TY - JOUR A1 - Bornemann, Folkmar A. A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Quantum-classical molecular dynamics as an approximation to full quantum dynamics JF - J. Chem. Phys. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/103/ U6 - https://doi.org/10.1063/1.471952 VL - 105 IS - 3 SP - 1074 EP - 1083 ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Adaptive Accuracy Control for Car-Parrinello Simulations JF - Num. Math. Y1 - 1999 UR - http://publications.imp.fu-berlin.de/88/ U6 - https://doi.org/10.1007/s002110050445 VL - 83 IS - 2 SP - 179 EP - 186 ER - TY - CHAP A1 - Schütte, Christof A1 - Bornemann, Folkmar A. ED - Deuflhard, Peter ED - Hermans, J. ED - Leimkuhler, Benedict ED - Marks, A. ED - Reich, Sebastian ED - Skeel, R. T1 - Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model T2 - Computational Molecular Dynamics Y1 - 1999 UR - http://publications.imp.fu-berlin.de/134/ VL - 4 SP - 380 EP - 395 PB - Springer ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - On the Singular Limit of the Quantum-Classical Molecular Dynamics Model JF - J. Appl. Math. Y1 - 1999 UR - http://publications.imp.fu-berlin.de/94/ U6 - https://doi.org/10.1137/S0036139997318834 VL - 59 IS - 4 SP - 1208 EP - 1224 ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - A mathematical investigation of the Car-Parrinello Method JF - Num. Math. Y1 - 1998 UR - http://publications.imp.fu-berlin.de/96/ U6 - https://doi.org/10.1007/s002110050316 VL - 78 IS - 3 SP - 359 EP - 376 ER - TY - JOUR A1 - Schütte, Christof A1 - Bornemann, Folkmar A. T1 - Homogenization Approach to Smoothed Molecular Dynamics JF - Nonlinear Analysis Y1 - 1997 UR - http://publications.imp.fu-berlin.de/99/ U6 - https://doi.org/10.1016/S0362-546X(97)00216-2 VL - 30 IS - 3 SP - 1805 EP - 1814 ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Homogenization of Hamiltonian Systems with a Strong Constraining Potential JF - Physica D Y1 - 1997 UR - http://publications.imp.fu-berlin.de/100/ U6 - https://doi.org/10.1016/S0167-2789(96)00245-X VL - 102 IS - 1-2 SP - 57 EP - 77 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Bornemann, Folkmar A. A1 - Schmidt, Burkhard A1 - Schütte, Christof T1 - An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics JF - Chem. Phys. Lett. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/101/ U6 - https://doi.org/10.1016/0009-2614(96)00471-X VL - 256 IS - 6 SP - 581 EP - 588 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Adaptive Accuracy Control for Microcanonical Car-Parrinello Simulations N2 - The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter $\mu$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP--solution from the BO-solution {\em decreases} like $\mu^{1/2}$ asymptotically. Since the computational effort {\em increases} like $\mu^{-1/2}$, the choice of $\mu$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls $\mu$: the parameter $\mu$ is repeatedly adapted during the simulation by choosing $\mu$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example. T3 - ZIB-Report - SC-96-20 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2311 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Krause, Rolf T1 - Classical and Cascadic Multigrid - A Methodical Comparison N2 - Using the full multigrid method {\em without} any coarse grid correction steps but with an a posteriori control of the number of smoothing iterations was shown by Bornemann and Deuflhard [1996] to be an optimal iteration method with respect to the energy norm. They named this new kind of multigrid iteration the {\em cascadic multigrid method}. However, numerical examples with {\em linear} finite elements raised serious doubts whether the cascadic multigrid method can be made optimal with respect to the {\em $L^2$-norm}. In this paper we prove that the cascadic multigrid method cannot be optimal for linear finite elements and show that the case might be different for higher order elements. We present a careful analysis of the two grid variant of the cascadic multigrid method providing a setting where one can understand the methodical difference between the cascadic multigrid method and the classical multigrid $V$-cycle almost immediately. As a rule of thumb we get that whenever the cascadic multigrid works the classical multigrid will work too but not vice versa. T3 - ZIB-Report - SC-96-25 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2368 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Homogenization of Highly Oscillatory Hamiltonian Systems N2 - The paper studies Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale. In particular, we are interested in the limit situation where the size $\epsilon$ of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of order ${\rm O}(1)$. The process of establishing an effective initial value problem for the limit positions will be called {\em homogenization} of the Hamiltonian system. This problem occurs in mechanics as the problem of realization of holonomic constraints, in plasma physics as the problem of guiding center motion, in the simulation of biomolecules as the so called smoothing problem. We suggest the systematic use of the notion of {\em weak convergence} in order to approach this problem. This methodology helps to establish unified and short proofs of the known results which throw light on the inherent structure of the problem. Moreover, we give a careful and critical review of the literature. T3 - ZIB-Report - SC-95-39 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2050 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - On the Singular Limit of the Quantum-Classical Molecular Dynamics Model N2 - \noindent In molecular dynamics applications there is a growing interest in so-called {\em mixed quantum-classical} models. These models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of quantum mechanics. A particularly extensively used model, the QCMD model, consists of a {\em singularly perturbed}\/ Schrödinger equation nonlinearly coupled to a classical Newtonian equation of motion. This paper studies the singular limit of the QCMD model for finite dimensional Hilbert spaces. The main result states that this limit is given by the time-dependent Born-Oppenheimer model of quantum theory---provided the Hamiltonian under consideration has a smooth spectral decomposition. This result is strongly related to the {\em quantum adiabatic theorem}. The proof uses the method of {\em weak convergence} by directly discussing the density matrix instead of the wave functions. This technique avoids the discussion of highly oscillatory phases. On the other hand, the limit of the QCMD model is of a different nature if the spectral decomposition of the Hamiltonian happens not to be smooth. We will present a generic example for which the limit set is not a unique trajectory of a limit dynamical system but rather a {\em funnel} consisting of infinitely many trajectories. T3 - ZIB-Report - SC-97-07 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2761 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - A Mathematical Investigation of the Car-Parrinello Method N2 - The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the {\em quantitative} influence of the involved parameter $\mu$, the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order ${\rm O}(\mu^{1/2})$ --- provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general. Finally, we show how to use the gained quantitative insight for an automatic control of the unphysical ``fake'' kinetic energy of the method. T3 - ZIB-Report - SC-96-19 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2302 ER - TY - GEN A1 - Schütte, Christof A1 - Bornemann, Folkmar A. T1 - Homogenization Approach to Smoothed Molecular Dynamics N2 - {\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.} T3 - ZIB-Report - SC-96-31 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2410 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schemann, Martin T1 - Adaptive Rothe's Method for the 2D Wave Equation N2 - The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved {\em virtually} using an adaptive state-of-the-art integrator. The {\em actual} realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus {\em perturbing} the virtual function space method. The admissible size of that perturbation can be computed {\em a priori} and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. T3 - ZIB-Report - SC-96-39 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2492 ER - TY - GEN A1 - Bornemann, Folkmar A. T1 - An Adaptive Multilevel Approach to Parabolic Equations II. N2 - In continuation of part I this paper develops a variable-order time discretization in Hilbert space based on a multiplicative error correction. Matching of time and space errors as explained in part I allows to construct an adaptive multilevel discretization of the parabolic problem. In contrast to the extrapolation method in time, which has been used in part I, the new time discretization allows to separate space and time errors and further to solve fewer elliptic subproblems with less effort, which is essential in view of the application to space dimension greater than one. Numerical examples for space dimension one are included which clearly indicate the improvement. T3 - ZIB-Report - SC-90-13 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-410 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Quantum-Classical Molecular Dynamics as an Approximation to Full Quantum Dynamics N2 - This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a {\em partial} classical limit of the full Schrödinger equation. This limit is achieved in two steps: separation of the full wavefunction and short wave asymptotics for its ``classical'' part. Both steps can be rigorously justified under certain smallness assumptions. Moreover, the results imply that neither the time-dependent self-consistent field method nor mixed quantum-semi-classical models lead to better approximations than QCMD since they depend on the separation step, too. On the other hand, the theory leads to a characterization of the critical situations in which the models are in danger of largely deviating from the solution of the full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of an Argon atom with a harmonic quantum oscillator. T3 - ZIB-Report - SC-95-26 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1922 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - A Mathematical Approach to Smoothed Molecular Dynamics: Correcting Potentials for Freezing Bond Angles N2 - The interaction potential of molecular systems which are typically used in molecular dynamics can be split into two parts of essentially different stiffness. The strong part of the potential forces the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles). However, the naive way of doing this via holonomic constraints is bound to produce incorrect results. The paper presents a mathematically rigorous discussion of the limit situation in which the stiffness of the strong part of the potential is increased to infinity. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given and its significant contribution is demonstrated in an illustrative example. It appears that this correcting potential is definitely not identical with the Fixman-potential as was repeatedly assumed in the literature. T3 - ZIB-Report - SC-95-30 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1960 ER - TY - GEN A1 - Bornemann, Folkmar A. T1 - On the Convergence of Cascadic Iterations for Elliptic Problems. N2 - We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them {\em cascadic iterations}. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of $H^2$-regular two-dimensional problems with quasi-uniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional $H^{1+\alpha}$-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method. T3 - ZIB-Report - SC-94-08 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1389 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Erdmann, Bodo A1 - Kornhuber, Ralf T1 - A Posteriori Error Estimates for Elliptic Problems. N2 - {\def\enorm {\mathop{\mbox{\boldmath{$|\!|$}}}\nolimits} Let $u \in H$ be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some $\tilde{u} \in {\cal S}$, $\cal S$ being a suitable finite element space. Efficient and reliable a posteriori estimates of the error $\enorm u - \tilde{u}\enorm $, measuring the (local) quality of $\tilde{u}$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.} T3 - ZIB-Report - SC-93-29 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1257 ER -