Eric Barth

Algorithms for constrained molecular dynamics

In molecular dynamics simulations, the fastest components of the potential field impose severe restrictions on the stability and hence the speed of computational methods. One possibility for treating this problem is to replace the fastest components with algebraic length constraints. In this article the resulting systems of mixed differential and algebraic equations are studied. Commonly used discretization schemes for constrained Hamiltonian systems are discussed. The form of the nonlinear equations is examined in detail and used to give convergence results for the traditional nonlinear solution technique SHAKE iteration and for a modification based on successive overrelaxation (SOR). A simple adaptive algorithm for finding the optimal relaxation parameter is presented. Alternative direct methods using sparse matrix techniques are discussed. Numerical results are given for the new techniques, which have been implemented in the molecular modeling software package CHARMM and show as much as twofold improvement over SHAKE iteration.

Generating generalized distributions from dynamical simulation

We present a general molecular-dynamics simulation scheme, based on the Nose thermostat, for sampling from arbitrary phase space distributions. We formulate numerical methods based on both Nose–Hoover and Nose–Poincare thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that is designed to accelerate sampling in molecular systems.

Algorithm 800: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. I: the square-reduced method

This article describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loans method. The transformation of the Hamiltonian matrix to a square-reduced form transforms a Hamiltonian eigenvalue problem of order 2n to a Hessenberg eigenvalue problem of order n. The eigenvalues of the Hamiltonian matrix are the square roots of those of the Hessenberg matrix. Symplectic scaling and norm scaling are provided, which, in some cases, improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.

A Time-Reversible Variable-Stepsize Integrator for Constrained Dynamics

This article considers the design and implementation of variable-timestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, and we consider time-reparameterization techniques employing a time-reversible (symmetric) integration method to solve the equations of motion. We give several numerical examples, including a simulation of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation.

Approach to Thermal Equilibrium in Biomolecular Simulation

The evaluation of molecular dynamics models incorporating temperature control methods is of great importance for molecular dynamics practitioners. In this paper, we study the way in which biomolecular systems achieve thermal equilibrium. In unthermostatted (constant energy) and Nose-Hoover dynamics simulations, correct partition of energy is not observed on a typical MD simulation timescale. We discuss the practical use of numerical schemes based on Nose-Hoover chains, Nose-Poincare and recursive multiple thermostats (RMT) [8], with particular reference to parameter selection, and show that RMT appears to show the most promise as a method for correct thermostatting. All of the MD simulations were carried out using a variation of the CHARMM package in which the Nose-Poincare, Nose-Hoover Chains and RMT methods have been implemented.

A Test Set for Molecular Dynamics Algorithms

This article describes a collection of model problems for aiding numerical analysts, code developers and others in the design of computational methods for molecular dynamics (MD) simulation. Common types of calculations and desirable features of algorithms are surveyed, and these are used to guide selection of representative models. By including essential features of certain classes of molecular systems, but otherwise limiting the physical and quantitative details, it is hoped that the test set can help to facilitate cross-disciplinary algorithm and code development efforts.

MDT - The Molecular Dynamits Test Set

Over the past two decades, computational scientists have turned increasing attention to the field of molecular modeling. Advances have been made in the design of efficient algorithms for problems such as fast summation methods for computing non-bonded atomic interactions, long-time numerical intergration of equations of motion, non-Newtonian dynamical formulations for simulation in a projects have often involved methematicians and/of computer scientists who, though adept at algorithm and software development, may possess limited or no pysicial or chemical knowledge.The molecular dynamics test set is a collection of model problems for aiding numerical analysts, code developers and others in the design of computational methods for molecular dynamics (MD) simulation. Common types of calculations and desirable features of algorithms have been considered, and these were used to guide selection of representative models. By including essential features of certain classes of molecular systems.but otherwise limiting the physical and quantitative details, it is hoped that the test set will help to facilitate cross-disciplinary algorithm and code development efforts.