Alignment methods for biased multicanonical sampling

2018

Eur. Phys. J. B

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We define the projected entropy ( ) S T at a given temperature T in the context of an Ising model transition matrix calculation as the entropy associated with the distribution of Markov-chain realizations in energy-magnetization, E H − , space. An even sampling of states is achieved by accumulating the results from multiple Markov chains while decrementing 1 T at a rate proportional to the inverse of the effective number, ( ) ( ) exp S T , of accessible projected states. Such a procedure is both highly accurate and far simpler to implement than a previously suggested method based on monitoring the evolution of the E H − distribution at each temperature. [1] We further demonstrate a transition matrix procedure that instead ensures uniform sampling in physical entropy. Introduction: This paper considers the determination of the statistical behavior of one or more global variables ( ) E α   that depend on numerous stochastically varying local quantities α  . While this information can be directly obtained by Monte-Carlo (randomly) sampling the parameter space, α  , the computation time is often excessive if the properties of interest are associated with low probability regions of ( ) E α   . Accordingly, numerically efficient Markov-chain based procedures such as the multicanonical [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] and Wang-Landau [13] [14] [15] methods bias the local variables α  through an acceptance rule such that instead of the individual components of α  , physically relevant ranges of the ( ) E α   are nearly uniformly sampled. Such methods can additionally be employed to construct a transition matrix T where ij T corresponds to the probability that a Markov chain state in a histogram bin i E transitions to bin j E in a single unbiased Markov step δα  recorded before the application of the acceptance rule. [1] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] The normalized eigenvector of T with unit eigenvalue, which can be obtained by repeatedly multiplying an initially random vector by T then coincides with the desired probability distribution ( ) p E . [27] The transition matrix method can also be accelerated by incorporating renormalization techniques that estimate the transition matrix associated with a system by iteratively convolving subsystem matrices. [28] [29] [30]

mentioning

confidence: 81%

A projected entropy controller for transition matrix calculations

2018

Eur. Phys. J. B

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“…While microcanonical and quasi-microcanonical calculations of the specific heat are intrinsically less accurate than those employing the canonical procedure, in other non-statistical mechanics contexts microcanonical methods can prove advantageous. [39] In Figure 7 the result of a typical microcanonical calculation in which the unnormalized value of microcanonical E is first lowered from 0 to 2056  (lower solid curve) is compared to that of a similar calculation in which microcanonical E is instead raised from 2056  to 0 (upper solid curve) and to the exact result [40] (dashed curve). The error of the two curves, which is smaller when the initial state is magnetized, is systematic but can be reduced by accumulating the transition matrix elements from both of the calculations.…”

Section: E mentioning

confidence: 99%

“…steps for each realization [37]. [38][35] These results clearly indicate that the precision of the adaptive method exceeds that of standard procedures for an equivalent number of realizations.While microcanonical and quasi-microcanonical calculations of the specific heat are intrinsically less accurate than those employing the canonical procedure, in other non-statistical mechanics contexts microcanonical methods can prove advantageous [39]. InFigure 7the result of a typical microcanonical calculation in which the unnormalized value of microcanonical E is first lowered from 0 to 2056  (lower solid curve) is compared to that of a similar calculation in which microcanonical E is instead raised from 2056  to 0 (upper solid curve) and to the exact result…”

mentioning

confidence: 99%

Dynamic canonical and microcanonical transition matrix analyses of critical behavior

Lee

2017

Eur. Phys. J. B

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By monitoring the sampling of states with different magnetizations in transition matrix procedures a family of accurate and easily implemented techniques are constructed that automatically control the variation of the temperature or energy as the calculation proceeds. The accuracy of the method for a single Markov chain exceeds that of standard transition matrix procedures that accumulate elements from multiple chains.

“…The time variable is a shifted time coordinate, , where is the real-time coordinate and is the group velocity. is the saturation power (3) where is the photon energy, is the effective cross section area of the active region, and is the differential modal gain. We model the gain by a linear function of the carrier density , i.e., with the carrier density at transparency.…”

Section: Soa Output Power Simulationsmentioning

confidence: 99%

“…An excellent review is given by Berg in [2]. The main advantage of the MMC method is that it allows calculation of extremely low pdf values as for example down to [3]. Such low values cannot be reached by unbiased Monte Carlo simulations within practical time limits, and it may even be difficult to derive the pdf from an analytic expression by double precision computation.…”

Section: Introductionmentioning

confidence: 99%

Multicanonical Evaluation of the Tails of the Probability Density Function of Semiconductor Optical Amplifier Output Power Fluctuations

Tromborg

Reimer

IEEE J. Quantum Electron.

2010

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Abstract-This paper presents a multicanonical Monte Carlo method for simulating the tails of a probability density function (pdf) of the filtered output power from a semiconductor optical amplifier down to values of the order of 10 40 . The influence of memory effects on the pdf is examined in order to demonstrate the manner in which the calculated pdf approaches the true pdf with increasing integration time. The simulated pdf is shown to be in good agreement with a second-order analytic expression for the pdf.Index Terms-Microcanonical Monte Carlo, noise, optical communications, semiconductor optical amplifiers (SOAs).