Refine
Year of publication
Document Type
- ZIB-Report (25)
- In Proceedings (8)
- Article (3)
- Book chapter (2)
- Doctoral Thesis (1)
Is part of the Bibliography
- no (39)
Keywords
Institute
- ZIB Allgemein (24)
- Numerical Mathematics (6)
- Computational Medicine (1)
For the simulation of one-dimensional flame configurations reliabl e numerical tools are needed which have to be both highly efficient (large num ber of parametric calculations) and at the same time accurate (in order t o avoid numerical errors). This can only be accomplished using fully adapt ive discretization techniques both in space and time together with a c ontrol of the discretization error. We present a method which accomplishes this task. It is based on a n adative MOL (method of lines) treatment. Space discretization is done by means of finite difference approxi mations on non-uniform grids. Time is discretized by the linearly-implicit Euler method. In order to control the discretization errors an extrapolation pro cedure is used in space and time. Results are presented for simple laser-induced ignition processes. The method, however, can be applied to other combustion processes, too.
Dieser Report enthält die Ergebnisse der Untersuchungen, die gemäss dem Forschungs-- und Entwicklungsvertrag Gravity zwischen dem GeoForschungsZentrum Potsdam und dem Konrad--Zuse--Zentrum Berlin vorgenommen wurden. Die damit vereinbarte wissenschaftliche Kooperation hat die folgenden Ziele: \item{die am GFZ vorhandenen Algorithmen und Softwaremodule auf ihre Effizienz hisichtlich Nutzung der Rechnerresourcen zu untersuchen und Lösungen für einen schnelleren Datendurchsatz zu entwickeln und zu implementieren,} \item{Methoden zur Regularisierung und Lösung schlecht konditionierter Normalgleichungssysteme (für Schwerefeldkoeffizienten) kritisch zu untersuchen und eine mathematisch objektive Strategie der Regularisierung zu entwickeln, und} \item{insbesondere im Hinblick auf die Anforderungen bei GRACE, verschiedene Bahnintegrationsverfahren hinsichtlich ihrer numerischen Genauigkeit und Einsatzmöglichkeiten zu untersuchen.}
Sensitivity analysis of linearly-implicit differential-algebraic systems by one-step extrapolation
(2002)
In this work we present an approach for the sensitivity analysis of linearly-implicit differential-algebraic equation systems. Solutions for both, states and sensitivities are obtained by applying an extrapolated linearly implicit Euler discretization scheme. This approach is compared to the widely used sensitivity extensions of multi-step BDF methods by means of case studies. Especially, we point out the benefit of this method in the context of dynamic optimization using the sequential approach.
The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and inexact Newton methods, where an inner iteration is used instead, are synoptically presented, both in affine invariant convergence theory and in numerical experiments. The three types of algorithms are: (a) affine covariant (formerly just called affine invariant) Newton algorithms, oriented toward the iterative errors, (b) affine contravariant Newton algorithms, based on iterative residual norms, and (c) affine conjugate Newton algorithms for convex optimization problems and discrete nonlinear elliptic PDEs.
The mathematical modeling of a special modular catalytic reactor kit leads to a system of partial differential equation in two space dimensions. As customary, this model contains unconfident physical parameters, which may be adapted to fit experimental data. To solve this nonlinear least squares problem we apply a damped Gauss-Newton method. A method of lines approach is used to evaluate the associated model equations. By an a priori spatial discretization a large DAE system is derived and integrated with an adaptive, linearly-implicit extrapolation method. For sensitivity evaluation we apply an internal numerical differentiation technique, which reuses linear algebra information from the model integration. In order not to interfere the control of the Gauss-Newton iteration these computations are done usually very accurately and, therefore, very costly. To overcome this difficulty, we discuss several accuracy adaptation strategies, e.g., a master-slave mode. Finally, we present some numerical experiments.
Bessel'scher Irrgarten
(2007)
Dieser Artikel berichtet über eine erfolgreiche Schüleraktivität, die seit Jahren am Zuse-Institut Berlin (ZIB) bei Besuchen von Schülergruppen erprobt und verfeinert worden ist. Das hier zusammengestellte Material ist gedacht als Basis für eine Unterrichtseinheit in Leistungskursen Mathematik an Gymnasien. Inhaltlich wird von einem zwar für Schüler (wie auch Lehrer) neuen, aber leicht fasslichen Gegenstand ausgegangen: der Drei-Term-Rekursion für Besselfunktionen. Die Struktur wird erklärt und in ein kleines Programm umgesetzt. Dazu teilen sich die Schüler selbstorganisierend in Gruppen ein, die mit unterschiedlichen Taschenrechnern "um die Wette" rechnen. Die Schüler und Schülerinnen erfahren unmittelbar die katastrophale Wirkung von an sich kleinen'' Rundungsfehlern, sie landen -- ebenso wie der Supercomputer des ZIB -- im Bessel'schen Irrgarten''. Die auftretenden Phänomene werden mathematisch elementar erklärt, wobei lediglich auf das Konzept der linearen Unabhängigkeit zurückgegriffen wird. Das dabei gewonnene vertiefte Verständnis fließt ein in die Konstruktion eines klassischen Algorithmus sowie eines wesentlich verbesserten Horner-artigen Algorithmus.
ZGUI-Handbuch
(1996)
In diesem Handbuch werden die Bausteine zum Aufbau einer graphischen Benutzeroberfläche mit {\tt ZGUI} beschrieben. Auf der einen Seite stehen die Tcl/Tk--Prozeduren, die die graphischen Elemente definieren. Die Beschreibung der Anwendung der Prozeduren und der Interaktionen der Elemente bildet den ersten Teil des Handbuches. Auf der anderen Seite stehen die Anforderungen an Anwendungen, die mit einer {\tt ZGUI}--Benutzeroberlfäche gesteuert werden sollen. Hier findet man im Handbuch die Beschreibung der Anwendungsprogrammierschnittstelle (application programming interface, API).
We present parallel formulations of the well established extrapolation algorithms EULSIM and LIMEX and its implementation on a distributed memory architecture. The discretization of partial differential equations by the method of lines yields large banded systems, which can be efficiently solved in parallel only by iterative methods. Polynomial preconditioning with a Neumann series expansion combined with an overlapping domain decomposition appears as a very efficient, robust and highly scalable preconditioner for different iterative solvers. A further advantage of this preconditioner is that all computation can be restricted to the overlap region as long as the subdomain problems are solved exactly. With this approach the iterative algorithms operate on very short vectors, the length of the vectors depends only on the number of gridpoints in the overlap region and the number of processors, but not on the size of the linear system. As the most reliable and fast iterative methods based on this preconditioning scheme appeared GMRES or FOM and BICGSTAB. To further reduce the number of iterations in GMRES or FOM we can reuse the Krylov-spaces constructed in preceeding extrapolation steps. The implementation of the method within the program LIMEX results in a highly parallel and scalable program for solving differential algebraic problems getting an almost linear speedup up to 64 processors even for medium size problems. Results are presented for a difficult application from chemical engineering simulating the formation of aerosols in industrial gas exhaust purification.
A new method for the numerical solution of highly nonlinear, coupled systems of parabolic differential equations in one space dimension is presented. The approach is based on a classical method of lines treatment. Time discretization is done by means of the semi--implicit Euler discretization. Space discretization is done with finite differences on non--uniform grids. Both basic discretizations are coupled with extrapolation techniques. With respect to time the extrapolation is of variable order whereas just one extrapolation step is done in space. Based on local error estimates for both, the time and the space discretization error, the accuracy of the numerical approximation is controlled and the discretization stepsizes are adapted automatically and simultaneously. Besides the local adaptation of the space grids after each integration step (static regridding), the grid may even move within each integration step (dynamic regridding). Thus, the whole algorithm has a high degree of adaptivity. Due to this fact, challenging problems from applications can be solved in an efficient and robust way.
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.
A fully adaptive method is presented for the numerical solution of highly nonlinear, coupled systems of parabolic differential equations in one space dimension. Time discretization is by means of the linearly--implicit Euler discretization. Space discretization is by finite differences on non--uniform grids. Both basic discretizations are combined with extrapolation. Based on local error estimates for both the time and the space discretization error, the accuracy of the numerical approximation is controlled and the discretization stepsizes are adapted automatically and simultaneously. The algorithm is implemented in a user friendly software package, PDEX1M. To be a powerful tool for users coming from applications the package has been equipped with some additional useful devices.
We study the parallelization of linearly--implicit extrapolation codes for the solution of large scale PDE systems and differential algebraic equations on distributed memory machines. The main advantage of these algorithms is that they enable adapativity both in time and space. Additive Krylov--Schwarz methods yield high parallel perfomance for such extrapolation methods. Our approach combines a slightly overlapping domain decomposition together with a polynomial block Neumann preconditioner and a reduced system technique. Furthermore we get important advantages through the explicit computation of the matrix--products of the preconditioner and the matrix of the linear system. The parallel algorithms exhibit scalability up to 64 processors already for medium--sized test problems. We show that the codes are really efficient in large application systems for chemical engineering problems.
Dynamical simulation of industrially relevant processes strongly advises the use of algorithms, which are {\em adaptive} both in time and in space discretization. The paper presents two alternatives: (a) a fully adaptive method of lines approach, which is based on finite difference methods and essentially applicable to 1D problems; (b) a fully adaptive Rothe method, which is based on a fast multilevel finite element method and applicable to 1D up to 3D.
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.
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.
New, highly efficient techniques for the numerical simulation and parameter identification of large chemical reaction systems are surveyed. The survey particularly addresses to chemists, which are understood to be potential users of the distributed software packages. These packages are written in the form of interactive dialogue systems - thus enabling chemists to concentrate on the chemistry of their problem. Large scale examples from chemical research environments are included.
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.
Die Autoren sind sich dar{ü}ber im klaren, daß zahlreiche Imponderabilien die Ergebnisse der vorgelegten Modellrechnungen beeinflussen k{ö}nnen. Immerhin repräsentiert jedoch das vorgelegte mathematische Modell die Summe unserer derzeitigen Kenntnisse zur Fragestellung AIDS--Epidemie; neu hinzukommende Erkenntnisse lassen sich rasch einarbeiten und in ihren Konsequenzen überschauen. Die Entwicklung neuer effizienter numerischer Methoden spielte eine Schlüsselrolle bei der tatsächlichen Simulation dieses umfangreichen realitätsnahen mathematischen Modells.
A numerical method for the treatment of moving discontinuities in the model equations of chemical engineering systems is presented. The derived model describing the effects of condensation and evaporation in a regenerative air to air heat exchanger yields an illustrative example for these so called moving boundary problems. The presented adaptive moving grid method is based on the algorithm {\sc Pdex} for parabolic partial differential equations. It is shown that the method is suited for problems where the arising discontinuities cause low rates of convergence if the equations are solved with a static grid.
Benchmarking of ODE methods has a long tradition. Several sets of test problems have been developed and new problems are still collected. So, a whole variety of problems can be used to check the efficiency of a method under investigation. In general, efficiency is measured by the amount of work whi ch is necessary to get a reliable solution for a prescribed accura cy. In order to quantify the term ``amount of work'' usually not only the computing time is measured but also the number of calls of functional units. To quantify the term ``quality of a numerical solution'' usually the $l_2$--norm of the true error at the final point of the integration interval is used. In our contribution we first discuss some general aspects of benchmarking. Then we present a new test frame which allows to solve typical benchmark problems with some of the state of the art integrators within a unified framework. Finally we show some results of our benchmark tests. Part of the test frame can be used interactively in the World Wide Web.
We study the parallelization of linearly--implicit extrapolation methods for the solution of large scale systems of differential algebraic equations arising in a method of lines (MOL ) treatment of partial differential equations. In our approach we combine a slightly overlapping domain decomposi tion together with a polynomial block Neumann preconditioner. Through the explicit computation of the matrix products of the pre conditioner and the system matrix a significant gain in overall efficiency is achieved for medium--sized problems. The parallel algorithm exhibits a good scalability up to 32 proces sors on a Cray T3E. Preliminary results for computations on a workstation cluster are reported.
Der Entwurf und die Implementierung des auf Tcl/Tk basierenden Werkzeugkastens ZGUI wird beschrieben und an einigen Beispielen erläutert. ZGUI unterstützt die Entwicklung einer graphischen Benutzeroberfläche (GUI) für die am ZIB erstellte numerische Software. Es sollen folgende Ziele erreicht werden: \begin{itemize} \item einfaches Ausprobieren anhand vordefinierter Testprobleme,\vspace*{-2mm} \item Kennenlernen numerischer Steuergrö\ss en und Verfahrensvarianten,\vspace*{-2mm} \item einfache Eingabe neuer Probleme,\vspace*{-2mm} \item einfache Nutzung graphischer Ausgabemöglichkeiten und\vspace*{-2mm} \item einheitliche Darstellung gleicher oder ähnlicher Optionen. \end{itemize}
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.
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.
Sensitivity analysis of linearly-implicit differential-algebraic systems by one-step extrapolation
(2004)
Dust formation {in brown dwarf atmospheres} is studied by utilizing a model for driven turbulence in the mesoscopic scale regime. We apply a pseudo-spectral method where waves are created and superimposed {within} a {limited} wavenumber interval. The turbulent kinetic energy distribution follows the Kolmogoroff spectrum which is assumed to be the most likely value. Such superimposed, stochastic waves may occur in a convectively active environment. They cause nucleation fronts and nucleation events and thereby initiate the dust formation process which { continues until} all condensible material is consumed. Small disturbances {are found to} have a large impact on the dust forming system. An initially dust-hostile region, which may originally be optically thin, becomes optically thick in a patchy way showing considerable variations in the dust properties during the formation process. The dust appears in lanes and curls as a result of the interaction with waves, i.e. turbulence, which form larger and larger structures with time. Aiming on a physical understanding of the variability of brown dwarfs, related to structure formation in substellar atmospheres, we work out first necessary criteria for small-scale closure models to be applied in macroscopic simulations of dust forming astrophysical systems.
Modelling, parameter identification, and simulation play an important rôle in Systems Biology. In recent years, various software packages have been established for scientific use in both licencing types, open source as well as commercial. Many of these codes are based on inefficient and mathematically outdated algorithms. By introducing the package BioPARKIN recently developed at ZIB, we want to improve this situation significantly. The development of the software BioPARKIN involves long standing mathematical ideas that, however, have not yet entered the field of Systems Biology, as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks. BioPARKIN originates from the package PARKIN, written by P.Deuflhard and U.Nowak, that has been applied successfully for parameter identification in physical chemistry for many years.