@misc{DeuflhardNowakPoehleetal., 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}, 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{Bornemann, 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}, 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{DeuflhardPotra, 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}, 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{HoppeKornhuber, 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}, 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{GajewskiEdDeuflhardMarkowiched, 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}, 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{Bornemann, 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}, 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{FiedlerScheurle, 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}, 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{LangWalter, 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}, 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{Schoeffel, 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}, 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{Griewank, 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}, 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} }