TY - GEN A1 - Deuflhard, Peter T1 - A Study of Lanczos-Type Iterations for Symmetric Indefinite Linear Systems N2 - The Lanczos iteration for symmetric indefinite linear systems seems to be well--known for quite a while. However, in order to modify it with the aim of improved performance, the present paper studies certain aspects in terms of an adjoint scalar three--term recurrence. Thus, at least a different view is opened. Moreover, an alternative $3n$--implementation in terms of the Euclidean orthogonal basis has been found that easily permits generalizations. The study is understood as a start--off for further numerical investigations and experiments. T3 - ZIB-Report - SC-93-06 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1029 ER - TY - GEN A1 - Schöpf, R. A1 - Deuflhard, Peter T1 - OCCAL: A mixed symbolic- numeric Optimal Control CALculator. N2 - 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. T3 - ZIB-Report - SC-91-13 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-638 ER - TY - GEN A1 - Deuflhard, Peter A1 - Leinen, P. A1 - Yserentant, Harry T1 - Concepts of an Adaptive Hierarchical Finite Element Code. N2 - 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. T3 - ZIB-Report - SC-88-05 KW - finite elements KW - hierarchical basis KW - adaptive mesh refinement KW - preconditioned conjugate gradient methods Y1 - 1988 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-131 ER - TY - GEN A1 - Deuflhard, Peter A1 - Wulkow, Michael T1 - Computational Treatment of Polyreaction Kinetics by Orthogonal Polynomials of a Discrete Variable. N2 - 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. T3 - ZIB-Report - SC-88-06 Y1 - 1988 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-140 ER - TY - GEN A1 - Deuflhard, Peter A1 - Potra, Florian T1 - A Refined Gauss-Newton- Mysovskii Theorem. N2 - 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. T3 - ZIB-Report - SC-91-04 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-541 ER - TY - GEN A1 - Wulkow, Michael A1 - Deuflhard, Peter T1 - Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions. N2 - 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. T3 - ZIB-Report - SC-90-01 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-292 ER - TY - GEN A1 - Deuflhard, Peter T1 - Global Inexact Newton Methods for Very Large Scale Nonlinear Problems. N2 - 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). T3 - ZIB-Report - SC-90-02 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-303 ER - TY - GEN A1 - Deuflhard, Peter A1 - Freund, R. A1 - Walter, Artur T1 - Fast Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems. N2 - 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. T3 - ZIB-Report - SC-90-05 KW - nonsymmetric linear systems KW - secant method KW - rank-1 update KW - Broyden's method KW - line search KW - GMRES Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-330 ER - TY - GEN A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Wulkow, Michael T1 - Recent Developments in Chemical Computing. N2 - 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. T3 - ZIB-Report - SC-90-07 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-359 ER - TY - GEN A1 - Deuflhard, Peter A1 - Potra, Florian T1 - Asymptotic Mesh Independence of Newton-Galerkin Methods via a Refined Mysovskii Theorem. N2 - 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. T3 - ZIB-Report - SC-90-09 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-379 ER -