41,194 research outputs found
Error analysis of coarse-grained kinetic Monte Carlo method
In this paper we investigate the approximation properties of the
coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice
stochastic dynamics. We provide both analytical and numerical evidence that the
hierarchy of the coarse models is built in a systematic way that allows for
error control in both transient and long-time simulations. We demonstrate that
the numerical accuracy of the CGMC algorithm as an approximation of stochastic
lattice spin flip dynamics is of order two in terms of the coarse-graining
ratio and that the natural small parameter is the coarse-graining ratio over
the range of particle/particle interactions. The error estimate is shown to
hold in the weak convergence sense. We employ the derived analytical results to
guide CGMC algorithms and we demonstrate a CPU speed-up in demanding
computational regimes that involve nucleation, phase transitions and
metastability.Comment: 30 page
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
Pseudo generators of spatial transfer operators
Metastable behavior in dynamical systems may be a significant challenge for a
simulation based analysis. In recent years, transfer operator based approaches
to problems exhibiting metastability have matured. In order to make these
approaches computationally feasible for larger systems, various reduction
techniques have been proposed: For example, Sch\"utte introduced a spatial
transfer operator which acts on densities on configuration space, while Weber
proposed to avoid trajectory simulation (like Froyland et al.) by considering a
discrete generator.
In this manuscript, we show that even though the family of spatial transfer
operators is not a semigroup, it possesses a well defined generating structure.
What is more, the pseudo generators up to order 4 in the Taylor expansion of
this family have particularly simple, explicit expressions involving no
momentum averaging. This makes collocation methods particularly easy to
implement and computationally efficient, which in turn may open the door for
further efficiency improvements in, e.g., the computational treatment of
conformation dynamics. We experimentally verify the predicted properties of
these pseudo generators by means of two academic examples
Highly accelerated simulations of glassy dynamics using GPUs: caveats on limited floating-point precision
Modern graphics processing units (GPUs) provide impressive computing
resources, which can be accessed conveniently through the CUDA programming
interface. We describe how GPUs can be used to considerably speed up molecular
dynamics (MD) simulations for system sizes ranging up to about 1 million
particles. Particular emphasis is put on the numerical long-time stability in
terms of energy and momentum conservation, and caveats on limited
floating-point precision are issued. Strict energy conservation over 10^8 MD
steps is obtained by double-single emulation of the floating-point arithmetic
in accuracy-critical parts of the algorithm. For the slow dynamics of a
supercooled binary Lennard-Jones mixture, we demonstrate that the use of
single-floating point precision may result in quantitatively and even
physically wrong results. For simulations of a Lennard-Jones fluid, the
described implementation shows speedup factors of up to 80 compared to a serial
implementation for the CPU, and a single GPU was found to compare with a
parallelised MD simulation using 64 distributed cores.Comment: 12 pages, 7 figures, to appear in Comp. Phys. Comm., HALMD package
licensed under the GPL, see http://research.colberg.org/projects/halm
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Leveraging legacy codes to distributed problem solving environments: A web service approach
This paper describes techniques used to leverage high performance legacy codes as CORBA components to a distributed problem solving environment. It first briefly introduces the software architecture adopted by the environment. Then it presents a CORBA oriented wrapper generator (COWG) which can be used to automatically wrap high performance legacy codes as CORBA components. Two legacy codes have been wrapped with COWG. One is an MPI-based molecular dynamic simulation (MDS) code, the other is a finite element based computational fluid dynamics (CFD) code for simulating incompressible Navier-Stokes flows. Performance comparisons between runs of the MDS CORBA component and the original MDS legacy code on a cluster of workstations and on a parallel computer are also presented. Wrapped as CORBA components, these legacy codes can be reused in a distributed computing environment. The first case shows that high performance can be maintained with the wrapped MDS component. The second case shows that a Web user can submit a task to the wrapped CFD component through a Web page without knowing the exact implementation of the component. In this way, a userâs desktop computing environment can be extended to a high performance computing environment using a cluster of workstations or a parallel computer
Derivation of Langevin Dynamics in a Nonzero Background Flow Field
We propose a derivation of a nonequilibrium Langevin dynamics for a large
particle immersed in a background flow field. A single large particle is placed
in an ideal gas heat bath composed of point particles that are distributed
consistently with the background flow field and that interact with the large
particle through elastic collisions. In the limit of small bath atom mass, the
large particle dynamics converges in law to a stochastic dynamics. This
derivation follows the ideas of [D. D\"urr, S. Goldstein, and J. L. Lebowitz,
1981 and 1983; P. Calderoni, D. D\"urr, and S. Kusuoka, 1989] and provides
extensions to handle the nonzero background flow. The derived nonequilibrium
Langevin dynamics is similar to the dynamics in [M. McPhie, et al., 2001]. Some
numerical experiments illustrate the use of the obtained dynamic to simulate
homogeneous liquid materials under flow.Comment: Minor revisions, refined discussion of the laminar bath approach and
non-Hamiltonian dynamics approach in Section 2. 41 pages, 8 figure
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