On the efficient numerical solution of lattice systems with low-order couplings

Ammon, Andreas; Genz, Alan; Hartung, Tobias; Jansen, Karl; Leövey, Hernan; Volmer, Julia

doi:10.1016/j.cpc.2015.09.004

Cited by 14 publications

(42 citation statements)

References 27 publications

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“…Comparing RNI directly with the for this model optimal MCMC Cluster algorithm shows an improvement of the error for RNI of several orders of magnitude even for a number of integration points where we are not yet in the intermediate, exponentially fast error scaling regime. Further applications of RNI to the anharmonic oscillator are reported in [9] and show extremely good results over a very broad range of parameters.…”

Section: Resultsmentioning

confidence: 98%

Applying recursive numerical integration techniques for solving high dimensional integrals

Ammon¹,

Genz²,

Hartung³

et al. 2017

Proceedings of 34th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2016)

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The error scaling for Markov-Chain Monte Carlo techniques (MCMC) with N samples behaves like 1/ √ N. This scaling makes it often very time intensive to reduce the error of computed observables, in particular for applications in lattice QCD. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling. One candidate for such an alternative integration technique is the method of recursive numerical integration (RNI). The basic idea of this method is to use an efficient low-dimensional quadrature rule (usually of Gaussian type) and apply it iteratively to integrate over high-dimensional observables and Boltzmann weights. We present the application of such an algorithm to the topological rotor and the anharmonic oscillator and compare the error scaling to MCMC results. In particular, we demonstrate that the RNI technique shows an error scaling in the number of integration points m that is at least exponential.34th annual International Symposium on Lattice Field Theory

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Section: Resultsmentioning

confidence: 98%

Applying recursive numerical integration techniques for solving high dimensional integrals

Ammon¹,

Genz²,

Hartung³

et al. 2017

Proceedings of 34th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2016)

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“…Hence neither term can suppress the error of the other which we interpret as the cause of the peak in the relative error. Figure 4 shows the same comparison as Figure 3 but computations were performed with 1024bit floating point arithmetic, 6 i.e., approximately 307 digit precision. Again, we observe that the polynomially exact method operates on machine precision (as to be expected).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…with c 1 := n j=1m j , c 2 = 2 −n e −nµ , and c 3 = (−1) n 2 −n e nµ . Mathematically speaking, (6) is an application of "Fubini" 1 and translation invariance of the Haar measure since det D only depends on n−1 j=0 U j . We will frequently assume this form of D in analytic computations and we have implemented this form of D in order to reduce computational overhead.…”

Section: One Dimensional Lattice Qcdmentioning

confidence: 99%

Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness

et al. 2016

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In this paper we describe a new integration method for the groups UðNÞ and SUðNÞ, for which we verified numerically that it is polynomially exact for N ≤ 3. The method is applied to the example of onedimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that also in other regions of parameter space our new method leads to errors which are reduced by orders of magnitude.

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“…For the future we would like to see QMC in new territories, to tackle a significantly wider range of more realistic problems. Some emerging new application areas of QMC include e.g., Bayesian inversion [8,72], stochastic wave propagation [20,21], quantum field theory [3,47], and neutron transport [23,36].…”

Section: Software Resourcesmentioning

confidence: 99%

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube [0, 1] s and in R s , and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension s under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when s is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications. * Frances Y. Kuo ( ):

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On the efficient numerical solution of lattice systems with low-order couplings

Cited by 14 publications

References 27 publications

Applying recursive numerical integration techniques for solving high dimensional integrals

Applying recursive numerical integration techniques for solving high dimensional integrals

Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

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