講義ノート物性研究電子版 Vol. 7, No. 2, (2018 年 11 月号 ) Metropolis [1] Buffon Neumann 20 importance Importance (Markov-chain Monte Carlo

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1 953 Metropolis [] Buffon Neumann 20 importance Importance (Markov-chain Monte Carlo : MCMC) (PMC) hukusima@phys.c.u-tokyo.ac.jp

2 : ( ) Importance ( ) MCMC PMC ( ( )) MCMC X MC MCMC 2 MCMC 2: 2 MCMC X π(x) π(x) X 2

3 MCMC MCMC 2. MCMC X {X} π(x) X (),, X (M) X (m) X (m ) X X W (X X ) MCMC π(x) W (X X ) 2 (A) : X π(x)w (X X ) = π(x )W (X X), () X X X ( ) (B) : 2 X X n P n (X) P n+ (X) P n+ (X) = X P n (X )W (X X) (2) π(x) P (X) π(x) = X π(x )W (X X) (3) X W (X X ) = (3) π(x) W (X X ) = π(x )W (X X), (4) X X W Perron-Frobenius 3

4 3: ( ) ( ) 2.2 Metropolis filter MCMC π(x)w (X X ) = π(x )W (X X) (5) (X X ) ( 3) () (reversible) (Metropolis filter) ( W M (X X ) = min, π(x ) π(x) ). (6) Importance,. Suwa-Todo [4, 5] MCMC 2.3 Simulated Annealing MCMC W (X X ) 4

5 MCMC (Simulated annealing) [2] π(x) X log π(x) MCMC T π T (X) = π/t (X) X π/t (X) (7) MCMC X T π T (X) T π(x) T T T D-wave 3 MCMC (Multicanonical method)[3] Simulated Tempering(ST)[4] (Exchange MC, multiplecoupled MCMC, paralle tempering)[9] 990 [6, 7] [6] ST 3. Simulated tempering Simulated tempering(st) [4, 5] E(X) : π(x) = exp ( βe(x)). (8) Z(T ) ST T T, T 2,, T R (X, T ) π ST (X, T r ) = Z ST exp ( β r E(X) + g r ) (9) 5

6 g r T r Z ST T r X T r π ST (T r ) = exp ( β r E(X) + g r ) = Z(T r)e gr (0) Z ST Z ST X g r = log Z(T r ) Step 0 (, T (0) ) Step T X MCMC Step 2 X T. (X, T r ) T n T r T r+ /2 2. T n ( A(T r, T n ) = min, π ) ST(X, T n ) π ST (X, T r ) ST g r ST R (T, T 2,, T R ) {X} = {X, X 2,, X R } R ST π EX ({X}) () R π EX ({X}) = π Tr (X r ) (2) r= π Tr (X) T r π Tr (X) X 2 T r T r+ X r X r+ (2) R EX R EX = π EX(X, X 2,, X r+, X r,, X R ) π EX (X, X 2,, X r, X r+,, X R ) = π r(x r+ )π r+ (X r ) π r (X r )π r+ (X r+ ) 6 (3)

7 min(, R EX ) R EX = exp ( ) (E(X q ) E(X q+ )), (4) T q T q+ 2 2 ( 4) π T (X) T r π Tr (X r ) ST g r ST X () X () 2 X () 3 X () 4 X () 5 X () T T 2 T 3 T 4 T 5 T 6 MC Step 4: R = MCMC ln Z(β) = β 0 dβ d dβ ln Z(β ) = β 0 dβ E(β ) (5) MCMC 7

8 e+0 e+08 e Exact MC # of dimers" 5: ( ) ( ) [3] ( ). 6 6 [3] 5( ) [6, 7], N [8], [6] [25] 4 Lifting Event-chain MC MCMC MCMC 8

9 Suwa-Todo [4] Peskun Lifting [8, 9, 2, 22, 23] MCMC Lifting Lifting Event-chain Monte Carlo(ECMC) ECMC [2] Lifting ECMC [22] Event [24] 5 4 [] N. Metropolis, A.W.Rosenbluth, M.N.Rosenbluth, A.H.Teller and E.Teller, J. Chem. Phys. 2, 087/09 (953) 9

10 [2] S. Kirkpatrick, C.D.Gelatt, and M.P.Vecchi, Optimization by simulated annealing, Science /680, 983. [3] B. A. Berg and T. Neuhaus: Phys. Lett. B267, 249 (99). [4] E. Marinari and G. Parisi, Europhys. Lett. 9, 45 (992). [5] A. P. Lyubartsev, A. A. Martinovski, S. V. Shevkunov, and P. N. Vorontsov-Velyaminow, J. Chem. Phys. 96, 776 (992). [6] Y. Iba: Extended Ensemble Monte Carlo, Int. J. Mod. Phys. C, 2, 623/652, (200). [7] J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer-Verlag New York (200). [8] K. Hukushima, Comp. Phys. Comm., 47, 77/82 (2002). [9] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn., 65, 604/6, (996). [0], II 2005 [] K. Hukushima, Phys. Rev. E 60, 3606, 999. [2] F. Wang, and D.P. Landau, Phys. Rev. Lett. 86, 2050, 200. [3] W.Krauth, Statistical Mechanics: Algorithms and Computations, Oxford Master Series in Physics (2006). [4] H. Suwa and S. Todo, Phys. Rev. Lett. 05, (200). [5] S. Todo and H. Suwa, J. Phys.: Conf. Ser. 473, 0203 (203). [6] K. Pinn and C.Wieczerkowski, Int. J. Mod. Phys. C9, 54 (998). [7] A. Kitajima and M. Kikuchi, PLos ONE 0 (205), DOI: 0.37/journal.pone [8] K. S. Turitsyn, M. Chertkov, and M. Vucelja, Physica D 240, 40 (20). [9] Y. Sakai and K. Hukushima, J. Phys. Soc. Jpn (203). [20] Y. Sakai and K. Hukushima, arxiv preprint arxiv: (205). [2] E. P. Bernard, W. Krauth, and D. B. Wilson, Phys. Rev. E 80, (2009). [22] E. P. Bernard and W. Krauth, Phys. Rev. Lett. 07, (20). [23] M. Michel, S. C. Kapfer, and W. Krauth, J. Chem. Phys. 40, 0546 (204). [24] Y. Nishikawa, M. Michel, W. Krauth, and K.Hukushima, Phys. Rev. E 92, (205). [25] Y. Igarashi, H. Takenaka, Y. Nakanishi-Ohno, M. Uemura, S. Ikeda, and M. Okada, J. Phys. Soc. Jpn. 87, (208) 0

講義ノート 物性研究 電子版 Vol. 7, No. 2, (2018 年 11 月号 ) Metropolis [1] Buffon Neumann 20 importance Importance (Markov-chain Monte Carlo

講義ノート物性研究電子版 Vol. 7, No. 2, (2018 年 11 月号 ) Metropolis [1] Buffon Neumann 20 importance Importance (Markov-chain Monte Carlo