65L05 Initial value problems
Refine
Year of publication
- 2002 (3) (remove)
Document Type
- ZIB-Report (3)
Language
- English (3)
Has Fulltext
- yes (3)
Is part of the Bibliography
- no (3)
Keywords
- Euler discretization (1)
- Hamilton systems (1)
- Hamiltonian dynamics (1)
- ILDM (1)
- Markov chain (1)
- Perro (1)
- QSSA (1)
- almost invariant sets (1)
- conformational dynamics (1)
- conformations (1)
Institute
- Computational Medicine (2)
- Numerical Mathematics (2)
- ZIB Allgemein (1)
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.
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.
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. SchĂĽtte. 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.