Parallel Tempering Algorithm in Monte Carlo Simula5on

Transcription

1 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)

2 Monte Carlo Algorithms Mo5va5on: Idea: Difficulty in direct sampling Construct a Markov chain with desired equilibrium distribu5on è Es5mate with Bayesian inference Underlying principle: Detailed balance condi5on with a certain transi5on probability π(x)p(x,y)=π(y)p(y,x)

3 Boltzmann Distribu5on Canonical ensemble for systems taking discrete values of energy The most common ensemble in sta5s5cal mechanics Probability distribu5on: P(E;T)= e E/ k B T /Z(T) Objec5ve: Employ Monte Carlo algorithms to calculate physical quan55es of interest

4 N- vector Model Mathema5cal model of ferromagne5sm in sta5s5cal mechanics Square/cubic lavce containing magne5zed spins with dimension N N = 1 è Ising model N = 2 è XY model N = 3 è Heisenberg model N = 4 è Standard model Physical Quan55es Hamiltonian: H= J <i,j> < s i, s j > Magne5za5on: M= i s i h^p://rutgersscholar.rutgers.edu/volume2/cowldevl/fig1.jpg

5 Metropolis Algorithm Transi5on probability: P flip =min {1, e E/ k B T } Flow 1. Generate an ini5al state randomly 2. Equilibra5on 5me, during which at each step: Choose a spin randomly and propose a trail flip Accept the flip with a probability P flip, or otherwise retain the original state

6 Metropolis Algorithm Flow (Cont d) 3. Sampling 5me, during which at each step: Choose a spin randomly and propose a trail flip Accept the flip with a probability P flip and store the physical quan5tes, or otherwise retain the original state 4. Calculate the average physical quan55es of interest

7 Kraken XT5 Located in ORNL Cray Linux Environment (CLE) computed node, each with 12 cores & 16 GB memory h^p:// high- right- 425.jpg

8 Experiment 1: 2D Ising 1 9 equilibra5on steps & 1 9 sampling steps 1 Mean Magne'za'on Per Spin N= Temperature ([J])

9 Drawback of Metropolis Algorithm Low convergence rate at low temperatures Reason: For lower temperature systems, For E>, P flip = min {1, e E/ k B T } For E<, P flip = min {1, e E/ k B T } =1 è trapped in energy minimum è fail to generate states according to Boltzmann distribu5on

10 Parallel Tempering Objec5ve: Run Metropolis Algorithm on different temperatures & allow exchange of states every certain amount of sampling steps è High- temperature configura5ons apply to low- temperature systems & rescue them from being trapped P exchange = min {1, e βδe } ;β= 1/ k B T

11 Experiment 2: 2D Ising model 1 9 equilibra5on steps & 1 9 sampling steps Varying number of evenly- distributed exchanges 1 Mean magne'za'on per spin Total replica exchange count exchanges 1 exchanges Temperature ([J])

12 Experiment 2: 2D Ising model 1 9 equilibra5on steps & 1 9 sampling steps Varying number of evenly- distributed exchanges 1 Mean magne'za'on per spin Total replica exchange count exchanges 1 exchanges Temperature ([J])

13 Experiment 2: 2D Ising model 1 9 equilibra5on steps & 1 9 sampling steps Varying number of evenly- distributed exchanges 1 Mean magne'za'on per spin Total replica exchange count 25 exchanges 1^4 exchanges 1^7 exchanges Temperature (J)

14 Experiment 3 Convergence of magne5c suscep5bility 1*1 2- D Ising Model (square lavce) Total equilibra5on step = 1^9 Total Monte Carlo sampling step = 1^9 Temperature Range K b T =.5 (J) ~ 5.5 (J) 96 processors covering the temperature range Second moment requires more 5me to converge

15 Experiment 3 Convergence of magne5c suscep5bility 2 Magne'c Suscep'bility S/S k b T (J)

16 Experiment 3 Convergence of magne5c suscep5bility 2 Magne'c Suscep'bility S/S S/P 1^ k b T (J)

17 Experiment 3 Convergence of magne5c suscep5bility 2 Magne'c Suscep'bility S/P 1^4 P/P 1^ k b T (J)

18 Experiment 3 Convergence of magne5c suscep5bility 2 Magne'c Suscep'bility P/P 1^ k b T (J)

19 Experiment 4 Geometric temperature spacing 1*1 2- D Ising Model (square lavce) Total equilibra5on step = 1^9 Total Monte Carlo sampling step = 1^9 Number of exchange = 1^6 Temperature Range K b T =.5 (J) ~ 5. (J) 96 processors covering the temperature range

20 Experiment 4 Geometric temperature spacing Magne'c suscep'bility per spin k b T (J) A/SE G/SE

21 Experiment 4 Geometric temperature spacing 1 Replica exchange difficulty throughout temperature range Exchange acceptance ra'o A/SE G/SE k b T (J)

22 Experiment 4 Geometric temperature spacing 1 Replica exchange difficulty throughout temperature range Exchange acceptance ra'o A/SE G/SE Ideal k b T (J)

23 Adap5ve temperature spacing E E E E MC MC MC MC

24 Adap5ve temperature spacing E E E E MC MC MC MC

25 Adap5ve temperature spacing E E E E MC MC MC MC

26 Adap5ve temperature spacing E1 T=1. E1 T=2. E1 T=3. E1 T=4. Adjust Adjust Adjust Adjust E2 T=1. E2 T=2.3 E2 T=2.7 E2 T=4. MC MC MC MC

27 Experiment 5 Adap5ve temperature spacing 1*1 2- D Ising Model (square lavce) Total equilibra5on step = 1^9 Total Monte Carlo sampling step = 1^9 Number of exchange = 1^4 Temperature Range K b T =.5 (J) ~ 5.5 (J) 96 processors covering the temperature range

28 Experiment 5 Adap5ve temperature spacing Replica exchange difficulty with/without adap've spacing Exchange acceptance ra'o Regular Adapt/ k b T (J)

29 Experiment 5 Adap5ve temperature spacing Magne'c Suscep'bility with/without adap've temp spacing 15 Regular Magne'c suscep'bility k b T (J)

30 Experiment 5 Adap5ve temperature spacing Magne'c Suscep'bility with regular & adap've spacing 15 Regular Adap5ve Magne'c suscep'bility k b T (J)

31 Implementa5on on other models 2- D Heisenberg Model 1*1 2- D Heisenberg Model (square lavce) Total equilibra5on step = 1^9 Total Monte Carlo sampling step = 1^9 Number of exchange = 1^4 Temperature Range K b T =.1 (J) ~ 4.25 (J) 18 processors covering the temperature range

32 Implementa5on on other models 2- D Heisenberg Model 12 1 Magne'c Suscep'bility S/P k b T (J)

33 Implementa5on on other models 2- D Heisenberg Model 12 1 Magne'c Suscep'bility S/P P/P k b T (J)

34 Implementa5on on other models 2- D Heisenberg Model 12 1 Magne'c Suscep'bility P/P A/P k b T (J)

35 Implementa5on on other models 2- D Heisenberg Model.8 Exchange acceptance rate Regular Adap5ve/ k b T (J)

36 Implementa5on on other models 3- D Heisenberg Model 25*25*25 3- D Heisenberg Model (square lavce) Total equilibra5on step = 1^9 Total Monte Carlo sampling step = 1^9 Number of exchange = 1^4 Temperature Range K b T =.3 (J) ~ 4.5 (J) 192 processors covering the temperature range

37 Implementa5on on other models 3- D Heisenberg Model 25 Magne'c suscep'bility Serial k b T (J)

38 Implementa5on on other models 3- D Heisenberg Model 25 Magne'c suscep'bility Serial Adap5ve k b T (J)

39 Future Direc5on: Interoperable Execu5ve Library (IEL) Sotware framework used for mul5- physics simula5ons Designed to execute & schedule in parallel a series of physics solvers Objec5ve: Run parallel tempering on different parameter spaces with data & informa5on change on shared boundaries