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

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

Extended sources of the main events of the Umbria-Marche (1997) seismic sequence inverted from geophysical data

The three largest events of the 1997 Umbria-Marche (Italy) sequence occurred on September 26, 1997 at 00:33 GMT (Event 1, MW=5.7) and 09:40 GMT (Event 2, MW=6.0) in the Colfiorito area and on the October 14, 1997 at 15:23 (Event 3, MW=5.6) in the Sellano area. The availability of different sets of geodetic and seismological data allowed several studies to characterize the extended sources of events 1-3. In this work, I review some of the studies that obtain the properties of the seismic sources by inversion of available data. Generally these studies assume the seismic sources as dislocations or distributions of equivalent point sources in elastic half-spaces. Following their chronological order, they model increasing complexities of the sources by using an increasing number of data. Some of the differences between results obtained, such as the top edge depth estimates, are shown to be due to the different approaches used. Commonly a 1-D crustal model is used in inverting strongmotion data. Instead homogeneous elastic half-spaces are mainly assumed in inverting geodetic data to obtain the three main sources of the 1997 Umbria-Marche sequence. Assuming the same crustal structure is important to make comparable results obtained analyzing seismological data or geodetic data separately, as it has been done till now for this sequence

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

A numerical study of the RG equation for the deformed O(3)O(3) nonlinear sigma model

The Renormalization Group equation describing the evolution of the metric of the nonlinear sigma model poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. In this article we briefly report some results obtained from the numerical study of the solutions in the case of a two dimensional target space (deformation of the $O(3)$ sigma model). In particular, our analysis shows that the so-called sausages define an attracting manifold in the $U(1)$-symmetric case, at one-loop level. Moreover, data from two-loop evolution are used to test the association put forward in Nucl. Phys., B406 (1993) 521 between the so-called $SSM_{\nu}$ field theory and a certain $U(1)$-symmetric, factorized scattering theory (FST).Comment: LaTex (RevTex style), 16 pages, 6 uuencoded figures. Minor technical changes

The quasi-static approximation of the spring-slider motion

International audienceThe spring-slider is a simple dynamical system consisting in a massive block sliding with friction and pulled through a spring at a given velocity. Understanding the block motion is fundamental for studying more complex phenomena of frictional sliding, such as the seismogenic fault motion. We analyze the dynamical properties of the system, subject to rate- and state-dependent friction laws and forced at a constant load velocity. In particular we study the limits within which the quasi-static model can be used. The latter model approximates the complete model of the system without taking into account the inertia effects. The system parameters are here found to be grouped into three characteristic times of the three dynamics present in the complete model. A necessary condition for the quasi-static approximation to hold is that the characteristic time of the inertial equation is much smaller than the other two characteristic times. We have studied a modification of one of the classical forms of the rate- and state-dependent friction laws. Subsequently we have developed a linear analysis in the neighbourhood of the equilibrium point of the system. For the quasi-static model we rigorously found, by means of a nonlinear analysis, a supercritical Hopf bifurcation, a dynamical property of the complete model. The classical form of the friction laws can be obtained as a particular case of the one we considered, but fails to preserve the Hopf bifurcation in the quasi-static approximation. We conclude that to have a good quasi-static approximation of the system, even in nonlinear conditions, the form of the friction laws considered is a critical factor

Creep and locking of a low-angle normal fault: Insights from the Altotiberina fault in the Northern Apennines (Italy)

While low-angle normal faults have been recognized worldwide from geological studies, whether these structures are active or capable of generating big earthquakes is still debated. We provide new constraints on the role and modes of the Altotiberina fault (ATF) in accommodating extension in the Northern Apennines. We model GPS velocities to study block kinematics, faults slip rates and interseismic coupling of the ATF, which is active and accounts, with its antithetic fault, for a large part of the observed chain normal 3 mm/yr tectonic extension. A wide portion of the ATF creeps at the long-term slip rate (1.7 \ub1 0.3 mm/yr), but the shallow locked portions are compatible with M > 6.5 earthquakes. We suggest that positive stress accumulation due to ATF creep is most likely released by more favorable oriented splay faults, whose rupture may propagate downdip along low-angle normal fault surface and reduce the probability of occurrence of a seismic rupture of the shallower locked portion

Modeling instantaneous dynamic triggering in a 3–D fault system: application to the June 2000 South Iceland seismic sequence

We present a model of seismogenesis on an extended 3–D fault subjected to the external perturbations of coseismic stress changes due to an earthquake occurred on another fault (the causative fault). As an application, we consider the spatio–temporal stress distribution produced by the MS = 6.6 June 17, 2000 mainshock in the South Iceland Seismic Zone (SISZ) on the Hvalhnúkur fault. The latter is located nearly 64 km from the causative fault and failed 26 s after the mainshock with an estimated magnitude Mw  [5, 5.5], providing an example of instantaneous dynamic triggering. The stress perturbations are computed by means of a discrete wavenumber and reflectivity code. The response of the perturbed fault is then analyzed solving the truly 3–D, fully dynamic (or spontaneous) problem, accounting for crustal stratification. In a previous study, the response of the Hvalhnúkur fault was analyzed by using a spring–slider fault model, comparing the estimated perturbed failure time with the observed origin time. In addition to the perturbed failure time, the present model can provide numerical estimates of many other dynamical features of the triggered event that can be compared with available observations: the rupture history of the whole fault plane and its final extension and the seismic moment of the 26 s event. We show the key differences existing between a mass–spring model and the present extended fault model, in particular we show the essential role of the load exerted by the other slipping points of the fault. By considering both rate– and state–dependent laws and non–linear slip–dependent law, we show how the dynamics of the 26 s fault strongly depends on the assumed constitutive law and initial stress conditions. In the case of rate– and state– dependent governing laws, assuming an initial effective normal stress distribution which is suitable for the SISZ and consistent with previously stated conditions of instantaneous dynamic triggering of the Hvalhnúkur fault, we obtain results in general agreement with observations

Climacostol reduces tumour progression in a mouse model of melanoma via the p53-dependent intrinsic apoptotic programme

Climacostol, a compound produced by the ciliated protozoan Climacostomum virens, displayed cytotoxic properties in vitro. This study demonstrates that it has anti-tumour potential. Climacostol caused a reduction of viability/proliferation of B16-F10 mouse melanoma cells, a rapidly occurring DNA damage, and induced the intrinsic apoptotic pathway characterised by the dissipation of the mitochondrial membrane potential, the translocation of Bax to the mitochondria, the release of Cytochrome c from the mitochondria, and the activation of Caspase 9-dependent cleavage of Caspase 3. The apoptotic mechanism of climacostol was found to rely on the up-regulation of p53 and its targets Noxa and Puma. In vivo analysis of B16-F10 allografts revealed a persistent inhibition of tumour growth rate when melanomas were treated with intra-tumoural injections of climacostol. In addition, it significantly improved the survival of transplanted mice, decreased tumour weight, induced a remarkable reduction of viable cells inside the tumour, activated apoptosis and up-regulated the p53 signalling network. Importantly, climacostol toxicity was more selective against tumour than non-tumour cells. The anti-tumour properties of climacostol and the molecular events associated with its action indicate that it is a powerful agent that may be considered for the design of pro-apoptotic drugs for melanoma therapy