On the non-convergence of the Wang-Landau algorithms with multiple random walkers

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and $1/t$ algorithms. The classical algorithms are modified by the use of $m$ independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number $\pi$ by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, $t$; then, the average over $m$ walkers is performed. It is observed that the error goes as $1/\sqrt{m}$. However, if the number of walkers increases above a certain critical value $m>m_x$, the error reaches a constant value (i.e. it saturates). This occurs for both algorithms; however, it is shown that for a given system, the $1/t$ algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value $m_x$, since it does not reduces the error in the calculation. Therefore, the number of walkers does not guarantee convergence.Comment: 10 pages, 12 figures, Regular Articl

Uncertainty quantification for radio interferometric imaging: II. MAP estimation

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, for massive data sizes, like those anticipated from the Square Kilometre Array (SKA), it will be difficult if not impossible to apply any MCMC technique due to its inherent computational cost. We formulate Bayesian inference problems with sparsity-promoting priors (motivated by compressive sensing), for which we recover maximum a posteriori (MAP) point estimators of radio interferometric images by convex optimisation. Exploiting recent developments in the theory of probability concentration, we quantify uncertainties by post-processing the recovered MAP estimate. Three strategies to quantify uncertainties are developed: (i) highest posterior density credible regions; (ii) local credible intervals (cf. error bars) for individual pixels and superpixels; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. Our MAP-based methods are approximately $10^5$ times faster computationally than state-of-the-art MCMC methods and, in addition, support highly distributed and parallelised algorithmic structures. For the first time, our MAP-based techniques provide a means of quantifying uncertainties for radio interferometric imaging for realistic data volumes and practical use, and scale to the emerging big-data era of radio astronomy.Comment: 13 pages, 10 figures, see companion article in this arXiv listin

Optical polarimetric monitoring of the type II-plateau SN 2005af

Aims. Core-collapse supernovae may show significant polarization that implies non-spherically symmetric explosions. We observed the type II-plateau SN 2005af using optical polarimetry in order to verify whether any asphericity is present in the supernova temporal evolution. Methods. We used the IAGPOL imaging polarimeter to obtain optical linear polarization measurements in R (five epochs) and V (one epoch) broadbands. Interstellar polarization was estimated from the field stars in the CCD frames. The optical polarimetric monitoring began around one month after the explosion and lasted ~30 days, between the plateau and the early nebular phase. Results. The weighted mean observed polarization in R band was [1.89 +/- 0.03]% at position angle (PA) 54 deg. After foreground subtraction, the level of the average intrinsic polarization for SN 2005af was ~0.5% with a slight enhancement during the plateau phase and a decline at early nebular phase. A rotation in PA on a time scale of days was also observed. The polarimetric evolution of SN 2005af in the observed epochs is consistent with an overall asphericity of ~20% and an inclination of ~30 deg. Evidence for a more complex, evolving asphericity, possibly involving clumps in the SN 2005af envelope, is found.Comment: 6 pages, 5 figures, to be published A&

Further Criteria for the Existence of Steady Line-Driven Winds

In Paper I, we showed that steady line-driven disk wind solutions can exist by using "simple" models that mimic the disk environment. Here I extend the concepts introduced in Paper I and discuss many details of the analysis of the steady/unsteady nature of 1D line-driven winds. This work confirms the results and conclusions of Paper I, and is thus consistent with the steady nature of the 1D streamline line-driven disk wind models of Murray and collaborators and the 2.5D line-driven disk wind models of Pereyra and collaborators. When including gas pressures effects, as is routinely done in time-dependent numerical models, I find that the spatial dependence of the nozzle function continues to play a key role in determining the steady/unsteady nature of supersonic line-driven wind solutions. I show here that the existence/nonexistence of local wind solutions can be proved through the nozzle function without integrating the equation of motion. This work sets a detailed framework with which we will analyze, in a following paper, more realistic models than the "simple" models of Paper I.Comment: 30 pages, 5 figures, accepted for publication by The Astrophysical Journa

Dissociative Adsorption: A Solvable Model

A model of "hot"-dimer deposition in one dimension, introduced by Pereyra and Albano, is modified to have an unbounded dissociation range. The resulting dynamical equations are solved exactly. A related k-mer dissociation model is also introduced and its solution obtained as a quadrature.Comment: TeX (plain

Self-similarity of single-channel transmission for electron transport in nanowires

We demonstrate that the single-channel transmission in the resonance tunneling regime exhibits self-similarity as a function of the nanowire length and the energy of incident electrons. The self-similarity is used to design the nonlinear transformation of the nanowire length and energy which, on the basis of known values of transmission for a certain region on the energy-length plane, yields transmissions for other regions on this plane. Test calculations with a one-dimensional tight-binding model illustrate the described transformations. Density function theory based transport calculations of Na atomic wires confirm the existence of the self-similarity in the transmission

Wang-Landau Algorithm: a Theoretical Analysis of the Saturation of the Error

In this work we present a theoretical analysis of the convergence of the Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced years ago to calculate the density of states in statistical models. We study the dynamical behavior of the error in the calculation of the density of states.We conclude that the source of the saturation of the error is due to the decreasing variations of the refinement parameter. To overcome this limitation, we present an analytical treatment in which the refinement parameter is scaled down as a power law instead of exponentially. An extension of the analysis to the N-fold way variation of the method is also discussed.Comment: 7 pages, 5 figure

Analysis of the convergence of the 1/t and Wang-Landau algorithms in the calculation of multidimensional integrals

In this communication, the convergence of the 1/t and Wang - Landau algorithms in the calculation of multidimensional numerical integrals is analyzed. Both simulation methods are applied to a wide variety of integrals without restrictions in one, two and higher dimensions. The errors between the exact and the calculated values of the integral are obtained and the efficiency and accuracy of the methods are determined by their dynamical behavior. The comparison between both methods and the simple sampling Monte Carlo method is also reported. It is observed that the time dependence of the errors calculated with 1/t algorithm goes as N^{-1/2} (with N the MC trials) in quantitative agreement with the simple sampling Monte Carlo method. It is also showed that the error for the Wang - Landau algorithm saturates in time evidencing the non-convergence of the methods. The sources for the error are also determined.Comment: 8 pages, 5 figure

The influence of the “hot”-dimer adsorption mechanism on the kinetics of a monomer-dimer surface reaction

“Hot” dimers are molecules which after adsorption dissociate and each of the remaining “hot” monomers fly apart up to a maximum distance R from the original adsorption site. The influence of the “hot”-dimer adsorption mechanism on relevant aspects of the bimolecular catalyzed reaction of the type A − (1/2)B2(“hot”) → AB is studied by means of the Monte-Carlo simulation technique. The temporal evolution of both the reactant's coverages as well as the rate of AB-production is evaluated and discussed. Due to the enhanced probability of “hot” species for encounters with other adsorbed particles, the rate of AB-production becomes faster when increasing R. This behavior may be relevant in the dynamic of some catalyzed reactions such as for example the oxidation of carbon monoxide on transition metal surfaces, i.e. A≡CO, B2≡O2, and AB≡CO2. Also the sticking coefficient of “hot” dimers and the average distance traveled by the “hot” monomers are evaluated and discussed.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicada