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Population annealing study of the frustrated Ising antiferromagnet - - PowerPoint PPT Presentation

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice Michal Borovsk y

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice

Michal Borovsk´ y

Department of Theoretical Physics and Astrophysics, University of P. J. ˇ Saf´ arik in Koˇ sice, Slovakia

18th November 2015

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 1/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Collaboration

  • Dr. Martin Weigel (Applied Mathematics Research

Centre, Coventry University, UK)

  • Dr. Lev Yu. Barash (Landau Institute for Theoretical

Physics, Chernogolovka, Russia)

  • Dr. Milan ˇ

Zukoviˇ c (UPJˇ S, Koˇ sice, Slovakia)

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 2/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 3/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 4/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Population annealing (PA)

Introduction

  • K. Hukushima and Y. Iba, Population Annealing and Its Application to a

Spin Glass, AIP Conf. Proc. 690, 200 (2003).

suitable for systems with rough free energy surfaces (spin glasses, frustrated spin systems, complex biomolecular systems, etc.) used as an alternative to parallel tempering combination of simulated annealing, population algorithms and sequential Monte Carlo method provides a good estimate of free energy

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 5/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Population annealing

Algorithm

initialize population of RK replicas at βK+1 = 0 for βk from βK to β0 with step ∆β = βk − βk+1 partition function ratio: Qk =

1 ˜ Rβk+1 ˜ Rβk+1

  • j−1

exp [−∆βEj] for all replicas do:

normalize weights: τj =

1 Qk exp [−∆βEj]

resampling: create N

  • Rβk/˜

Rβk+1

  • τj
  • copies of replica

(N [a] - Poisson random variate with mean value a)

calculate new size of a population ˜ Rβk equilibrate replicas for θk Monte Carlo sweeps calculate observables and the free energy: −βk ˜ F(βk) = ln Ω +

k

  • l=K

ln Ql

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 6/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 7/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

CPU vs. GPU

Performance comparison

2004 2006 2008 2010 2012 2014 10

1

10

2

10

3

10

4

year peak performance in GFLOPS

GPUs NVIDIA GeForce (fp64) GPUs NVIDIA Tesla (fp32) GPUs NVIDIA Tesla (fp64) GPUs NVIDIA GeForce (fp32) CPUs Intel (fp32) CPUs Intel (fp64)

rok CPU - Intel GPU - NVIDIA GeForce GPU - NVIDIA Tesla 2004 Pentium 4 570J (3.8GHz) 6800 GT

  • 2006

Core 2 Duo E6700 (2.66GHz) 7950 GT

  • 2008

Core 2 Quad Q9400 (2.66GHz) 9800 GT (112 CUDA cores @ 600MHz) C870 (128@600MHz) 2010 Core i7-980 (3.33GHz) GTX 480 (448@607MHz) C2070 (448@575MHz) 2012 Core i7-3770K (3.5GHz) GTX 680 (1536@1006MHz) K20 (2496@706MHz) 2013

  • GTX 780 Ti (2880@875MHz)

K40 (2880@705MHz) 2014 Core i7-4790K (4GHz) GTX Titan Z(5760@705MHz) K80 (4992@562MHz) GTX 980 (2048@1126MHz) Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 8/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

GPU CUDA architecture

Schematic depiction

  • M. Weigel, Journal of Computational Physics 231 (2012) 30643082

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 9/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

CUDA program

GPU program:

Host code - ANSI C Device code - ANSI C extended by keywords for kernels (parallel functions) and data structures

NVIDIA C compiler (nvcc) Program execution:

THREAD (WARP) BLOCK GRID

SIMT - ”single instruction

multiple threads”

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 10/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Parallelizing the PA algorithm

2 levels of parallelism:

  • ver replicas (τi, Q) → 1 thread = 1 replica
  • ver spins of each replica (MC update,E,M) → 1 block of

threads - 8x8x8 block-wise coalesced array of spin values; 1 block = 1 replica use of parallel reduction algorithm for summing over replicas/spin values/local energy contributions parallel generation of long sequences of pseudo-random numbers - ”cuRAND” Philox 4x32 10 (p = 2128 ≈ 1038) Boltzmann factor tabulation - texture memory

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 11/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 12/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Stacked triangular Ising antiferromagnet

Sublattice partition and hamiltonian

Sublattice:

  • 1 - 2 - 3
  • 4 - 5 - 6

J1 J2

Hamiltonian:

H = −J1

  • i,j

SiSj − J2

  • i,j

SiSk Si = ±1 . . . Ising spin variable J1 < 0 . . . antiferromagnetic intralayer (interchain) interaction J2 < 0 . . . antiferromagnetic interlayer (intrachain) interaction

Geometrical frustration:

?

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 13/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Stacked triangular Ising antiferromagnet

Kinetic freezing in a standard MCMC simulation R.R. Netz and A.N. Berker, Phys. Rev. Lett. 66, 377 (1991). J1 = J2, 24x24x32 spins (Lz = 32 layers), 105 MCMC sweeps (+20% for equilibration), oz =

Lz

  • k=1

(−1)kSk, snapshot at kBT/|J1| = 0.01

1 2 3 4 −2 −1.5 −1 −0.5 kB T / |J1| E / N |J1| 0.4 0.8 1.2 C / N kB

0.25 0.5 −2.002 −2 −1.998 −1.996 −1.994 kB T / |J1| E / N |J1|

intrachain staggered magnetization oz

−32 −24 −16 −8 8 16 24 32

Spin orientation in selected chain

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 14/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 15/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

MCMC and PA comparison

GS energy and configuration J1 = J2, 24x24x32 spins (Lz = 32 layers), snapshot at kBT/|J1| = 0.1

0.1 0.2 0.3 0.4 0.5 −2 −1.999 −1.998 −1.997 −1.996 −1.995 −1.994 −1.993 kB T / |J1| E / N |J1| MCMC simulation GS energy PA, R = 10

4, θ = 102, ∆β = 0.01

PA, R = 10

4, θ = 102, ∆β = 0.005

PA, R = 10

5, θ = 102, ∆β = 0.005

intrachain staggered magnetization oz

−32 −24 −16 −8 8 16 24 32

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 16/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

MCMC and PA comparison

Family entropy

  • W. Wang, J. Machta, and H. G. Katzgraber, Phys. Rev. E 92, 013303

(2015) Family entropy: Sf = −

i νi ln νi

νi . . . fraction of the population with origin in the i-th replica eSf . . . effective number of surviving families equilibration requirement: eSf ≥ 100 (or Sf 4.6)

1 2 3 4 1 2 3 4 5 6 7 8 9 kB T / |J1| Sf PA, R = 10

4, θ = 102, ∆β = 0.01

PA, R = 10

4, θ = 102, ∆β = 0.005

PA, R = 10

5, θ = 102, ∆β = 0.005

Number of unique GS configurations: 171 (0.171% of the population size, eSf = 3.7375) 23 (0.23%, eSf = 2.1845) 32 (0.32%, eSf = 1.5857)

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 17/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

PA algorithm performance

Nvidia GTX Titan 24x24x32 R = 103 R = 104 R = 105 θ tSF[ns] tSF[ns] tSF[ns] 100 8.235 7.714 7.933 101 1.024 0.953 0.961 102 0.308 0.276 0.269 103 0.240 0.209 0.259 104 0.233 0.208 0.245 GPU memory used 17.62 MB 176.24 MB 1762.39 MB

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 18/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Outline

1 Population annealing 2 GPU realization of PA 3 Stacked triangular Ising antiferromagnet 4 Results 5 Conclusions and perspective

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 19/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Conclusions and perspective

Conclusions:

we created optimized parallel GPU program of the PA algorithm for the frustrated stacked triangular Ising antiferromagnet system reached GS (Wannier-like phase with antiferromagnetically

  • rdered spin chains) even for relatively small R and θ

equilibration criterion was not met in all simulations for a low-T region

Perspective:

choice of more effective high quality PRNG parallel resampling of replicas in the GPU global memory adaptive inverse temperature step ∆βk - histogram overlap asynchronous multispin coding - bitwise operations multi-histogram reweighting

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Thank you for your attention.

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 21/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 22/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

Population annealing

Weighted averaging

  • J. Machta, Population annealing with weighted averages: A Monte Carlo

method for rough free energy landscapes, Phys.Rev.E 82, 026704 (2010) for not sufficient values of parameters ˜ Rk, ∆β, θk ⇒ bias lets consider a set of the M independent runs of the algorithm with

  • bservables ˜

Ar(β) and free energies ˜ Fr(β) weighted averaging: ¯ A(β) =

M

  • r=1

˜ Ar(β)ωr(β), where ωr(β) =

exp[−β ˜ Fr(β)]

M

  • r=1

exp[−β ˜ Fr (β)]

. unbiased free energy: −β ¯ F(β) = ln

  • 1

M M

  • r=1

exp

  • −β ˜

Fr(β)

  • weighted averaging errors - bootstrapping
  • ptimization - minimize Var(−β ˜

F) using the same computational resources

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

GS configurations

J2 / |J1| h / |J1|

  • 1

2 4 6 8

F ( ) Fi-(F-A) ( ) Fi-A ( )

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 24/30

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Enthalpy per spin

1 2 3 4 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

kB T / |J1| E / N |J1|

(c)

0.5 1 1.5 2 2.5 3 3.5 4 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

kB T / |J1| enthalpy per spin

h/|J1| = 0 1 1.5 2 2.5 3 4 5 6 7 7.5

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Heat capacity

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2

kB T / |J1| C / N kB

(d)

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1 1.2

kB T / |J1| C / N kB

h/|J1| = 0 1 1.5 2 2.5 3 4 5 6 7 7.5

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Total magnetization per spin

1 2 3 4 0.2 0.4 0.6 0.8 1 kB T / |J1| m

(a)

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

kB T / |J1| m

h/|J1| = 0 1 1.5 2 2.5 3 4 5 6 7 7.5

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Magnetic susceptibility

1 2 3 4 0.2 0.4 0.6 0.8 1

kB T / |J1| χ

(b)

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6

kB T / |J1|

χ

h/|J1| = 0 1 1.5 2 2.5 3 4 5 6 7 7.5

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Ground state configurations

h/|J1| = 1 h/|J1| = 4

[oz

(A), oz (B), oz (C)] =

[−7.6667, −20.3333, 27.3333]

−32 −24 −16 −8 8 16 24 32

(a)

  • xy = 0.0061
  • z = 1

~

[oz

(A), oz (B), oz (C)] = [−32, 32, 0]

(b)

  • z = 2/3

~

  • xy = 1

h/|J1| = 7 h/|J1| = 7.5

[oz

(A), oz (B), oz (C)] = [6, −24, 18]

−32 −24 −16 −8 8 16 24 32

(c)

  • z = 1/2

~

  • xy = 1

[oz

(A), oz (B), oz (C)] = [12, 20, −32]

(d)

  • z = 2/3

~

  • xy = 1

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Population annealing GPU realization of PA Stacked triangular Ising antiferromagnet Results Conclusions and perspective

h > 0

Ground state configurations - degeneracy

1 2 3 4

Michal Borovsk´ y — Population annealing study of the frustrated Ising antiferromagnet on the stacked triangular lattice 30/30