DOCSLIB.ORG

Monte Carlo Methods for Rough Energy Landscapes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Monte Carlo Methods for Rough Energy Landscapes Monte Carlo Methods for Rough Energy Landscapes Jon Machta University of Massachusetts Amherst, Santa Fe Institute SFI, August 9, 2013 Collaborators • Stephan Burkhardt • Chris Pond • Jingran Li • Richard Ellis Motivation • How to sample equilibrium states of systems with rugged free energy landscapes, e.g. spin glasses, configurational glasses, proteins, combinatorial optimization problems. Problem • Markov chain Monte Carlo (MCMC) at a single temperature such as the Metropolis algorithm gets stuck in local minima. Problem • Markov chain Monte Carlo (MCMC) at a single temperature such as the Metropolis algorithm gets stuck in local minima. Outline • Population Annealing • Parallel Tempering • PA vs PT • Conclusions 5 spin glasses and perhaps other systems with rough free 0.1 energy landscapes. By using weighted averages over an ensemble of runs, biases inherent in a single run can be 0.01 can be made small and high precision results can be ob- tained. The method can be optimized by minizing the ⇤ 0.001 variance of the free energy estimator. If the variance of the free energy estimator is less than unity, high precision 4 results can be obtained from a relatively small ensemble 10 of runs. The method is comparable in efficiency to par- allel tempering and is well suited to parallelization. 105 0.1 0.2 0.3 0.4 Var ⇥F FIG. 3: The variance of the free energy Var(βF˜) vs. the prob- ability of a small overlap, ⇥, for a sample⇥ of 25 disorder real- izations. 50 40 30 20 Acknowledgments 10 3813 3814 3815 3816 3817 This work was supported in part from NSF grant FIG. 4: The histogram of the dimensionless free energy, βF . DMR-0907235. I am grateful to Helmut Katzgraber and − PopulationThe solidAnnealing line is the best fit to a Gaussian distribution. Burcu Yucesoy for helpful discussions. [1] K. Hukushima and Y. Iba, in THE MONTE CARLO [7] C. Jarzynski, Phys. Rev. E 56, 5018 (1997). METHOD IN THE PHYSICAL SCIENCES: Celebrating [8] M. E. J. Newman and G. T. Barkema, Monte Carlo the 50th Anniversary of the Metropolis Algorithm, edited Methods in Statistical Physics (Oxford, Oxford, 1999). by J. E. Gubernatis (AIP, 2003), vol. 690, pp. 200–206. [9] H. G. Katzgraber and A. P. Young, Phys. Rev. B 65, [2] R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 57, 214402 (2002). 2607 (1986). [10] F. Krzakala and O. C. Martin, Phys. Rev. Lett. 85, 3013 [3] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, (2000). • Modification of simulated annealing1604 (1996).for equilibrium [11] M. Palassini and A. P. Young, Phys. Rev. Lett. 85, 3017 [4] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science (2000). sampling. 220, 671 (1983). [12] E. Marinari and F. Zuliani, J. Phys. A: Math. Gen. 32, [5] A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 7447 (1999). A population of replicas is cooled61, according 2635 (1988). to an • [6] J. B. Anderson, J. Chem. Phys. 63, 1499 (1975). annealing schedule. Each replica is acted on by a MCMC (e.g. Metropolis) at the current temperature. • During each temperature step, the population is differentially resampled according to Boltzmann weights to maintain equilibrium. Population Annealing E1 E2 E3 E4 E5 ER R replicas β=1/kT ... Population Annealing E1 E2 E3 E4 E5 ER R replicas β=1/kT ... βʹ Population annealing = simulated annealing with differential reproduction (resampling) of replicas Temperature Step E1 E2 E3 E4 E5 ER β ... exp [ (β β)Ej] ⇥j(β, β)= − − Q(β, β) Replica j is reproduced nj times where nj is an integer random R variate with mean τj. j=1 exp [ (β β)Ej] Q(β, β)= − − R Temperature Step E1 E2 E3 E4 E5 ER β ... βʹ exp [ (β β)Ej] ⇥j(β, β)= − − Q(β, β) Replica j is reproduced nj times where nj is an integer random R variate with mean τj. j=1 exp [ (β β)Ej] Q(β, β)= − − R Resampling e.g. n1 =1, n2 =0, n3 =3, n4 =1 exp [ (β β)Ej] ⇥j(β, β)= − − Q(β, β) Replica j is reproduced nj times where nj is an integer random R variate with mean τj. j=1 exp [ (β β)Ej] Q(β, β)= − − R Nearest integer resampling Prob(n = ⌧ )=⌧ ⌧ Minimizes variance of nj. b c d e Maximizes number of ancestors with Prob(n = ⌧ )=⌧ ⌧ descendants. d e b c Population size fluctuates. Prob(n ⌧ > 1) = 0 − Population Annealing is related to... ✦Simulated annealing ✦Sequential Monte Carlo - See e.g. “Sequential Monte Carlo Methods in Practice”, A. Doucet, et. al. (2001) - aka Particle Filters - Nested Sampling, Skilling • Go with the Winners, Grassberger (2002) • Diffusion (quantum) Monte Carlo • Nonequilibrium Equality for Free Energy Differences, Jarzynski (1997) • Histogram Re-weighting, Swendsen and Ferrenberg (1988) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i 1 R I(β, β0, λ)=E log( Y ) R j j=1 X JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i 1 R I(β, β0, λ)=E log( Y ) R j j=1 X dI(β, β0, λ) A 0 = h iR dλ λ=0 JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i 1 R 1 VarY 1 I(β, β0, λ)=E log( Yj) = log(EY ) +O R 2 3/2 j=1 − 2R (EY ) R X ✓ ◆ dI(β, β0, λ) A 0 = h iR dλ λ=0 JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i 1 R 1 VarY 1 I(β, β0, λ)=E log( Yj) = log(EY ) +O R 2 3/2 j=1 − 2R (EY ) R X ✓ ◆ dI(β, β0, λ) 1 A 0 = = E0A +O h iR dλ R λ=0 ✓ ◆ JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Suppose the population at � is a correct, iid sample: Y =exp[ (β0 β)E + λA ] i − − i i 1 R 1 VarY 1 I(β, β0, λ)=E log( Yj) = log(EY ) +O R 2 3/2 j=1 − 2R (EY ) R X ✓ ◆ dI(β, β0, λ) 1 A 0 = = E0A +O h iR dλ R λ=0 ✓ ◆ A bias inversely proportional to the population size results from the re-weighting to �´ JM, R. Ellis, JSP 144, 544 (2011) Convergence to Equilibrium Resampling complicates the analysis... Conjecture: Fixing other parameters of population annealing, observable converge to their equilibrium values inversely in population size. Quantifying Convergence to Equilibrium • Do many runs for each of many population sizes. Expensive. • Is there an analog to the autocorrelation function and the exponential autocorrelation time? Quantifying Convergence to Equilibrium • Idea: resample small populations from a single run with a large population (bootstrap). – Choose a small population of parents and measure the observable in their descendants. – Repeat many times for different populations sizes. based on 300000*100*10 realization 1 realization 2 log10|<q2>-<q2>exact| log systematic error -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 3.5 4.0 4.5 5.0 log10(R) log population size • Problem: A small sample within a large population is not the same as an isolated small population because the sample within the large populations exists in a more competitive environment. realization 2 =bootstrap log10(|q2R-q2exact|) logsystematicerror 100runs =many runs 100000subsets -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 3.0 3.2 3.4 3.6 3.8 log10(R) log population size 3 the multinomial distribution p [R; n1,...,nR; ⇢1/R, . , ⇢R/R] for R trials. In this implementation the population size is fixed. Other valid resampling methods are available. For example, the number of copies of replica j can be chosen as a PoissonDirect random variable Estimate with mean proportional of Free to ⇢j(β, β0 ),Energies in which case the population size fluctuates [13]. For large R and small (β0 β), the resampled distribution is close to an equilibrium ensemble at the new temperature, − β0. However, the regions of the equilibrium distribution for β0 that di↵er significantly from the equilibrium distribution for β are not well sampled, leading to biasesR exp in the [ population(β β)E at β]0. In addition, due to resampling, the replicas are no longer independent. To mitigate bothj=1 of these problem, thej equilibrating subroutine is now applied. Finally, Q(β, β)= − − observables are measured by averaging over the population.R The entire algorithm consists of S 1 steps: in step k the temperature is lowered from βS k to βS k 1 via resampling − − − − followed by the application of the equilibratingK subroutine and data collection at temperature βS k 1. Population annealing permits one to estimate free energy di↵erences. If the annealing schedule− begins− at infinite βkF (βk)= ln Q(β⇥+1, β⇥)+βK F (βK ) ˜ temperature corresponding− to βS = 0, then it yields an estimate of the absolute free energy F (βk) at every temperature ⇥=k in the annealing schedule. The following calculation shows that the normalization factor Q(β, β0) is an estimator of the ratio of the partition functions at the two temperatures: β0E Z(β ) e− γ Derivation: 0 = γ Z(β) P Z(β) βEγ (β0 β)E e− = e− − γ ( ) Z(β) γ X (β0 β)E = e− − γ h iβ R 1 (β0 β)E e− − j = Q(β, β0).
Load more

Recommended publications
  • Parallel-Tempered Stochastic Gradient Hamiltonian Monte Carlo for Approximate Multimodal Posterior Sampling
    1st Symposium on Advances in Approximate Bayesian Inference, 20181–6 Parallel-tempered Stochastic Gradient Hamiltonian Monte Carlo for Approximate Multimodal Posterior Sampling ∗ Rui Luo [email protected] ∗ Qiang Zhang [email protected] Yuanyuan Liu [email protected] American International Group, Inc. Abstract We propose a new sampler that integrates the protocol of parallel tempering with the Nose-Hoover´ (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially facilitates deep Bayesian learning on large datasets where complex multi- modal posteriors and mini-batch gradient are encountered. 1. Introduction In Bayesian inference, one of the fundamental problems is to efficiently draw i.i.d. samples from the posterior distribution π¹θj Dº given the dataset D = fxg, where θ 2 D denotes the variable of interest. Provided the prior distribution π¹θº and the likelihood per datum `¹θ; xº, the posterior to be sampled can be formulated as Ö π¹θj Dº = π¹θº `¹θ; xº: (1) x 2D To facilitate posterior sampling, the framework of Markov chain Monte Carlo (MCMC) has been established, which has initiated a broad family of methods that generate Markov chains to propose new sample candidates and then apply tests of acceptance in order to guarantee the condition of detailed balance. Methods like the Metropolis-Hastings (MH) algorithm (Metropolis et al., 1953; Hastings, 1970), the Gibbs sampler (Geman and Geman, 1984), and the hybrid/Hamiltonian Monte Carlo (HMC) (Duane et al., 1987; Neal, 2011) are famous representatives for the MCMC family where different generating procedures of Markov chains are adopted; each of those methods has achieved great success on various tasks in statistics and related fields.
    [Show full text]
  • Chapter 2 Monte Carlo Simulations
    Chapter 2 Monte Carlo Simulations David J. Earl and Michael W. Deem Summary AdescriptionofMonteCarlomethodsforsimulationofproteinsisgiven. Advantages and disadvantages of the Monte Carlo approach are presented. The the- oretical basis for calculating equilibrium properties of biological molecules by the Monte Carlo method is presented. Some of the standard and some of the more re- cent ways of performing Monte Carlo on proteins are presented. A discussion of the estimation of errors in properties calculated by Monte Carlo is given. Keywords: Markov chain Metropolis algorithm Monte Carlo Protein simula- · · · tion Stochastic methods · 1 Introduction The term Monte Carlo generally applies to all simulations that use stochastic meth- ods to generate new configurations of a system of interest. In the context of mole- cular simulation, specifically, the simulation of proteins, Monte Carlo refers to importance sampling, which we describe in Sect. 2, of systems at equilibrium. In general, a Monte Carlo simulation will proceed as follows: starting from an initial configuration of particles in a system, a Monte Carlo move is attempted that changes the configuration of the particles. This move is accepted or rejected based on an ac- ceptance criterion that guarantees that configurations are sampled in the simulation from a statistical mechanics ensemble distribution, and that the configurations are sampled with the correct weight. After the acceptance or rejection of a move, one calculates the value of a property of interest, and, after many such moves, an accu- rate average value of this property can be obtained. With the application of statistical mechanics, it is possible to calculate the equilibrium thermodynamic properties of the system of interest in this way.
    [Show full text]
  • LAMMPS – an Object Oriented Scientific Application
    LAMMPS – An Object Oriented Scientific Application Dr. Axel Kohlmeyer (with a little help from several friends) Associate Dean for Scientific Computing College of Science and Technology Temple University, Philadelphia http://sites.google.com/site/akohlmey/ [email protected] Workshop on Advanced Techniques in Scientific Computing LAMMPS is a Collaborative Project A few lead developers and many significant contributors: ● Steve Plimpton, Aidan Thompson, Paul Crozier, Axel Kohlmeyer - Roy Pollock (LLNL), Ewald and PPPM solvers - Mike Brown (ORNL), GPU package - Greg Wagner (Sandia), MEAM package for MEAM potential - Mike Parks (Sandia), PERI package for Peridynamics - Rudra Mukherjee (JPL), POEMS package for rigid body motion - Reese Jones (Sandia), USER-ATC package for coupling to continuum - Ilya Valuev (JIHT), USER-AWPMD package for wave-packet MD - Christian Trott (Sandia), USER-CUDA package, KOKKOS package - A. Jaramillo-Botero (Caltech), USER-EFF electron force field package - Christoph Kloss (JKU), LIGGGHTS package for DEM and fluid coupling - Metin Aktulga (LBL), USER-REAXC package for C version of ReaxFF - Georg Gunzenmuller (EMI), USER-SPH package - Ray Shan (Sandia), COMB package, QEQ package - Trung Nguyen (ORNL), RIGID package, GPU package - Francis Mackay and Coling Denniston (U Western Ontario), USER-LB Workshop on Advanced Techniques in Scientific Computing LAMMPS is an Extensible Project ● ~2900 C/C++/CUDA files, 120 Fortran files, about 900,000 lines of code in core executable ● Only about 200 files are essential, about
    [Show full text]
  • Desmond Users Guide Release 3.4.0 / 0.7
    Desmond Users Guide Release 3.4.0 / 0.7 D. E. Shaw Research December 18, 2012 CONTENTS 1 Front matter 3 1.1 Notice..................................................3 1.2 Copyright.................................................3 1.3 Trademarks................................................3 2 Preface 5 2.1 Intended audience............................................5 2.2 Prerequisites...............................................5 2.3 Format conventions...........................................5 2.4 About the equations...........................................6 3 Key Concepts 7 3.1 What is Desmond?............................................7 3.2 Forces..................................................8 3.3 Particles.................................................9 3.4 Force fields................................................9 3.5 Space................................................... 10 3.6 Time................................................... 10 3.7 Dynamics................................................. 11 3.8 Using Desmond............................................. 12 4 Running Desmond 15 4.1 About configuration........................................... 15 4.2 Invoking Desmond............................................ 16 4.3 Running Desmond in parallel...................................... 19 4.4 Configuring Desmond applications................................... 20 4.5 Naming output files........................................... 25 4.6 Configuring the built-in plugins..................................... 25 4.7 Configuring
    [Show full text]
  • Hybrid Parallel Tempering and Simulated Annealing Method
    Applied Mathematics and Computation 212 (2009) 216–228 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Hybrid parallel tempering and simulated annealing method Yaohang Li a,*, Vladimir A. Protopopescu b, Nikita Arnold c, Xinyu Zhang d, Andrey Gorin e a Department of Computer Science, North Carolina A&T State University, Greensboro, NC 27411, USA b Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA c Department of Soft Matter Physics, Institute of Experimental Physics, Johannes Kepler Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria d Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA e Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA article info abstract Keywords: In this paper, we propose a new hybrid scheme of parallel tempering and simulated Markov chain Monte Carlo annealing (hybrid PT/SA). Within the hybrid PT/SA scheme, a composite system with multi- Simulated annealing ple conformations is evolving in parallel on a temperature ladder with various transition Parallel tempering step sizes. The simulated annealing (SA) process uses a cooling scheme to decrease the temperature values in the temperature ladder to the target temperature. The parallel tem- pering (PT) scheme is employed to reduce the equilibration relaxation time of the compos- ite system at a particular temperature ladder configuration in the SA process. The hybrid PT/SA method reduces the waiting time in deep local minima and thus leads to a more effi- cient sampling capability on high-dimensional complicated objective function landscapes.
    [Show full text]
  • The American Ceramic Society 38Th International Conference
    The American Ceramic Society 38th International Conference & Exposition on Advanced Ceramics and Composites ABSTRACT BOOK January 26–31, 2014 Daytona Beach, Florida Introduction This volume contains abstracts for more than 900 presentations during the 38th International Conference & Exposition on Advanced Ceramics and Composites in Daytona Beach, Florida. The abstracts are reproduced as submitted by authors, a format that provides for longer, more detailed descriptions of papers. The American Ceramic Society accepts no responsibility for the content or quality of the abstract content. Abstracts are arranged by day, then by symposium and session title. An Author Index appears at the back of this book. The Meeting Guide contains locations of sessions with times, titles and authors of papers, but not presentation abstracts. How to Use the Abstract Book Refer to the Table of Contents to determine page numbers on which specific session abstracts begin. At the beginning of each session are headings that list session title, location and session chair. Starting times for presentations and paper numbers precede each paper title. The Author Index lists each author and the page number on which their abstract can be found. Copyright © 2014 The American Ceramic Society (www.ceramics.org). All rights reserved. MEETING REGULATIONS The American Ceramic Society is a nonprofit scientific organization that facilitates the exchange of knowledge meetings and publication of papers for future reference. The Society owns and retains full right to control its publications and its meetings. The Society has an obligation to protect its members and meetings from intrusion by others who may wish to use the meetings for their own private pro- motion purpose.
    [Show full text]
  • Comparing Monte Carlo Methods for Finding Ground States of Ising Spin
    Comparing Monte Carlo methods for finding ground states of Ising spin glasses: Population annealing, simulated annealing, and parallel tempering Wenlong Wang,1, ∗ Jonathan Machta,1, 2, y and Helmut G. Katzgraber3, 4, 2 1Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 USA 2Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501 USA 3Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA 4Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, USA Population annealing is a Monte Carlo algorithm that marries features from simulated-annealing and parallel-tempering Monte Carlo. As such, it is ideal to overcome large energy barriers in the free-energy landscape while minimizing a Hamiltonian. Thus, population-annealing Monte Carlo can be used as a heuristic to solve combinatorial optimization problems. We illustrate the capabilities of population-annealing Monte Carlo by computing ground states of the three-dimensional Ising spin glass with Gaussian disorder, while comparing to simulated-annealing and parallel-tempering Monte Carlo. Our results suggest that population annealing Monte Carlo is significantly more efficient than simulated annealing but comparable to parallel-tempering Monte Carlo for finding spin-glass ground states. I. INTRODUCTION primary purpose of this paper is to introduce population annealing as an effective method for finding ground states Spin glasses present one of the most difficult chal- of frustrated disordered spin systems. lenges in statistical physics [1]. Finding spin-glass ground Population-annealing Monte Carlo [18{22] is closely re- states is important in statistical physics because some lated to simulated annealing and also shares some simi- properties of the low-temperature spin-glass phase can larities with parallel tempering.
    [Show full text]
  • Multiscale Implementation of Infinite-Swap Replica Exchange Molecular Dynamics
    Multiscale implementation of infinite-swap replica exchange molecular dynamics Tang-Qing Yua, Jianfeng Lub,c,d, Cameron F. Abramse, and Eric Vanden-Eijndena,1 aCourant Institute of Mathematical Sciences, New York University, New York, NY 10012; bDepartment of Mathematics, Duke University, Durham, NC 27708; cDepartment of Physics, Duke University, Durham, NC 27708; dDepartment of Chemistry, Duke University, Durham, NC 27708; and eDepartment of Chemical and Biological Engineering, Drexel University, Philadelphia, PA 19104 Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved August 9, 2016 (received for review March 28, 2016) Replica exchange molecular dynamics (REMD) is a popular method values permits accelerated sampling. These replicas are then to accelerate conformational sampling of complex molecular systems. evolved in parallel using standard MD, and their temperatures are The idea is to run several replicas of the system in parallel at different periodically swapped. If the swaps are performed in a thermody- temperatures that are swapped periodically. These swaps are typically namically consistent way, the correct canonical averages at any of attempted every few MD steps and accepted or rejected according to a these temperatures can be sampled from the pieces of the replica Metropolis–Hastings criterion. This guarantees that the joint distribu- trajectories during which they feel this temperature. In the stan- tion of the composite system of replicas is the normalized sum of the dard implementation of REMD, this is done by attempting swaps symmetrized product of the canonical distributions of these replicas at at a predefined frequency and accepting or rejecting them according – the different temperatures. Here we propose a different implementa- to a Metropolis Hastings criterion.
    [Show full text]
  • Parallel Tempering: Theory, Applications, and New Perspectives PCCP
    INVITED ARTICLE www.rsc.org/pccp Parallel tempering: Theory, applications, and new perspectives PCCP David J. Earlab and Michael W. Deema a Departments of Bioengineering and Physics & Astronomy, Rice University, 6100 Main Street MS142, Houston, Texas, 77005, USA. E-mail: [email protected] b Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford, UK OX1 3NP. E-mail: [email protected] Received 13th July 2005, Accepted 11th August 2005 First published as an Advance Article on the web 5th September 2005 We review the history of the parallel tempering simulation method. From its origins in data analysis, the parallel tempering method has become a standard workhorse of physicochemical simulations. We discuss the theory behind the method and its various generalizations. We mention a selected set of the many applications that have become possible with the introduction of parallel tempering, and we suggest several promising avenues for future research. 1. Introduction is also the case that parallel tempering can make efficient use of large CPU clusters, where different replicas can be run in The origins of the parallel tempering, or replica exchange, parallel. An additional benefit of the parallel tempering meth- simulation technique can be traced to a 1986 paper by Swend- 1 od is the generation of results for a range of temperatures, sen and Wang. In this paper, a method of replica Monte Carlo which may also be of interest to the investigator. It is now was introduced in which replicas of a system of interest were widely appreciated that parallel tempering is a useful and simulated at a series of temperatures.
    [Show full text]
  • PLUMED Tutorial
    PLUMED Tutorial A portable plugin for free-energy calculations with molecular dynamics CECAM, Lausanne, Switzerland September 28, 2010 - October 1, 2010 This document and the relative computer exercises have been written by: Massimiliano Bonomi Davide Branduardi Giovanni Bussi Francesco Gervasio Alessandro Laio Fabio Pietrucci Tutorial website: http://sites.google.com/site/plumedtutorial2010/ PLUMED website: http://merlino.mi.infn.it/∼plumed PLUMED users Google group: [email protected] PLUMED reference article: M. Bonomi, D. Branduardi, G. Bussi, C. Camilloni, D. Provasi, P. Rai- teri, D. Donadio, F. Marinelli, F. Pietrucci, R.A. Broglia and M. Par- rinello, PLUMED: a portable plugin for free-energy calculations with molecular dynamics, Comp. Phys. Comm. 2009 vol. 180 (10) pp. 1961- 1972. 1 Contents 1 Compilation 4 1.1 PLUMED compilation . 4 1.1.1 Compile PLUMED with GROMACS . 5 1.1.2 Compile PLUMED with QUANTUM ESPRESSO 9 1.1.3 Compile PLUMED with LAMMPS . 10 2 Basics: monitoring simulations 13 2.1 Syntax for collective variables . 14 2.2 Monitoring a CV . 16 2.3 Postprocessing with driver ............... 19 3 Basics: biasing simulations 23 3.1 Restrained/steered molecular dynamics . 23 3.1.1 An umbrella sampling calculation. Alanine dipep- tide. 23 3.1.2 A steered molecular dynamics example: targeted MD. 27 3.1.3 A programmed steered MD with steerplan. 30 3.1.4 Soft walls . 32 3.2 Committment analysis . 33 3.3 Metadynamics . 35 3.4 Restarting metadynamics . 38 3.5 Free-energy reconstruction . 38 3.6 Well-tempered metadynamics . 41 4 Parallel machines 43 4.1 Exploiting MD parallelization .
    [Show full text]
  • Conditions for Rapid Mixing of Parallel and Simulated Tempering on Multimodal Distributions
    The Annals of Applied Probability 2009, Vol. 19, No. 2, 617–640 DOI: 10.1214/08-AAP555 © Institute of Mathematical Statistics, 2009 CONDITIONS FOR RAPID MIXING OF PARALLEL AND SIMULATED TEMPERING ON MULTIMODAL DISTRIBUTIONS BY DAWN B. WOODARD,SCOTT C. SCHMIDLER AND MARK HUBER Duke University We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, which are applica- ble to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct conse- quence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models, and for the mean-field Ising model. 1. Introduction. Stochastic sampling methods have become ubiquitous in statistics, computer science and statistical physics. When independent samples from a target distribution are difficult to obtain, a widely-applicable alternative is to construct a Markov chain having the target distribution as its limiting distri- bution. Sample paths from simulating the Markov chain then yield laws of large numbers and often central limit theorems [20, 26], and thus are widely used for Monte Carlo integration and approximate counting. Such Markov chain Monte Carlo (MCMC) methods have revolutionized computation in Bayesian statistics [9], provided significant breakthroughs in theoretical computer science [10]and become a staple of physical simulations [2, 18]. A common difficulty arising in the application of MCMC methods is that many target distributions arising in statistics and statistical physics are strongly multi- modal; in such cases, the Markov chain can take an impractically long time to reach stationarity.
    [Show full text]
  • Qualitative Properties of Parallel Tempering and Its Infinite Swapping
    Qualitative properties of parallel tempering and its in…nite swapping limit Paul Dupuis Division of Applied Mathematics Brown University CERMICS workshop: Computational statistics and molecular simulation with Jim Doll (Department of Chemistry, Brown University) Yufei Liu, Pierre Nyquist, Nuria Plattner, David Lipshutz 2 February 2016 Paul Dupuis (Brown University) 2 February 2016 Outline 1 General perspective 2 Problem of interest 3 Large deviation theory for empirical measure 4 An accelerated algorithm–parallel tempering 5 Three uses of large deviation: First large deviations analysis and the in…nite swapping limit Approximation of risk-sensitive functionals–a second rare event issue The particle/temperature empirical measure–a diagnostic for convergence 6 Implementation issues and partial in…nite swapping 7 References Paul Dupuis (Brown University) 2 February 2016 Advantages: generally works directly with quantities of interest–not a surrogate Disadvantages: is an asymptotic theory (is it the right asymptotic for the problem at hand?); often requires solution to a variational problem Large relative variance of standard Monte Carlo when estimating small probabilities Poor communication/ergodicity properties when using MCMC for stationary distributions For purposes of design and qualitative understanding of the methods, need some way to characterize the impact of these events Large deviation theory gives such information General perspective When using Monte Carlo, there are several ways “rare events” can adversely a¤ect performance or
    [Show full text]