TY - GEN A1 - Deuflhard, Peter A1 - Dellnitz, Michael A1 - Junge, Oliver A1 - Schütte, Christof T1 - Computation of Essential Molecular Dynamics by Subdivision Techniques I: Basic Concept N2 - The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the {\em essential dynamics\/} of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times of conformational changes need to be actually computed. In the language of dynamical systems, such information is contained in the natural invariant measure (infinite relaxation time) or in almost invariant sets ("large" finite relaxation times). The paper suggests the direct computation of these objects via eigenmodes of the associated Frobenius-Perron operator by means of a multilevel subdivision algorithm. The advocated approach is different to both Monte-Carlo techniques on the one hand and long term trajectory simulation on the other hand: in our setup long term trajectories are replaced by short term sub-trajectories, Monte-Carlo techniques are just structurally connected via the underlying Frobenius-Perron theory. Numerical experiments with a first version of our suggested algorithm are included to illustrate certain distinguishing properties. A more advanced version of the algorithm will be presented in a second part of this paper. T3 - ZIB-Report - SC-96-45 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2553 ER - TY - GEN A1 - Dellnitz, Michael A1 - Junge, Oliver T1 - On the Approximation of Complicated Dynamical Behavior N2 - We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow to approximate SBR-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius-Perron operator, and two essentially different mathematical concepts are used: the idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from Ergodic Theory concerning the approximation of SBR-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples. T3 - ZIB-Report - SC-96-35 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2456 ER - TY - GEN A1 - Dellnitz, Michael A1 - Junge, Oliver T1 - An Adaptive Box Refinement in Subdivision Techniques for the Approximation of Dynamical Behavior N2 - Recently subdivision techniques have been introduced in the numerical investigation of complicated temporal behavior of dynamical systems. In this article we intertwine the subdivision process with the computation of invariant measures and propose an adaptive scheme for the box refinement which is based on the combination of these methods. Using this new algorithm the numerical effort for the computation of box coverings is in general significantly reduced, and we illustrate this fact by several numerical examples. T3 - ZIB-Report - SC-96-36 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2464 ER -