biased pn-Junctions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-389}, number = {SC-90-10}, year = {1990}, abstract = {The potential distribution of reverse biased pn-junctions can be described by a double obstacle problem for the Laplacian. This problem is solved by a self adaptive Finite Element Method involving automatic termination criteria for the iterative solver, local error estimation and local mesh refinement. Special attention is paid to the efficient resolution of the geometries typically arising in semiconductor device simulation. The algorithm is applied to a reverse biased pn- junction with multi-step field plate and stop- electrode to illustrate its efficiency and reliability.}, language = {en} } @misc{GatermannHohmann1990, author = {Gatermann, Karin and Hohmann, Andreas}, title = {Symbolic Exploitation of Symmetry in Numerical Path-following.}, number = {SC-90-11}, year = {1990}, abstract = {Parameter-dependent systems of nonlinear equations with symmetry are treated by a combination of symbolic and numerical computations. In the symbolic part of the algorithm the complete analysis of the symmetry occurs, and it is here where symmetrical normal forms, symmetry reduced systems, and block diagonal Jacobians are computed. Given a particular problem, the symbolic algorithm can create and compute through the list of possible bifurcations thereby forming a so-called tree of decisions correlated to the different types of symmetry breaking bifurcation points. The remaining part of the algorithm deals with the numerical pathfollowing based on the implicit reparametrisation as suggested and worked out by Deuflhard/Fiedler/Kunkel. The symmetry preserving bifurcation points are computed using recently developed augmented systems incorporating the use of symmetry. {\bf Keywords:} pathfollowing, mixed symbolic-numeric algorithm, parameter-dependent, nonlinear systems, linear representations.}, language = {en} } @misc{Deuflhard1990, author = {Deuflhard, Peter}, title = {Zuses Werk weiterdenken. (Vortrag)}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4717}, number = {TR-90-07}, year = {1990}, language = {de} } @misc{DeuflhardPotra1990, author = {Deuflhard, Peter and Potra, Florian}, title = {Asymptotic Mesh Independence of Newton-Galerkin Methods via a Refined Mysovskii Theorem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-379}, number = {SC-90-09}, year = {1990}, abstract = {The paper presents a theoretical characterization of the often observed asymptotic mesh independence of Newton's method, which means that Newton's method applied to discretized operator equations behaves essentially the same for all sufficiently fine discretizations. The theory does not need any uniform Lipschitz assumptions that were necessary in comparable earlier treatments. The refined Newton-Mysovskii theorem, which will be of interest in a wider context, gives both existence and uniqueness of the solution and quadratic convergence for sufficiently good starting points. Attention is restricted to Galerkin approximations even though similar results should hold for finite difference methods - but corresponding proofs would certainly be more technical. As an illustrative example, adaptive 1-D collocation methods are discussed.}, language = {en} } @misc{ChevalierMelenkWarnatz1990, author = {Chevalier, C. and Melenk, Herbert and Warnatz, J.}, title = {Automatic Generation of Reaction Mechanisms for Description of Oxidation of Higher Hydrocarbons.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-361}, number = {SC-90-08}, year = {1990}, abstract = {Oxidation mechanisms even for rather simple hydrocarbons like heptane consist due to the occurrence of many isomeric structures of thousands of reactions of hundreds of species. The automatic generation of these reaction mechanisms using artificial intelligence means is described. Results are presented for n-heptane-air mixtures, where a hand-written reaction mechanism tested against experimental data is available.}, language = {en} } @misc{DeuflhardHohmann1990, author = {Deuflhard, Peter and Hohmann, Andreas}, title = {Einf{\"u}hrung in die Numerische Mathematik.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4704}, number = {TR-90-06}, year = {1990}, abstract = {Diese Vorlesung ist eine Einf{\"u}hrung in die Numerische Mathematik als einem der drei Bereiche (neben den Naturwissenschaften und der Informatik) des am besten mit dem Begriff Scientific Computing charakterisierten Forschungsgebietes. Aufgabe dieser relativ jungen Wissenschaft ist die Entwicklung von Rechenverfahren f{\"u}r Probleme aus den Naturwissenschaften mit Hilfe mathematischer Methoden.}, language = {de} } @misc{DeuflhardNowakWulkow1990, author = {Deuflhard, Peter and Nowak, Ulrich and Wulkow, Michael}, title = {Recent Developments in Chemical Computing.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-359}, number = {SC-90-07}, year = {1990}, abstract = {The paper surveys three aspects of chemical computing, which seem to play a role in recent developments. First, extrapolation methods for the numerical treatment of differential- algebraic equations are introduced. The associated extrapolation code LIMEX has reached a certain level of sophistication, which makes it a real competitor to the elsewhere widely used multi-step code DASSL of Petzold. Second, adaptive methods of lines for partial differential equations such as those arising in combustion problems are treated. Both static and dynamic regridding techniques are discussed in some detail. Finally, some new ideas about the treatment of the kinetic equations arising from polymer reactions are presented. The new feature of the suggested approach is the application of a Galerkin procedure using sets of orthogonal polynomials over a discrete variable (which, of course, in the case of polymer reactions is the polymer degree). The new approach may open the door to a new reliable low dimensional treatment of complex polymer reactions.}, language = {en} } @misc{DeuflhardFreundWalter1990, author = {Deuflhard, Peter and Freund, R. and Walter, Artur}, title = {Fast Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-330}, number = {SC-90-05}, year = {1990}, abstract = {A family of secant methods based on general rank-1 updates has been revisited in view of the construction of iterative solvers for large non- Hermitian linear systems. As it turns out, both Broydens "good" and "bad" update techniques play a special role - but should be associated with two different line search principles. For Broydens "bad" update technique, a minimum residual principle is natural - thus making it theorectically comparable with a series of well-known algorithms like GMRES. Broydens "good" update technique, however, is shown to be naturally linked with a minimum "next correction" principle - which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized PDE's of convection diffusion type in 2-D with internal layers give a first impression of the possible power of the derived "good" Broyden variant. {\bf Key Words:} nonsymmetric linear system, secant method, rank-1 update, Broydens method, line search, GMRES. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20.}, language = {en} } @misc{Wang1990, author = {Wang, Daoliu}, title = {On Symplectic Difference Schemes for Hamiltonian Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-345}, number = {SC-90-06}, year = {1990}, abstract = {Symplectic difference schemes have been shown to be a right formalism for numerical computation of Hamiltonian systems. They are suitable to long time computation and of good qualitative properties. These properties are ensured by the fact that a symplectic difference scheme approximating to a time-independent Hamiltonian system can be regarded as a perturbed time-dependent Hamiltonian system of the original one. That is, a solution of a symplectic difference scheme is a solution of a certain perturbed time dependent Hamiltonian system evaluated at discrete (time) points. This is the main result of the paper. Moreover, linear symplectic difference schemes approximating to a linear time-independent Hamiltonian system can be regarded as a perturbed time-independent Hamiltonian system. So it has all properties that a linear Hamiltonian system has. Based on these results, stochastic webs and chaos in symplectic difference schemes are also discussed. They will appear in numerical simulation for Hamiltonian systems, even with one degree of freedom.}, language = {en} } @misc{OPUS4-984, title = {Jahresbericht 1989}, issn = {0934-5892}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9847}, number = {1989}, year = {1990}, language = {de} } @misc{Bornemann1990, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations I. General Theory \& 1D-Implementation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-320}, number = {SC-90-04}, year = {1990}, abstract = {A new adaptive multilevel approach for parabolic PDE's is presented. Full adaptivity of the algorithm is realized by combining multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization. In the theoretical part of the paper the existence of asymptotic expansions in terms of time-steps for single-step methods in Hilbert space is established. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The theoretical presentation is independent of space dimension. In this part I of the paper details of the algorithm and numerical examples are given for the 1D case only. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.}, language = {en} } @misc{Hege1990, author = {Hege, Hans-Christian}, title = {Datenabh{\"a}ngigkeitsanalyse und Programmtransformationen auf CRAY-Rechnern mit dem Fortran-Pr{\"a}prozessor fpp}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4685}, number = {TR-90-04}, year = {1990}, abstract = {The FORTRAN preprocessor fpp in the newly introduced Autotasking System of CRAY Research allows automatic vectorization and parallelization on basis of a data dependence analysis. An introduction into data dependence analysis is given, showing how data dependence graphs unveil opportunities for program transformations like vectorization and concurrentization. The report contains a complete description of the preprocessors functionality, its options and directives for increasing the effectiveness of the dependence analyzer and steering the code transformations. Finally, some advice is given for the practical use of fpp on CRAY computers.}, language = {de} } @misc{OPUS4-467, title = {CRAY-Handbuch. Einf{\"u}hrung in die Benutzung der CRAY X-MP unter UNICOS 5.1}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4679}, number = {TR-90-03}, year = {1990}, abstract = {Das CRAY-Handbuch des ZIB beschreibt nach einer grundlegenden bersicht detailliert die Handhabung von UNICOS-Dateien, die Verwendung von UNICOS-Kommandos und schließlich die (f{\"u}r Benutzer allein zugelassene) Ausf{\"u}hrung von Batchjobs auf der CRAY. Neben der Erreichbarkeit der CRAY {\"u}ber das TCP/IP-Netz werden die Compiler FORTRAN, PASCAL und C beschrieben und schließlich auf die Optimierung der Rechenzeit ebenso wie auf die Fehlersuche eingegangen. Ein umfangreicher Anhang f{\"u}hrt bereitgestellte Programmpakete, weiterf{\"u}hrende Literatur, Zahlenbereiche und Zeichendarstellung an der CRAY auf.}, language = {de} } @misc{Gatermann1990, author = {Gatermann, Karin}, title = {Symbolic solution of polynomial equation systems with symmetry.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-311}, number = {SC-90-03}, year = {1990}, abstract = {Systems of polynomial equations often have symmetry. The Buchberger algorithm which may be used for the solution ignores this symmetry. It is restricted to moderate problems unless factorizing polynomials are found leading to several smaller systems. Therefore two methods are presented which use the symmetry to find factorizing polynomials, decompose the ideal and thus decrease the complexitiy of the system a lot. In a first approach projections determine factorizing polynomials as input for the solution process, if the group contains reflections with respect to a hyperplane. Two different ways are described for the symmetric group Sm and the dihedral group Dm. While for Sm subsystems are ignored if they have the same zeros modulo G as another subsystem, for the dihedral group Dm polynomials with more than two factors are generated with the help of the theory of linear representations and restrictions are used as well. These decomposition algorithms are independent of the finally used solution technique. We used the REDUCE package Groebner to solve examples from CAPRASSE, DEMARET and NOONBURG which illustrate the efficiency of our REDUCE program. A short introduction to the theory of linear representations is given. In a second approach problems of another class are transformed such that more factors are found during the computation; these transformations are based on the theory of linear representations. Examples illustrate these approaches. The range of solvable problems is enlarged significantly.}, language = {en} } @misc{Deuflhard1990, author = {Deuflhard, Peter}, title = {Global Inexact Newton Methods for Very Large Scale Nonlinear Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-303}, number = {SC-90-02}, year = {1990}, abstract = {Newton methods for nonlinear problems are known to require the solution of a sequence of linear problems of the same type. For very large scale problems, as understood herein, the arising linear systems can only be solved by iterative methods. Then Newtons iteration appears as outer iteration. The question of interest will be to control the accuracy of the inner iteration such that the convergence speed of Newtons method is preserved. The purpose of the paper is to combine the concept of inexact Newton methods with the concept of the affine invariant exact Newton methods - which is important for problems with ill- conditioned Jacobian matrices (such as typical 2-D or 3-D discretized partial differential equations).}, language = {en} } @misc{MaierhoeferSkorobohatyj1990, author = {Maierh{\"o}fer, Gerhard and Skorobohatyj, Georg}, title = {Implementierung von parallelen Versionen der Gleichungsl{\"o}ser EULEX und EULSIM auf Transputern.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4666}, number = {TR-90-02}, year = {1990}, abstract = {Im vorliegenden Bericht wird die Parallelisierung zweier numerischer Algorithmen zur L{\"o}sung gew{\"o}hnlicher Differentialgleichungssysteme 1.Ordnung (explizite und semi-implizite Euler- Diskretisierung und \$h\$-Extrapolation) beschrieben. Implementiert wurden die Algorithmen mit OCCAM2 unter TDS (Transputer Development System) mit bis zu 4 Transputern T800. Meßwerte f{\"u}r die erreichten Beschleunigungen werden anhand mehrer Beispiele von Differentialgleichungs-systemen angegeben. {\bf Schl{\"u}sselw{\"o}rter:} Adaptive, parallele Systeme; OCCAM2; Transputer; numerische Gleichungsl{\"o}ser; Euler-Diskretisierung; \$h\$-Extrapolation.}, language = {de} } @misc{Gatermann1990, author = {Gatermann, Karin}, title = {Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4652}, number = {TR-90-01}, year = {1990}, abstract = {\$G\$-invariant cubature formulas for numerical integration over n-dimensional, \$G\$- invariant integration regions are computed symbolically. The nodes are the common zeros of some \$d\$-orthogonal polynomials which build an \$H\$-basis of an ideal. Approaches for these polynomials depending on parameters are made with the help of the theory of linear representations of a group \$G\$. This theory is also used for the effective computation of necessary conditions which determines the parameters. Another approach uses invariant theory and gr{\"o}bner bases.}, language = {de} } @misc{Yserentant1989, author = {Yserentant, Harry}, title = {Two Preconditioners Based on the Multi-Level Splitting of Finite Element Spaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-274}, number = {SC-89-09}, year = {1989}, abstract = {The hierarchical basis preconditioner and the recent preconditioner of BRAMBLE, PASCIAK and XU are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; our theory is based exclusively on local properties like the shape regularity of the finite elements.}, language = {en} } @misc{Hoppe1989, author = {Hoppe, Ronald H. W.}, title = {Numerical Solution of Multicomponent Alloy Solidification by Multi-Grid Techniques.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-285}, number = {SC-89-10}, year = {1989}, abstract = {The solidification of an \$ N \$-component alloy is described by an initial boundary value problem for a system of degenerate parabolic equations modelling heat conduction and mass diffusion. Discretizing implicitly in time and by piecewise linear finite elements in the space variables, at each time step the solution of a system of quasivariational inequalities is required. For the numerical solution of that system, a multi-grid algorithm is developed by making use of game theoretic concepts and duality arguments from convex analysis. Finally, the efficiency of the algorithm is demonstrated by displaying numerical results for a ternary alloy.}, language = {en} } @misc{Maierhoefer1989, author = {Maierh{\"o}fer, Gerhard}, title = {Ein paralleler adaptiver Algorithmus f{\"u}r die numerische Integration.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-265}, number = {SC-89-08}, year = {1989}, abstract = {Der Bericht ist die Ausarbeitung eines Vortrages, der auf dem Transputer Anwender Treffen (TAT'89) im September 1989 gehalten wurde. Es wird die Parallelisierung und Implementierung eines adaptiven Algorithmus zur numerischen Integration (Romberg Quadratur) beschrieben. Ausgew{\"a}hlte Meßergebnisse sind enthalten. {\bf Keywords:} Numerischer Algorithmus, Romberg Quadratur, paralleler adaptiver Algorithmus, dynamische Lastverteilung und Prozessorzahl, lokaler Speicher, Nearest-Neighbour-Architektur, Transputer, TDS, OCCAM2.}, language = {de} } @misc{MaierhoeferSkorobohatyj1989, author = {Maierh{\"o}fer, Gerhard and Skorobohatyj, Georg}, title = {Implementierung des parallelen TRAPEX auf Transputern.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4649}, number = {TR-89-08}, year = {1989}, abstract = {Der folgende Bericht ist eine Erg{\"a}nzung des ZIB Technical Report TR 88-05. Entsprechend wird hier nicht auf die grunds{\"a}tzlichen Fragen der Parallelisierbarkeit des sequentiellen TRAPEX eingegangen. Diese sind im TR 88-5 er{\"o}rtert, die dort beschriebenen Algorithmen (vertikale und horizontale Parallelisierung) werden auch f{\"u}r die Transputerarchitektur verwendet. Meßergebnisse sind im letzten Teil angef{\"u}gt.}, language = {de} } @misc{GrammelMaierhoeferSkorobohatyj1989, author = {Grammel, Martin and Maierh{\"o}fer, Gerhard and Skorobohatyj, Georg}, title = {Trapex in POOL-Implementierung eines numerischen Algorithmus in einer parallelen objektorientierten Sprache.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4696}, number = {TR-90-05}, year = {1989}, abstract = {Die Parallelisierung eines vorhandenen sequentiellen Programmes erfolgt im allgemeinen in der Weise, daß es auf Grund einer Analyse der Datenabh{\"a}ngigkeiten in mehrere parallel ausf{\"u}hrbare Teile zerlegt wird, die ihrerseits sequentiell ablaufen und untereinander Daten austauschen. Eine parallele Programmiersprache sollte diese Vorgehensweise unterst{\"u}tzen, indem sie Sprachmittel zur Verf{\"u}gung stellt, mittels derer sich die Modularisierung und Kommunikation bequem formulieren l{\"a}ßt. Nun l{\"a}ßt sich prinzipiell jede vorhandene sequentielle Programmiersprache um solche Konstrukte erweitern; g{\"u}nstiger in Hinblick auf die Strukturierung eines parallelen Programmes erscheint es aber m{\"o}glicherweise, eine solche Programmiersprache zu verwenden, die die genannten Konzepte als elementare zur Verf{\"u}gung stellt. Dies ist offensichtlich bei objektorientierten Programmiersprachen der Fall. Im vorliegenden Bericht wird an Hand eines Beispiels - TRAPEX -, das in POOL-T implementiert wurde, untersucht, inwieweit eine objektorientierte Sprache zur effektiven Programmierung eines numerischen Algorithmus geeignet ist. {\bf Key Words:} parallele objektorientierte Programmiersprache, Message Passing, Abstract Data Typing, Klassenhierarchi, Modul-Import-Mechanismus, Client/Server-Prinzip, Load Balancing, adaptive numerische Algorithmen, Ordnungs- und Schrittweitensteuerung, Romberg-Quadratur.}, language = {de} } @misc{Bornemann1989, author = {Bornemann, Folkmar A.}, title = {Adaptive multilevel discretization in time and space for parabolic partial differential equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4638}, number = {TR-89-07}, year = {1989}, abstract = {The present paper developes an adaptive multilevel approach for parabolic PDE's - as a first step, for one linear scalar equation. Full adaptivity of the algorithm is conceptually realized by simultaneous multilevel discretization in both time and space. Thus the approach combines multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization such as the hierarchical basis method. The algorithmic approach is theoretically backed by careful application of fundamental results from semigroup theory. These results help to establish the existence of asymptotic expansions (in terms of time-steps) in Hilbert space. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The arising space grids are not required to satisfy any quasi- uniformity assumption. Even though the theoretical presentation is independent of space dimension details of the algorithm and numerical examples are given for the 1-D case only. For the 1-D elliptic solver, which is used, an error estimator is established, which works uniformly well for a family of elliptic problems. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.}, language = {en} } @misc{HuamoShuli1989, author = {Huamo, Wu and Shuli, Yang}, title = {MmB-A New Class of Accurate High Resolution Schemes for Conservation Laws in Two Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-248}, number = {SC-89-06}, year = {1989}, abstract = {In this paper we present the MmB schemes, which preserve the local maximum and minimum bounds of the initial data in the smallest union of mesh elements of previous time step containing the domain of dependence of the solution on the mesh element with center at point \$ P \$\ under consideration. In 1-D, the MmB schemes are almost identical with TVD schemes. As well-known, there is no second-order TVD scheme in 2-D, nevertheless, we present here two classes of 2-D second-order accurate MmB-schemes. It is proved that 1-D discrete MmB (or TVD) and 1-D semi-discrete TVD schemes may have second-order accuracy at (nonsonic) critical points, but cannot be of uniformly second-order accurate in the whole neighborhood of the critical points. New accurate high resolution flux limiters are suggested. Numerical results for 1-D and 2-D test problems are given. {\bf Keywords:} Difference scheme, TVD, MmB, flux limiter.}, language = {en} } @misc{BuddeWulkow1989, author = {Budde, Uwe and Wulkow, Michael}, title = {Computation of Molecular Weight Distributions for Free Radical Polymerization Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-253}, number = {SC-89-07}, year = {1989}, abstract = {Modeling of free radical polymerization leads to very large and usually stiff systems of ordinary differential equations which cannot be solved directly in an efficient way. This paper presents the application of a new approach called discrete Galerkin method to a realistic example - the polymerization of methyl methacrylate(MMA). The method is characterized by a Galerkin approximation on the basis of orthogonal polynomials of a discrete variable which represents the polymer degree. It allows the efficient computation of solutions of complete kinetic schemes with time- or moment-dependent reaction coefficients by reducing the complexity to a few differential equations. The approximation error can be controlled by an error estimation. In the case of MMA polymerization a reduction of computational effort by a factor of about 25 compared to a standard method can be obtained for the quasi-steady-state approximation of the model. In addition solutions of the instationary kinetic scheme can be easily computed.}, language = {en} } @misc{Huamo1989, author = {Huamo, Wu}, title = {On the Possible Accuracy of TVD Schemes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-212}, number = {SC-89-03}, year = {1989}, abstract = {The paper presents a detailed analysis of the possible accuracy available for TVD schemes in one dimension with emphasis to the semi-discrete 1-D TVD schemes. The analysis shows that the widely accepted statement [1] of degeneration of accuracy at critical points for TVD schemes should be corrected. We have theorem: TVD schemes using flux limiters \$ \varphi \$ of the form [1], [2] may be second-order accurate at critical points if \$ \varphi \$ (3) + \$ \varphi \$(-1) = 2, but cannot be uniformly second-order accurate in the whole neighborhood of critical point. If \$ \varphi \$(1) = 1, then the TVD schemes are second-order accurate in the region of smooth solutions sufficiently far from the critical points. Two ways are suggested to improve the accuracy. Numerical example is given. {\bf Keywords:} Semi-discrete schemes, TVD, flux limiter, degeneration of accuracy.}, language = {en} } @misc{Moeller1989, author = {M{\"o}ller, H. Michael}, title = {Multivariate Rational Interpolation: Reconstruction of Rational Functions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-227}, number = {SC-89-04}, year = {1989}, abstract = {In this paper we consider the problem of reconstructing a multivariate rational function, when only its values at sufficiently many points are known. We use for the reconstruction of bivariate rational functions a bivariate rational interpolation operator investigated by Siemaszko [7] and a new one, compare both by examples in a Computer Algebra system, and present their multivariate generalizations. {\bf Keywords:} Multivariate rational interpolation, reconstruction, symbolic computation.}, language = {en} }