2,310 research outputs found
The adaptive Verlet method
This is the published version, also available here: http://dx.doi.org/10.1137/S1064827595284658.We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the flow, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog--Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scalar nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector field (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem
Adaptive multi-stage integrators for optimal energy conservation in molecular simulations
We introduce a new Adaptive Integration Approach (AIA) to be used in a wide
range of molecular simulations. Given a simulation problem and a step size, the
method automatically chooses the optimal scheme out of an available family of
numerical integrators. Although we focus on two-stage splitting integrators,
the idea may be used with more general families. In each instance, the
system-specific integrating scheme identified by our approach is optimal in the
sense that it provides the best conservation of energy for harmonic forces. The
AIA method has been implemented in the BCAM-modified GROMACS software package.
Numerical tests in molecular dynamics and hybrid Monte Carlo simulations of
constrained and unconstrained physical systems show that the method
successfully realises the fail-safe strategy. In all experiments, and for each
of the criteria employed, the AIA is at least as good as, and often
significantly outperforms the standard Verlet scheme, as well as fixed
parameter, optimized two-stage integrators. In particular, the sampling
efficiency found in simulations using the AIA is up to 5 times better than the
one achieved with other tested schemes
Accurate and efficient splitting methods for dissipative particle dynamics
We study numerical methods for dissipative particle dynamics (DPD), which is
a system of stochastic differential equations and a popular stochastic
momentum-conserving thermostat for simulating complex hydrodynamic behavior at
mesoscales. We propose a new splitting method that is able to substantially
improve the accuracy and efficiency of DPD simulations in a wide range of the
friction coefficients, particularly in the extremely large friction limit that
corresponds to a fluid-like Schmidt number, a key issue in DPD. Various
numerical experiments on both equilibrium and transport properties are
performed to demonstrate the superiority of the newly proposed method over
popular alternative schemes in the literature
Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDE)
are proposed which (i) are ergodic with respect to the (known) equilibrium
distribution of the SDE and (ii) approximate pathwise the solutions of the SDE
on finite time intervals. Both these properties are demonstrated in the paper
and precise strong error estimates are obtained. It is also shown that the
Metropolized integrator retains these properties even in situations where the
drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for
SDEs typically become unstable and fail to be ergodic.Comment: 46 pages, 5 figure
From Classical to Quantum and Back: Hamiltonian Adaptive Resolution Path Integral, Ring Polymer, and Centroid Molecular Dynamics
Path integral-based simulation methodologies play a crucial role for the
investigation of nuclear quantum effects by means of computer simulations.
However, these techniques are significantly more demanding than corresponding
classical simulations. To reduce this numerical effort, we recently proposed a
method, based on a rigorous Hamiltonian formulation, which restricts the
quantum modeling to a small but relevant spatial region within a larger
reservoir where particles are treated classically. In this work, we extend this
idea and show how it can be implemented along with state-of-the-art path
integral simulation techniques, such as ring polymer and centroid molecular
dynamics, which allow the approximate calculation of both quantum statistical
and quantum dynamical properties. To this end, we derive a new integration
algorithm which also makes use of multiple time-stepping. The scheme is
validated via adaptive classical--path-integral simulations of liquid water.
Potential applications of the proposed multiresolution method are diverse and
include efficient quantum simulations of interfaces as well as complex
biomolecular systems such as membranes and proteins
Using Bayes formula to estimate rates of rare events in transition path sampling simulations
Transition path sampling is a method for estimating the rates of rare events
in molecular systems based on the gradual transformation of a path distribution
containing a small fraction of reactive trajectories into a biased distribution
in which these rare trajectories have become frequent. Then, a multistate
reweighting scheme is implemented to postprocess data collected from the staged
simulations. Herein, we show how Bayes formula allows to directly construct a
biased sample containing an enhanced fraction of reactive trajectories and to
concomitantly estimate the transition rate from this sample. The approach can
remediate the convergence issues encountered in free energy perturbation or
umbrella sampling simulations when the transformed distribution insufficiently
overlaps with the reference distribution.Comment: 11 pages, 8 figure
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