@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} }