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Title: Numerical methods for molecular dynamics

Abstract

This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods which are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. Wemore » first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.« less

Authors:
Publication Date:
Research Org.:
Illinois Univ., Urbana, IL (United States)
Sponsoring Org.:
USDOE; USDOE, Washington, DC (United States)
OSTI Identifier:
5436878
Report Number(s):
DOE/ER/25099-1
ON: DE92010934
DOE Contract Number:
FG02-91ER25099
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; MOLECULES; DYNAMICS; PARTIAL DIFFERENTIAL EQUATIONS; NUMERICAL SOLUTION; BIOPHYSICS; BOLTZMANN EQUATION; EQUATIONS OF MOTION; ITERATIVE METHODS; PROGRESS REPORT; DIFFERENTIAL EQUATIONS; DOCUMENT TYPES; EQUATIONS; MECHANICS; 664000* - Atomic & Molecular Physics- (1992-); 990200 - Mathematics & Computers

Citation Formats

Skeel, R.D.. Numerical methods for molecular dynamics. United States: N. p., 1991. Web. doi:10.2172/5436878.
Skeel, R.D.. Numerical methods for molecular dynamics. United States. doi:10.2172/5436878.
Skeel, R.D.. Tue . "Numerical methods for molecular dynamics". United States. doi:10.2172/5436878. https://www.osti.gov/servlets/purl/5436878.
@article{osti_5436878,
title = {Numerical methods for molecular dynamics},
author = {Skeel, R.D.},
abstractNote = {This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods which are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. We first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.},
doi = {10.2172/5436878},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Jan 01 00:00:00 EST 1991},
month = {Tue Jan 01 00:00:00 EST 1991}
}

Technical Report:

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  • This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods whichmore » are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. We first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.« less
  • The objective is to find numerical algorithms suitable for large parallel computers that can much more efficiently model the dynamics of macromolecules such as proteins, DNA, and lipids. Emphasis is on the use of integration schemes, notably symplectic schemes, that can use large time steps to produce qualitatively correct simulations for long-time integrations. The goal is to obtain the desired information with the least computational effort, and the methodology is to use mathematical analysis and computational experiments on model problems. The techniques developed are to be tested on realistic molecular models as part of a different, complementary research project involvingmore » software development. Among the techniques to be considered, the better known ones are multiple time steps, constraint dynamics, and fast Coulomb solvers.« less
  • The aim of this research is to explore ideas for more efficient numerical integrators for molecular dynamics (MD) and where needed to develop appropriate theoretical tools. Emphasis is on macromolecules and techniques suitable for implementation in the biomolecular dynamics program NAMD. Listed on this report are the main accomplishments during the past two or so years. First are listed algorithmic developments suitable for implementation. Second are listed more theoretical developments helpful for further exploration of new algorithms.
  • An important area of research in computational biochemistry is the design of molecules for specific applications. The design of these molecules depends on the accurate determination of their three-dimensional structure or conformation. Under the assumption that molecules will settle into a configuration for which their energy is at a minimum, this design problem can be formulated as a global optimization problem. The solution of the molecular conformation problem can then be obtained, at least in principle, through any number of optimization algorithms. Unfortunately, it can easily be shown that there exist a large number of local minima for most moleculesmore » which makes this an extremely difficult problem for any standard optimization method. In this study, we present results for various optimization algorithms applied to a molecular conformation problem. We include results for genetic algorithms, simulated annealing, direct search methods, and several gradient methods. The major result of this study is that none of these standard methods can be used in isolation to efficiently generate minimum energy configurations. We propose instead several hybrid methods that combine properties of several local optimization algorithms. These hybrid methods have yielded better results on representative test problems than single methods.« less