@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{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{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{DeuflhardNowak1986, author = {Deuflhard, Peter and Nowak, Ulrich}, title = {Efficient Numerical Simulation and Identification of Large Chemical Reaction Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14}, number = {SC-86-01}, year = {1986}, abstract = {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.}, language = {en} } @misc{Deuflhard1987, author = {Deuflhard, Peter}, title = {Uniqueness Theorems for Stiff ODE Initial Value Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53}, number = {SC-87-03}, year = {1987}, abstract = {The paper presents a new uniqueness theory for ODE initial value problems, derived in view of numerical stiff integration. The theory supplies stepsize bounds for stiff integrators that can easily be estimated in extrapolation methods. The additional devices lead to a significant speed-up of computations - in particular in combustion PDE problems.}, language = {en} } @misc{Deuflhard1987, author = {Deuflhard, Peter}, title = {A Note on Extrapolation Methods for Second Order ODE Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-79}, number = {SC-87-05}, year = {1987}, abstract = {A slight modification of the extended Stoermer discretization for non self-adjoint second order ODE systems is derived on the basis of a simple stability analysis. This discretization easily extends to implicit ODE systems, which are known to arise e.g. in mechanical engineering. In addition, a special variant of semi-implicit Euler discretization is proposed, which essentially treats the state variables explicitly, but their derivatives implicitly. Numerical tests over critical parameter values of the van der Pol oscillator illustrate the domain of efficiency of the suggested discretizations.}, language = {en} } @misc{SchuetteFischerHuisingaetal.1999, author = {Sch{\"u}tte, Christof and Fischer, Alexander and Huisinga, Wilhelm and Deuflhard, Peter}, title = {A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3889}, number = {SC-98-45}, year = {1999}, abstract = {Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator \$T\$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of \$T\$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix \$P\$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of \$T\$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.}, language = {en} } @misc{EhrigNowakDeuflhard1997, author = {Ehrig, Rainald and Nowak, Ulrich and Deuflhard, Peter}, title = {Massively Parallel Linearly-Implicit Extrapolation Algorithms as a Powerful Tool in Process Simulation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3127}, number = {SC-97-43}, year = {1997}, abstract = {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.}, language = {en} } @misc{LipnikovDeuflhard1997, author = {Lipnikov, Konstantin and Deuflhard, Peter}, title = {Domain Decomposition with Subdomain CCG for Material Jump Elliptic Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3057}, number = {SC-97-36}, year = {1997}, abstract = {A combination of the cascadic conjugate gradient (CCG) method for homogeneous problems with a non-overlapping domain decomposition (DD) method is studied. Mortar finite elements on interfaces are applied to permit non-matching grids in neighboring subdomains. For material jump problems, the method is designed as an alternative to the cascadic methods.}, language = {en} } @misc{EhrigDeuflhard1997, author = {Ehrig, Rainald and Deuflhard, Peter}, title = {GMERR - an Error Minimizing Variant of GMRES}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3323}, number = {SC-97-63}, year = {1997}, abstract = {The paper analyzes a recently proposed iterative error minimizing method for the solution of linear systems. Sufficient and necessary conditions for convergence are studied, which show that the method essentially requires normal matrices. An efficient implementation similar to GMRES has been worked out in detail. Numerical tests on general non--normal matrices, of course, indicate that this approach is not competitive with GMRES. Summarizing, if error minimizing is important, one should rather choose CGNE. A computational niche for GMERR might be problems, where normal but non--symmetric matrices occur, like dissipative quantum mechanics.}, language = {en} } @misc{BeckDeuflhardHiptmairetal.1997, author = {Beck, Rudolf and Deuflhard, Peter and Hiptmair, Ralf and Hoppe, Ronald H. W. and Wohlmuth, Barbara}, title = {Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3356}, number = {SC-97-66}, year = {1997}, abstract = {The focus of this paper is on the efficient solution of boundary value problems involving the double-- curl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling time--harmonic problems or in the context of eddy--current computations. Their discretization is based on on N\'ed\'elec's {\bf H(curl}; \$\Omega\$)--conforming edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the curl--operator and its orthogonal complement. Only on the latter we have proper ellipticity of the problem. Yet, exploiting the existence of computationally available discrete potentials for edge element spaces, we can switch to an elliptic problem in potential space to deal with nullspace of curl. Thus both cases become amenable to strategies of error estimation and multigrid solution developed for second order elliptic problems. The efficacy of the approach is confirmed by numerical experiments which cover several model problems and an application to waveguide simulation.}, language = {en} } @misc{DeuflhardSeebass1998, author = {Deuflhard, Peter and Seebass, Martin}, title = {Adaptive Multilevel FEM as Decisive Tools in the Clinical Cancer Therapy Hyperthermia}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3739}, number = {SC-98-30}, year = {1998}, abstract = {The paper surveys recent progress in a joint mathematical-medical project on cancer therapy planning. Within so-called regional hyperthermia the computational task is to tune a set of coupled radiofrequency antennas such that a carefully measured tumor is locally heated, but any outside hot spots are avoided. A mathematical model of the whole clinical system -- air, applicator with antennas, water bolus, individual patient body -- involves Maxwell's equations in inhomogeneous media and a parabolic bioheat transfer equation, which represents a simplified model of heat transfer in the human body (ignoring strong blood vessel heat transport). Both PDEs need to be computed fast and to medical reliability (!) on a workstation within a clinical environment. This requirement triggered a series of new algorithmic developments to be reported here, among which is an adaptive multilevel FEM for Maxwell's equations, which dominates the numerical simulation time. In total, however, the main bulk of computation time (see Table 3 in Section 4 below) still goes into segmentation -- a necessary preprocessing step in the construction a 3D virtual patient from the input of a stack of 2D computed tomograms (left out here).}, language = {en} } @misc{DeuflhardWeiserSeebass1998, author = {Deuflhard, Peter and Weiser, Martin and Seebass, Martin}, title = {A New Nonlinear Elliptic Multilevel FEM Applied to Regional Hyperthermia}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3785}, number = {SC-98-35}, year = {1998}, abstract = {In the clinical cancer therapy of regional hyperthermia nonlinear perfusion effects inside and outside the tumor seem to play a not negligible role. A stationary model of such effects leads to a nonlinear Helmholtz term within an elliptic boundary value problem. The present paper reports about the application of a recently designed adaptive multilevel FEM to this problem. For several 3D virtual patients, nonlinear versus linear model is studied. Moreover, the numerical efficiency of the new algorithm is compared with a former application of an adaptive FEM to the corresponding instationary model PDE.}, language = {en} } @misc{WeiserSchielaDeuflhard2003, author = {Weiser, Martin and Schiela, Anton and Deuflhard, Peter}, title = {Asymptotic Mesh Independence of Newton's Method Revisited}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7352}, number = {03-13}, year = {2003}, abstract = {The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems.}, language = {en} } @misc{Deuflhard2002, author = {Deuflhard, Peter}, title = {A Comparison of Related Concepts in Computational Chemistry and Mathematics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7138}, number = {02-46}, year = {2002}, abstract = {This article studies the relation of the two scientific languages Chemistry and Mathematics via three selected comparisons: (a) QSSA versus dynamic ILDM in reaction kinetics, (b) lumping versus discrete Galerkin methods in polymer chemistry, and (c) geometrical conformations versus metastable conformations in drug design. The common clear message from these comparisons is that chemical intuition may pave the way for mathematical concepts just as chemical concepts may gain from mathematical precisioning. Along this line, significant improvements in chemical research and engineering have already been possible -- and can be further expected in the future from the dialogue between the two scientific languages.}, language = {en} } @misc{Deuflhard2002, author = {Deuflhard, Peter}, title = {Adaptive Pseudo-transient Continuation for Nonlinear Steady State Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6814}, number = {02-14}, year = {2002}, abstract = {Pseudo--transient continuation methods are quite popular for the numerical solution of steady state problems, typically in PDEs. They are based on an embedding into a time dependent initial value problem. In the presence of dynamical invariants the Jacobian matrix of the nonlinear equation system is bound to be singular. The paper presents a convergence analysis which takes this property into account -- in contrast to known approaches. On the basis of the new analysis adaptive algorithms are suggested in detail. These include a variant with Jacobian approximations as well as inexact pseudo--transient continuation, both of which play an important role in discretized PDEs. Numerical experiments are left to future work.}, language = {en} } @misc{Deuflhard2002, author = {Deuflhard, Peter}, title = {From Molecular Dynamics to Conformational Dynamics in Drug Design}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6878}, number = {02-20}, year = {2002}, abstract = {Computational drug design studies molecular recognition in the {\em virtual lab}. The arising Hamiltonian dynamics is known to be chaotic and ill-conditioned already after picoseconds, whereas times are \$msec\$ up to \$min\$. Classical molecular dynamics with long term trajectory computation gives, at best, information about time and statistical ensemble averages. The present paper surveys a recent new modeling approach called {\em conformational dynamics}, which is due to the author and Ch. Sch{\"u}tte. This approach achieves information about the dy time scales by telescoping a short term deterministic model with a statistical model. Examples of small biomolecules are included.}, language = {en} } @misc{HochmuthDeuflhard2003, author = {Hochmuth, Reinhard and Deuflhard, Peter}, title = {Multiscale Analysis for the Bio-Heat Transfer Equation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7302}, number = {03-08}, year = {2003}, abstract = {The bio-heat transfer equation is a macroscopic model for describing the heat transfer in microvascular tissue. In [{\sl Deuflhard, Hochmuth 2002}] the authors applied homogenization techniques to derive the bio-heat transfer equation as asymptotic result of boundary value problems which provide a microscopic description for microvascular tissue. Here those results are generalized to a geometrical setting where the regions of blood are allowed to be connected. Moreover, asymptotic corrector results are derived.}, language = {en} } @misc{DeuflhardHochmuth2002, author = {Deuflhard, Peter and Hochmuth, Reinhard}, title = {Multiscale Analysis of Thermoregulation in the Human Microvascular System}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6985}, number = {02-31}, year = {2002}, abstract = {The bio-heat transfer equation is a macroscopic model for describing the heat transfer in microvascular tissue. So far the deduction of the Helmholtz term in the bio-heat transfer equation is not co role. In view of a future numerical application of this new mathematical model to treatment planning in hyperthermia we derive asymptotic estimates for first and second order correctors.}, language = {en} } @misc{DeuflhardNowakWeiser2002, author = {Deuflhard, Peter and Nowak, Ulrich and Weiser, Martin}, title = {Affine Invariant Adaptive Newton Codes for Discretized PDEs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7005}, number = {02-33}, year = {2002}, abstract = {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.}, language = {en} } @misc{DeuflhardWeber2003, author = {Deuflhard, Peter and Weber, Marcus}, title = {Robust Perron Cluster Analysis in Conformation Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7415}, number = {03-19}, year = {2003}, abstract = {The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.}, language = {en} } @misc{DeuflhardSchuette2003, author = {Deuflhard, Peter and Sch{\"u}tte, Christof}, title = {Molecular Conformation Dynamics and Computational Drug Design}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7427}, number = {03-20}, year = {2003}, abstract = {The paper surveys recent progress in the mathematical modelling and simulation of essential molecular dynamics. Particular emphasis is put on computational drug design wherein time scales of \$msec\$ up to \$min\$ play the dominant role. Classical long-term molecular dynamics computations, however, would run into ill-conditioned initial value problems already after time spans of only \$psec=10^{-12} sec\$. Therefore, in order to obtain results for times of pharmaceutical interest, a combined deterministic-stochastic model is needed. The concept advocated in this paper is the direct identification of metastable conformations together with their life times and their transition patterns. It can be interpreted as a {\em transfer operator} approach corresponding to some underlying hybrid Monte Carlo process, wherein short-term trajectories enter. Once this operator has been discretized, which is a hard problem of its own, a stochastic matrix arises. This matrix is then treated by {\em Perron cluster analysis}, a recently developed cluster analysis method involving the numerical solution of an eigenproblem for a Perron cluster of eigenvalues. In order to avoid the 'curse of dimension', the construction of appropriate boxes for the spatial discretization of the Markov operator requires careful consideration. As a biomolecular example we present a rather recent SARS protease inhibitor.}, language = {en} }