Markov Chain Monte Carlo Simulations and Their Statistical Analysis

Berg, Bernd A.

doi:10.1142/5602

Cited by 365 publications

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“…Error bars as shown in figures are calculated using jackknife bins (e.g., chapter 2.7 of [23]) with their number given by the first value in column four (always 32), while the second value was also used for the number of equilibrium sweeps (without measurements) performed after the recursion. Weighted by the number of their completed cycles, the results from two or more runs are combined for for the final analysis (compare chapter 2.1.2 of [23]). …”

Section: Numerical Resultsmentioning

confidence: 99%

“…The goodness of our fit is Q = 0.64 (see, e.g., chapter 2.8 of Ref. [23] for the definition and a discussion of Q), and its estimate for the specific heat density is c 0 = 0.0001961 (26). This is 10% higher than the c 0 value reported by the Wuppertal group [7], where lattices up to size 18 3 were used.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For a second order phase transition the integrated autocorrelation time τ int scales approximately ∼ N 2 s and we estimate from our own simulations on smaller lattices that in units of those measurements τ int ≈ 7 000 for N τ = 6 and N s = 48. A MC segment of the length of τ int delivers one statistically independent event (e.g., chapter 4.1.1 of [23]). Therefore, the run of [18] would in case of a second order transition be based on less than one event and strong metastabilities would be expected as soon as the Markov chain approaches the scaling region.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Nonperturbative U(1) gauge theory at finite temperature

Berg

Bazavov

2006

Phys. Rev. D

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For compact U(1) lattice gauge theory (LGT) we have performed a finite size scaling analysis on Nτ N 3 s lattices for Nτ fixed by extrapolating spatial volumes of size Ns ≤ 18 to Ns → ∞. Within the numerical accuracy of the thus obtained fits we find for Nτ = 4, 5 and 6 second order critical exponents, which exhibit no obvious Nτ dependence. The exponents are consistent with 3d Gaussian values, but not with either first order transitions or the universality class of the 3d XY model. As the 3d Gaussian fixed point is known to be unstable, the scenario of a yet unidentified non-trivial fixed point close to the 3d Gaussian emerges as one of the possible explanations.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

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Nonperturbative U(1) gauge theory at finite temperature

Berg

Bazavov

2006

Phys. Rev. D

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“…Reweighting of the β = 0 random configurations to β = 0.2 is shown to fail. These are assignments a0301 02 and a0303 02 of [3].…”

Section: Sampling Reweighting and Important Configurationsmentioning

confidence: 99%

“…The first part of this article gives a treatment of the MCMC fundamentals that is largely based on the author's book [3] on the subject, which gives many more details and contains extensive additional material. The book comes with computer code that can be downloaded from the web.…”

Section: Introductionmentioning

confidence: 99%

Markov Chain Monte Carlo Methods for Simulations of Biomolecules

Berg

Lecture Notes in Physics

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The computer revolution has been driven by a sustained increase of computational speed of approximately one order of magnitude (a factor of ten) every five years since about 1950. In natural sciences this has led to a continuous increase of the importance of computer simulations. Major enabling techniques are Markov Chain Monte Carlo (MCMC) and Molecular Dynamics (MD) simulations.This article deals with the MCMC approach. First basic simulation techniques, as well as methods for their statistical analysis are reviewed. Afterwards the focus is on generalized ensembles and biased updating, two advanced techniques, which are of relevance for simulations of biomolecules, or are expected to become relevant with that respect. In particular we consider the multicanonical ensemble and the replica exchange method (also known as parallel tempering or method of multiple Markov chains).

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Comparison of decadal global water vapor changes derived from independent satellite time series

et al. 2014

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Citation:Mieruch, S., M. Schröder, S. Noël, and J. Schulz (2014), Comparison of decadal global water vapor changes derived from independent satellite time series, J. Geophys. Res. Atmos., 119, 12,489-12,499, doi:10.1002 Abstract We analyze trends in total column water vapor (TCWV) retrieved from independent satellite observations and retrieval schemes. GOME-SCIAMACHY (Global Ozone Monitoring Experiment-SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY) measurements are carried out in the visible part of the solar spectrum and present a partly cloud-corrected climatology that is available over land and ocean. The HOAPS (Hamburg Ocean Atmosphere Parameters and Fluxes from Satellite Data) product, provided by EUMETSAT's Satellite Application Facility on Climate Monitoring is based on passive microwave observations from the Special Sensor Microwave/Imager. It also includes the TCWV from cloudy pixels but is only available over oceans. The common observation time period is between 1996 and 2005. Due to the relatively short length of the period, the strong interannual variability with strong contributions from El Niño and La Niña events and the strong anomaly at the start of the common period, caused by the 1997/1998 El Niño, the observed trends should not be interpreted as long-term climate trends. After subtraction of average seasonality from monthly gridded data, a linear model and a level shift model have been fitted to the HOAPS and GOME-SCIAMACHY data, respectively. Autocorrelation and cross correlation of fit residuals are accounted for in assessing uncertainties in trends. The trends observed in both time series agree within uncertainty margins. This agreement holds true for spatial patterns, magnitudes, and global averages. The consistency increases confidence in the reliability of the trends because the methods, spectral range, and observation technique as well as the satellites and their orbits are completely independent of each other. The similarity of the trends in both data sets is an indication of sufficient stability in the observations for the time period of ≈ 10 years.

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Markov Chain Monte Carlo Simulations and Their Statistical Analysis

Cited by 365 publications

References 0 publications

Nonperturbative U(1) gauge theory at finite temperature

Nonperturbative U(1) gauge theory at finite temperature

Markov Chain Monte Carlo Methods for Simulations of Biomolecules

Comparison of decadal global water vapor changes derived from independent satellite time series

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