Parallel Tempering Algorithm in Monte Carlo Simulation
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1 Parallel Tempering Algorithm in Monte Carlo Simulation Tony Cheung (CUHK) Kevin Zhao (CUHK) Mentors: Ying Wai Li (ORNL) Markus Eisenbach (ORNL) Kwai Wong (UTK/ORNL)
2 Metropolis Algorithm on Ising Model Reason: difficulty of direct sampling Objective: compute average physical quantities of interest Idea: generate microstates according to Boltzmann distribution (canonical ensemble) after sufficient number of steps Boltzmann distribution: P s; T = exp ( βe s) Z Underlying principle: detailed balance, β = 1 KT
3 Metropolis Algorithm on Ising Model Simulation process 1. Randomly initialize the model 2. Choose a spin at random & make a trial flip 3. Accept the flip with probability P flip = min*1, exp β E +, β = 1 KT 4. If the flip is accepted, determine the desired physical quantities 5. Repeat steps 2-4 to obtain a sufficient number of microstates 6. Calculate the ensemble average of quantities
4 Parallel Tempering Recall: P flip = min*1, exp β E +, β = 1 Drawback: Low temperature Unlikely to accept flips with positive energy difference Trapped in energy local minimum Motivation: Run Metropolis algorithm on different temperatures & allow exchange of microstates High-temperature configuration at low-temperature system The probability of accepting an exchange is given by P excange = min*1, exp β E + KT
5 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N=
6 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N= N=
7 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N=9 N=
8 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N=249 N=1^
9 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N=1^4 N=1^
10 Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to N= N=1^
11 Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to N=
12 Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to N= N=1^
13 Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to N=1^4 N=1^
14 Total running time Running Time Dependence on # of Replica Exchanges Monte Carlo Simulation takes up most of the time Parallel tempering becoming the dominant factor ^4 1^5 1^6 1^7 Number of replica exchange (total MC step fixed)
15 Coming soon Parallel tempering Metropolis running with: Various temperature spacing (# of processors) Different exchange patterns Geometric temperature sequence Implementation on other models Goal: optimize the algorithm Better convergence with less time Self adjusting algorithms
Parallel Tempering Algorithm in Monte Carlo Simula5on
Parallel Tempering Algorithm in Monte Carlo Simula5on Tony Cheung (CUHK) Kevin Zhao (CUHK) Mentors: Ying Wai Li (ORNL) Markus Eisenbach (ORNL) Kwai Wong (UTK/ORNL) Monte Carlo Algorithms Mo5va5on: Idea:
Parallel Tempering And Adaptive Spacing. For Monte Carlo Simulation
Parallel Tempering And Adaptive pacing For Monte Carlo imulation Authors: iu Wun Cheung Department of Mathematics, CUHK Zhao, Yiwei Department of Mathematics, CUHK upervisors: Dr. Markus isenbach ORNL
Copyright 2001 University of Cambridge. Not to be quoted or copied without permission.
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