158,406 research outputs found
Convergence and Refinement of the Wang-Landau Algorithm
Recently, Wang and Landau proposed a new random walk algorithm that can be
very efficiently applied to many problems. Subsequently, there has been
numerous studies on the algorithm itself and many proposals for improvements
were put forward. However, fundamental questions such as what determines the
rate of convergence has not been answered. To understand the mechanism behind
the Wang-Landau method, we did an error analysis and found that a steady state
is reached where the fluctuations in the accumulated energy histogram saturate
at values proportional to . This value is closely related to
the error corrections to the Wang-Landau method. We also study the rate of
convergence using different "tuning" parameters in the algorithm.Comment: 6 pages, submitted to Comp. Phys. Com
Streamline integration as a method for structured grid generation in X-point geometry
We investigate structured grids aligned to the contours of a two-dimensional
flux-function with an X-point (saddle point). Our theoretical analysis finds
that orthogonal grids exist if and only if the Laplacian of the flux-function
vanishes at the X-point. In general, this condition is sufficient for the
existence of a structured aligned grid with an X-point. With the help of
streamline integration we then propose a numerical grid construction algorithm.
In a suitably chosen monitor metric the Laplacian of the flux-function vanishes
at the X-point such that a grid construction is possible.
We study the convergence of the solution to elliptic equations on the
proposed grid. The diverging volume element and cell sizes at the X-point
reduce the convergence rate. As a consequence, the proposed grid should be used
with grid refinement around the X-point in practical applications. We show that
grid refinement in the cells neighboring the X-point restores the expected
convergence rate
Bayesian ensemble refinement by replica simulations and reweighting
We describe different Bayesian ensemble refinement methods, examine their
interrelation, and discuss their practical application. With ensemble
refinement, the properties of dynamic and partially disordered (bio)molecular
structures can be characterized by integrating a wide range of experimental
data, including measurements of ensemble-averaged observables. We start from a
Bayesian formulation in which the posterior is a functional that ranks
different configuration space distributions. By maximizing this posterior, we
derive an optimal Bayesian ensemble distribution. For discrete configurations,
this optimal distribution is identical to that obtained by the maximum entropy
"ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble
refinement enhances the sampling of relevant configurations by imposing
restraints on averages of observables in coupled replica molecular dynamics
simulations. We show that the strength of the restraint should scale linearly
with the number of replicas to ensure convergence to the optimal Bayesian
result in the limit of infinitely many replicas. In the "Bayesian inference of
ensembles" (BioEn) method, we combine the replica and EROS approaches to
accelerate the convergence. An adaptive algorithm can be used to sample
directly from the optimal ensemble, without replicas. We discuss the
incorporation of single-molecule measurements and dynamic observables such as
relaxation parameters. The theoretical analysis of different Bayesian ensemble
refinement approaches provides a basis for practical applications and a
starting point for further investigations.Comment: Paper submitted to The Journal of Chemical Physics (15 pages, 4
figures); updated references; expanded discussions of related formalisms,
error treatment, and ensemble refinement with and without replicas; appendi
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