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)
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
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
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
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N=.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N= N=9.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N=9 N=249.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N=249 N=1^4.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N=1^4 N=1^7.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Mean Magnetization Per Spin dependence of mean magnetization per spin Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 1 N= N=1^7.75.5.25.5.8 1.1 1.4 1.7 2 2.3 2.6 2.9
Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 3 25 2 15 1 N= 5.5 1 1.5 2 2.5 3
Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 3 25 2 15 1 N= N=1^4 5.5 1 1.5 2 2.5 3
Energy Standard Deviation dependence of energy fluctuation (SD) Total # of MC steps fixed to be 1 9 ; equilibration time set to 1 9 3 25 2 15 1 N=1^4 N=1^7 5.5 1 1.5 2 2.5 3
Total running time Running Time Dependence on # of Replica Exchanges 5 4 3 2 1 Monte Carlo Simulation takes up most of the time Parallel tempering becoming the dominant factor 9 49 249 1249 1^4 1^5 1^6 1^7 Number of replica exchange (total MC step fixed)
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