PACS-CS Collaboration Member S. Aoki, N. Ishii, N. Ishizuka, D. - PowerPoint PPT Presentation

PACS-CS の取り組み Ref: PRD79.034503 (2009) 浮田 尚哉 ( 筑波大学計算科学研究センター ) for PACS-CS Collaboration 「ストレンジを含むクォーク多体系分野の理論的将来を考える」研究会 ２００９年２月２７－２８日 @ KKR ホテル熱海（熱海市）

1 PACS-CS Collaboration Member S. Aoki, N. Ishii, N. Ishizuka, D. Kadoh, K. Kanaya, Y. Kuramashi, Y. Namekawa, Y. Taniguchi, A. Ukawa, N. Ukita, T. Yoshie, T. Yamazaki (University of Tsukuba) K.-I. Ishikawa, M. Okawa, Y. Shimizu (Hiroshima Univ.) T. Izubuchi (BNL, Kanazawa Univ.)

Introduction 2 PACS-CS Project: N f = 2 + 1 Simulations at the Physical Point on large enough lattices • O ( a ) improved Wilson quark action with nonperturbative c SW (CP-PACS and JLQCD Collaborations, 2006) • Iwasaki gauge action (Iwasaki, 1983) • β = 1 . 90 ( a = 0 . 0907(13) fm) • 32 3 × 64 lattice ( V ≈ (2 . 9fm) 3 ) • m π = 156 ∼ 702 MeV • m π L � 2 . 3 u-d quarks : Domain-Decomposed HMC (DDHMC) algorithm (L¨ uscher, 2003) + Hasenbusch trick (Hasenbusch, 2001; Hasenbusch, Jansen, 2003) + · · · s quark : UV-filterd Polynomial HMC (UVPHMC) algorithm (JLQCD Collaborations, 2002) on the PACS-CS ( 2560nodes , 14 . 3TFLOPS ) and the T2K (95+140+61 TFLOPS ).

Introduction 3 Comparison with other groups for N f = 2 + 1 simulations: m min Collaboration Dirac op. m π (MeV) L a (fm) π � 156 0.09 [ M Ω ] PACS-CS clover 2.3 BMW(’08) stout clover � 190 4 0.065-0.125 [ M Ξ ] MILC(’04,’07) staggered � 240 4 0.06-0.18 [ F π ] NPLQCD(’07) staggered+DW � 290 3.7 0.13 [ r 0 ] RBC/UKQCD(’08) DW � 330 4.6 0.1 [ M Ω ] JLQCD/TWQCD(’08) Overlap � 310 2.8 0.108 [ r 0 ]

Introduction 4 The light hadron spectrum of QCD at a = 0 . 09 fm 2 meson octet baryon decuplet baryon ∗ Ω 1.5 ∗ Ξ Σ Ξ m[GeV] ∆ Σ Λ 1 * φ N K ρ 0.5 K chpt fse ( m π ,m K ,m Ω -input ) π 0

Introduction 5 The light hadron spectrum of QCD : PACS-CS (a=0.09fm) and BMW ( a → 0) 2 meson octet baryon decuplet baryon ∗ Ω 1.5 ∗ Ξ Σ Ξ m[GeV] ∆ Σ Λ 1 * φ N K ρ 0.5 PACS-CS ( m π ,m K ,m Ω -input ) K BMW (m π ,m K ,m Ξ -input) π 0

6 Plan to this talk : • Introduction • Algorithm for N f = 2 part : DDHMC, Hasenbusch trick, Solver • Simulation Parameters and Data Set • Run Status • ChPT Analysis : SU(2) and SU(3) ChPT up to NLO • The Physical Point Simulation • Conclusion

Algorithm for N f = 2 + 1 part DDHMC, Hasenbusch trick, Solver

DDHMC Algorithm (L¨ uscher, 2003) 7 Key to reduce the cost : Domain Decomposition & Multi Time Step Integrator • Domain Decomposition splitting lattice sites into even & odd domains as a preconditionor for N f = 2 O(a)-improved Wilson-Dirac op. domain size we use is 8 4 Even Odd Even „ D EE „ D EE « « „ « D − 1 D EO 0 1 EE D EO D = = D − 1 D OE D OO 0 D OO OO D OE 1 | det D | 2 = | det D EE | 2 | det D OO | 2 | det(1 − D − 1 EE D EO D − 1 ) | 2 OO D OE | {z } | {z } UV part ≡ DIR : IR part

8 After this preconditioning, we have a partition function, Z D EE | 2 | det ¯ e − SG | det(1 + T ) | 2 | det ¯ D OO | 2 | 2 Z = | det D IR | {z } | {z } | {z } U Gauge part UV part IR part Z  ff − 1 2Tr( P 2 ) − S ′ = exp G [ U ] − S UV [ U, φ ] − S IR [ U, χ ] . P,U,φ,χ Molecular dynamics equation : set random φ, χ ˙ U = P , ˙ P = F G [ U ] + F UV [ U, φ ] + F IR [ U, χ ] • Multi time step integrator for Gauge, UV and IR parts (Sexton and Weingarten, 1992) Relative magnitudes of force terms, F G , F IR , F UV : || F G || : || F UV || : || F IR || ≈ 16 : 4 : 1 . We choose the associated step sizes, δτ G , δτ UV , δτ IR such that δτ G || F G || ≈ δτ UV || F UV || ≈ δτ IR || F IR || , δτ G = τ/N 0 N 1 N 2 , δτ UV = τ/N 1 N 2 , δτ IR = τ/N 2 , N 0 = N 1 = 4 .

9 For strange quark, we employ UVPHMC algorithm (CP-PACS and JLQCD Collaborations, 2006) where the domain decomposition is not used. ∥ F s ∥ ≈ ∥ F IR ∥ ⇒ δτ s = δτ IR . ⇓ ⃝ m π � 296 MeV : DDHMC + UVPHMC algorithm works stable. × m π = 156 MeV : large fluctuation of ∥ F IR ∥ , slow to keep simulation stable. • Hasenbusch trick (Hasenbusch, 2001; Hasenbusch, Jansen, 2003) IR = D IR ( κ → κ ′ = ρκ ) , eg. ρ = 0 . 9995 to shift to the heavier mass D ′ ˛ «˛ „ D IR 2 | det D IR | 2 = | det D ′ ˛ ˛ IR | ˛ det ⇒ F IR ′ , F IR/IR ′ ˛ ˛ D ′ ˛ IR Step sizes, δτ G , δτ UV , δτ IR ′ , δ IR/IR ′ , are controlled by ( N 0 , N 1 , N 2 , N 3 ) . N 0 = N 1 = 4 , N 2 and N 3 are chosen to reduce the fluctuation of ∥ F IR ′ ∥ , ∥ F IR/IR ′ ∥ .

Solver for m π � 296 MeV : Dx = b 10 For DDHMC algorithm ( m π � 296MeV), • IR solver : SAP(single prec.) preconditioned GCR(double prec.) (L¨ uscher, 2004) • UV solver : SSOR(single prec.) preconditioned GCR(double prec.) • Stopping condition : | Dx − b | / | b | ≤ 10 − 14 for H, 10 − 9 for F

Solver for m π = 156 MeV : Dx = b 11 For MPDDHMC algorithm ( m π = 156MeV), ⋆ Chronological guess for IR part (Brower, Ivanenko, Levi, Orginos, 1997) ⋆ nested BiCGStab solver for IR and UV part : • Outer solver(double prec.) : Solve Dx = b with preconditioner M ≈ D − 1 with strict stopping condition 10 − 14 for F • Inner solver(single prec.) : Solve M ≈ D − 1 with appropriate precoditioner “ “ tol outer , 10 − 6 ” , 10 − 3 ” err outer with automatic tolerance control tol inner = min max ⋆ Deflation technique (Morgan, Wilcox, 2002; L¨ uscher, 2007) • inner BiCGStab stagnant → GCRO-DR (Parks et al, 2006) (Generalized Conjugate Residual with implicit inner Orthogonalization and Deflated Restarting)

Simulation Cost 12 10.0 N f =2+1 8.0 100 configs Tflops years a=0.1fm 6.0 L=3fm ( m π /m ρ ) − 3 Cost ∝ for HMC , 4.0 ( m π /m ρ ) − 2 ∝ for MPDDHMC . HMC physical pt κ ud =0.13770 2.0 κ ud =0.13781 (Ukawa, 2002; PACS-CS Collaboration, 2007) 0.0 0 0.2 0.4 0.6 0.8 1 m π /m ρ Physical Point simulations require HMC : O (100) Tflops computer , MPDDHMC : O (10) Tflops computer.

Simulation Parameters and Data Set

Simulation Parameters and Data Set 13 κ ud 0.13700 0.13727 0.13754 0.13754 0.13770 0.13781 0.137785 κ s 0.13640 0.13640 0.13640 0.13660 0.13640 0.13640 0.13660 Algorithm DDHMC DDHMC DDHMC DDHMC DDHMC MPDDHMC MPDDHMC τ 0.5 0.5 0.5 0.5 0.25 0.25 0.25 (4,4,10) (4,4,14) (4,4,20) (4,4,28) (4,4,16) (4,4,4,6) (4,4,2,4,4) ( N 0 ,N 1 ,N 2 ,N 3 ,N 4) (4,4,6,6) ρ 1 − − − − − 0.9995 0.9995 ρ 2 − − − − − − 0.9990 N poly 180 180 180 220 180 200 220 Replay on on on on on off off MD time 2000 2000 2250 2000 2000 1400 1000 m ud [ MeV ] 67 45 24 21 12 3.5 3.6 m π [ MeV ] 702 570 411 385 296 156 162 m π /m ρ 0.64 0.57 0.46 0.45 0.35 0.20 0.23 CPU time [h]/ τ 0.29 0.44 1.3 1.1 2.7 7.1 6.0 shifted hopping parameter κ ′ ud = ρ 1 κ ud ∼ 0 . 1377

Run Status dH and Force histories, Effective masses

dH History 14 m π = 570MeV m π = 296MeV m π = 156MeV 5 5 5 4 4 4 3 3 3 2 2 2 dH dH dH 1 1 1 0 0 0 -1 -1 -1 -2 -2 -2 1000 1500 2000 2500 1000 1500 2000 2500 500 550 600 650 700 750 800 850 900 τ τ traj. acc(HMC)=0.87 acc(HMC)=0.84 acc(HMC)=0.88 replay trick ∼ 0 . 1% replay trick ∼ 3% rate( | dH | > 2) ∼ 3%

Force History 15 m π = 570MeV m π = 296MeV m π = 156MeV 10 10 10 F0 F1 F2 F0 1 F3 F1 F2 1 1 0.1 0.01 0.1 0.1 0.001 0 5000 10000 15000 20000 4e+05 4.5e+05 5e+05 5.5e+05 6e+05 6.5e+05 7e+05 1e+06 1.2e+06 1.4e+06 1.6e+06 F 0 : Gauge + clv F 0 : Gauge + clv F 1 : UV F 1 : UV F 2 : IR ′ F 2 : IR F 3 : IR/IR ′

Effective mass : Meson 16 k ud = 0 . 13727 k ud = 0 . 13770 k ud = 0 . 13781 m π = 570MeV m π = 296MeV m π = 156MeV φ φ φ 0.5 0.5 * * * K K K 0.5 ρ ρ ρ 0.4 0.4 m Meson m Meson m Meson η η η 0.4 s s s 0.3 0.3 K K K 0.2 0.2 0.3 π π π 0.1 0.1 0.2 0 10 20 30 0 10 20 30 0 10 20 30 Nt Nt Nt Fit Range [ t min , t max ] : Pseudoscalar [13 − 30] , Vector [10 − 20]

Effective mass : Baryon 17 k ud = 0 . 13727 k ud = 0 . 13770 k ud = 0 . 13781 m π = 570MeV m π = 296MeV m π = 156MeV 0.9 0.9 0.95 0.85 0.85 Ω Ω Ω 0.9 0.8 0.8 * * * Ξ Ξ Ξ 0.85 0.75 0.75 * * * 0.7 Σ 0.8 Σ Σ 0.7 m B m B m B ∆ 0.65 ∆ ∆ 0.65 0.75 Ξ Ξ Ξ 0.6 0.6 Σ 0.7 Σ Σ 0.55 Λ 0.55 Λ Λ 0.5 0.65 0.5 N N N 0.45 0.6 0.45 0.4 0 10 20 5 15 25 0 5 10 15 20 25 0 5 10 15 20 25 Nt Nt Nt Fit Range [ t min , t max ] : Decuplet [13 − 30] , Octet [10 − 20]

Edinburgh Plot 18 1.5 1.4 1.3 m N / m ρ 1.2 κ s =0.13640 κ s =0.13660 Phisical Point 1.1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m π / m ρ

m 2 π /m ud , f K /f π vs m ud 19 4.2 1.3 2 /m AWI m π f K /f π ud 4.0 1.2 CP-PACS/JLQCD κ s =0.13640 PACS-CS κ s =0.13640 3.8 1.1 PACS-CS κ s =0.13660 Experiment 3.6 1.0 CP-PACS/JLQCD κ s =0.13640 PACS-CS κ s =0.13640 PACS-CS κ s =0.13660 3.4 0.9 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 AWI AWI m m ud ud

SU(3) ChPT analysis