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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.
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.
Automatic Generation of Reaction Mechanisms for Description of Oxidation of Higher Hydrocarbons.
(1990)
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.
Diese Vorlesung ist eine Einführung 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 für Probleme aus den Naturwissenschaften mit Hilfe mathematischer Methoden.
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.
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.
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.
Jahresbericht 1989
(1990)
An Adaptive Multilevel Approach to Parabolic Equations I. General Theory & 1D-Implementation.
(1990)
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.
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.
Das CRAY-Handbuch des ZIB beschreibt nach einer grundlegenden bersicht detailliert die Handhabung von UNICOS-Dateien, die Verwendung von UNICOS-Kommandos und schließlich die (für Benutzer allein zugelassene) Ausführung von Batchjobs auf der CRAY. Neben der Erreichbarkeit der CRAY über 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 führt bereitgestellte Programmpakete, weiterführende Literatur, Zahlenbereiche und Zeichendarstellung an der CRAY auf.
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.
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).
Implementierung von parallelen Versionen der Gleichungslöser EULEX und EULSIM auf Transputern.
(1990)
Im vorliegenden Bericht wird die Parallelisierung zweier numerischer Algorithmen zur Lösung gewöhnlicher 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 für die erreichten Beschleunigungen werden anhand mehrer Beispiele von Differentialgleichungs-systemen angegeben. {\bf Schlüsselwörter:} Adaptive, parallele Systeme; OCCAM2; Transputer; numerische Gleichungslöser; Euler-Diskretisierung; $h$-Extrapolation.
$G$-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 gröbner bases.
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.
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.
Der folgende Bericht ist eine Ergänzung des ZIB Technical Report TR 88-05. Entsprechend wird hier nicht auf die grundsätzlichen Fragen der Parallelisierbarkeit des sequentiellen TRAPEX eingegangen. Diese sind im TR 88-5 erörtert, die dort beschriebenen Algorithmen (vertikale und horizontale Parallelisierung) werden auch für die Transputerarchitektur verwendet. Meßergebnisse sind im letzten Teil angefügt.
Die Parallelisierung eines vorhandenen sequentiellen Programmes erfolgt im allgemeinen in der Weise, daß es auf Grund einer Analyse der Datenabhängigkeiten in mehrere parallel ausführbare Teile zerlegt wird, die ihrerseits sequentiell ablaufen und untereinander Daten austauschen. Eine parallele Programmiersprache sollte diese Vorgehensweise unterstützen, indem sie Sprachmittel zur Verfügung stellt, mittels derer sich die Modularisierung und Kommunikation bequem formulieren läßt. Nun läßt sich prinzipiell jede vorhandene sequentielle Programmiersprache um solche Konstrukte erweitern; günstiger in Hinblick auf die Strukturierung eines parallelen Programmes erscheint es aber möglicherweise, eine solche Programmiersprache zu verwenden, die die genannten Konzepte als elementare zur Verfügung 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.
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. Ausgewählte Meßergebnisse sind enthalten. {\bf Keywords:} Numerischer Algorithmus, Romberg Quadratur, paralleler adaptiver Algorithmus, dynamische Lastverteilung und Prozessorzahl, lokaler Speicher, Nearest-Neighbour-Architektur, Transputer, TDS, OCCAM2.
Adaptive multilevel discretization in time and space for parabolic partial differential equations.
(1989)
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.
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.
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.
The SPARC processor is a RISC (Reduced Instruction Set Computer) microcomputer, built into the SUN4 workstations. Since RISC processors are very well-suited for LISP processing, the implementation of a dialect of LISP (Portable Standard LISP, PSL) boded well for a great speed-up in comparison with other types of microcomputers. A first approach was done at The RAND Corporation in Santa Monica, which was derived from classical processor types like MC68000 or VAX. At the Konrad- Zuse-Zentrum für Informationstechnik Berlin (ZIB) that initial implementation was redesigned in order to adapt PSL to the specific features of the SPARC processor. The present implementation, in some parts, is very close to Cray PSL version also done in ZIB. Some timing informations are given in the appendix.
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.
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.
We propose an anisotropic refinement strategy which is specially designed for the efficient numerical resolution of internal and boundary layers. This strategy is based on the directed refinement of single triangles together with adaptive multilevel grid orientation. It is demonstrated by several numerical examples that compared to usual methods, the new anisotropic refinement ends up in more stable and more accurate solutions at much less computational cost. {\bf Keywords:} Adaptive finite elements, directed refinement, adaptive grid orientation, convection diffusion equation, internal and boundary layers.
KASKADE Programmer's Manual.
(1989)
KASKADE User's Manual.
(1989)
Jahresbericht 1988
(1989)
In this paper we introduce the concept of restricted singular values (RSV's) of matrix triplets. A theorem concerning the RSV's of a general matrix triplet $ (A,B,C) $, where $ A \in C^{m\times n} $, $B\in C^{m\times p} $ and $ C\in C^{q\times n} $, which is called restricted singular value decomposition (RSVD) of matrix triplets, is derived. This result generalizes the wellknown SVD, GSVD and the recently proposed product induced SVD (PSVD). Connection of RSV's with the problem of determination of matrix rank under restricted perturbation is also discussed. {\bf Keywords:} Matrix rank, singular values, generalized singular values, product induced singular values, restricted singular values, matrix decompositions.
Das voliegende Skriptum entstand aus einer Vorlesung, die ich im WS 87/88 an der Freien Universität Berlin im Fachbereich Mathematik gehalten habe. Mein ursprüngliches Vorlesungsmanuskript wurde von Herrn F. Bornemann in weiten Teilen überarbeitet, reorganisiert und substantiell ergänzt. Der Inhalt stammt größtenteils aus Originalarbeiten jüngeren Datums. Darüberhinaus finden sich zahlreiche Teile, die aus meiner jahrelangen Beschäftigung mit dem Thema entstanden aber unpubliziert geblieben sind. Das Skriptum erhebt nicht den Anspruch, ein Lehrbuch zu sein. Es war zunächst als Ausarbeitung für meinen studentischen Hörerkreis sowie als internes Arbeitspapier für das ZIB bestimmt. Die Kunde von der bloßen Existenz eines solchen Skriptums hat jedoch zu einer derart regen Nachfrage geführt, daß es hiermit als Technischer Report des ZIB einer breiteren ffentlichkeit zugänglich gemacht werden soll. In der vorliegenden Form richtet es sich in erster Linie an Mathematiker; es soll sich jedoch auch für Naturwissenschaftler und Ingenieure eignen, die sich einen Einblick in den theoretischen und algorithmischen Hintergrund der von ihnen verwendeten wissenschaftlichen Software verschaffen wollen.
A new approach for the discretisation of hyperbolic conservation laws via a finite element method is developed and analysed. Appropriate forms of the Eulers equation of gas dynamic are considered to employ the algorithm in a reasonable way for this system of nonlinear equations. Both mathematical and physical stability results are obtained. A main part of the paper is devoted to the convergence proof with energy methods under strong regularity of the solution of a scalar nonlinear conservation law. Some hints on the implementation and numerical results for the calculation of transonic gasflow through a Laval nozzle are given. The necessary amount of numerical work is compared to an established finite difference method and the efficiency of the algorithm is shown. A survey on recent literature about finite element methods for hyperbolic problem is included.
A Numerical Algorithm for Computing the Restricted Singular Value Decomposition of Matrix Triplets.
(1989)
This paper presents a numerical algorithm for computing the restricted singular value decomposition of matrix triplets (RSVD). It is shown that one can use unitary transformations to separate the regular part from a general matrix triplet. After preprocessing on the regular part, one obtains a matrix triplet consisting of three upper triangular matrices of the same dimensions. The RSVD of this special matrix triplet is computed using the implicit Kogbetliantz technique. The algorithm is well suited for parallel computation. {\bf Keywords:} Restricted singular values, matrix triplets, unitary transformations, implicit Kogbetliantz technique.
Ausgangspunkt bei der Behandlung konvektiv dominierter, elliptischer Probleme sind die bekannten hierarchischen Finite-Element-Methoden für den rein elliptischen Fall. Als stabile Erweiterung des Standard-Galerkin-Verfahrens wird das Stromlinien-Diffusions-Verfahren durch physikalische Überlegungen motiviert und kurz diskutiert. Anschließend zeigen wir, daß diese Methode erst in Verbindung mit einer hier erstmals vorgestellten lokalen Ausrichtung der Kanten wirksam eingesetzt werden kann. Zusammen mit einer ebenfalls neu entwickelten richtungsorientierten Verfeinerungsstrategie erhält man eine erheblich stabilere, genauere und schnellere Auflösung von Grenzschichten als mit herkömmlichen Methoden.
This paper describes some ways of transforming a sequential adaptive algorithm for numerical evaluation of an integral (Romberg- Quadrature with polynomial Extrapolation method) to a parallel one, such as have been implementad by the authors. We developed an algorithm which preserves the sequential adaptivity and is capable of running on various architectures, dynamically controlling the number of avtive processors depending on the problem. To study the time behaviour, we used the simulator SUSI(SUprenum SImulatur) which is able to simulate SUPRENUM-like architectures. Results are given in part 3. {\bf Keywords:} Romberg quadrature; numerical integration; parallel adaptive algorithm; SUSI; simulation; SUPRENUM; MIMD-Fortran; computer architecture; granularity; load balacing; master-slave-principle.
Induction heating of large steel slabs can be described by a coupled system of nonlinear evolution equations of Stefan type representing the temporal and spatial distribution of the induced magnetic field and the generated temperature within the slab. Discretizing these equations implicitly in time and by finite differences in space, at each time step the solution of a system of difference inclusions is required. For the solution of that system two multi-grid algorithms are given which combined with a nested iteration type continuation strategy to proceed in time result in computationally highly efficient schemes for the numerical simulation of the induction heating process. {\bf Keywords:} induction heating, system of two coupled Stefan equations, multi-grid algorithms. {\bf Subject Classification:} AMS(MOS): 35K60, 35R35, 65H10, 65N05, 65N20, 78A25, 78A55.
The paper presents a new application of computer algebra to the treatment of steady states of reaction systems. The method is based on the Buchberger algorithm. This algorithm was modified such that it can exploit the special structure of the equations derived from reaction systems, so even large systems can be handled. In contrast to numerical approximation techniques, the algebraic solution gives a complete and definite overview of the solution space and it is even applicable when parameter values are unknown or undetermined. The algorithm, its adaptation to the problem class and its application to selected examples are presented.
Computational Treatment of Polyreaction Kinetics by Orthogonal Polynomials of a Discrete Variable.
(1988)
The paper presents a new approach to the computational treatment of polyreaction kinetics. This approach is characterized by a Galerkin method based on orthogonal polynomials of a discrete variable, the polymer degree (or chain length). In comparison with the known competing approaches (statistical moment treatment, Galerkin methods for continuous polymer models), the suggested method is shown to avoid the disadvantages and preserve the adventages of either of them. The basic idea of the method is the construction of a discrete inner product associated with a reasonably chosen probability density function. For the so-called Schulz-Flory distribution one thus obtains the discrete Laguerre polynomials, whereas the Poisson distribution leads to the Charlier polynomials. Numerical experiments for selected polyreaction mechanisms illustrate the efficiency of the proposed method.
The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods.
Jahresbericht 1987
(1988)
The Buchberger algorithm is a basic tool for the solution of systems of polynomial equations in an environment of computer algebra applications. A model for overlapped processing of different steps of the algorithm is presented, which uses the data structure of the polynomials (distributive representation) for synchronization. The model can be applied for multi processors with fast access to shared data. It is tested with Cray X-MP multi processors based on a parallel version of Portable Standard Lisp (PSL 3.4).
This document describes operating procedures for running REDUCE specific to the CRAY 1 and CRAY X-MP computers running the Operating System UNICOS. The document was derived from the corresponding document for Vax/UNIX prepared by A. C. Hearn and L. R. Seward, The Rand Corporation, Santa Monica, (CP85).
This guide describes the CRAY/UNICOS REDUCE distribution tape and the procedures for installing, testing and maintaining REDUCE on a CRAY 1 or CRAY X-MP running UNICOS. This document was derived from the corresponding document for Vax/UNIX prepared by A. C. Hearn and L.R. Seward, The Rand Corporation, Santa Monica, publication CP84.
TDLG 3.1 dient zur Darstellung von dreidimensionalen Objekten, die sich aus offenen Linienzügen, Dreieck- und Viereckelementen zusammensetzen. Dabei können verschiedene Visualisierungstechniken benutzt werden: Drahtmodell, Schattierung mit und ohne Lichtquelle, Hidden Surface Removal. Durch Verwendung von Konvertierungsprogrammen können auch IGES- und VDAFS- Dateien dargestellt werden.
Gröbner bases are the main tool for solving systems of algebraic equations and some other problems in connection with polynomial ideals using Computer Algebra Systems. The procedure for the computation of Gröbner bases in REDUCE 3.3 has been modified in order to solve more complicated algebraic systems of equations by some general improvements and by some tools based on the specific resources of the CRAY X-MP. We present this modification and illustrate it by examples.
We model a symmetric system of coupled oscillators as a graph with symmetry group $\gamma$. Each vertex of the graph represents an "oscillator" or a "cell" of reactants. The magnitude (concentration) of the reactants in the $ i $ th cell is represented by a vector $ x^i $. The edges represent the coupling of the cells. The cells are assumed to evolve by identical reaction-diffusion equation which depends on the sum of the reactants in the nearest neighbors. Thus the dynamics of the system is described by a nonlinear differential system \begin{flushleft} \[ \mbox {(*) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \dot{x}^i = f (x^i,\sum_{j \in N_i} x^j), \mbox { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \] \end{flushleft} where the sum ranges over the set $ N_i $ of neighbors of cell $ i $ . If $ f $ also has a symmetry (e.g., oddness), there are geometric conditions on the graph such that the nonlinear system $ (*) $ decouples globally into a product flow on certain sums of isotropy subspaces. Thus we may detect higher-dimensional tori of solutions of $ (*) $ which are not amenable to other types of analysis. We present a number of examples, such as bipartite graphs, complete graphs, the square, the octahedron, and a 6-dimensional cube.