ITR: Non-equilibrium surface growth and the scalability of parallel discrete- event simulations for large asynchronous systems NSF DMR-0113049 http://www.rpi.edu/~korniss/Research/gk_research.html 1
PIs: Gyorgy Gyorgy Korniss (Rensselaer), Mark Novotny Korniss (Rensselaer), Mark Novotny PIs: ( Mississippi State U.) ( Mississippi State U.) postdoc: Alice : Alice Kolakowska Kolakowska (Mississippi State U.) (Mississippi State U.) postdoc graduate student: H. Guclu Guclu (Rensselaer) (Rensselaer) graduate student: H. undergraduate students: Katie Barbieri Barbieri, John Marsh, Brad , John Marsh, Brad undergraduate students: Katie McAdams (Rensselaer); Shannon Wheeler (MSState MSState) ) McAdams (Rensselaer); Shannon Wheeler ( collaborators: P.A. Rikvold Rikvold (Florida State U.), Z. (Florida State U.), Z. Toroczkai Toroczkai collaborators: P.A. (CNLS, Los Alamos), B.D. Lubachevsky Lubachevsky, Alan Weiss , Alan Weiss (CNLS, Los Alamos), B.D. (Lucent/Bell Labs) (Lucent/Bell Labs) Funded by NSF DMR/ITR , The Research Corporation, DOE (NERSC, SCRI/CSIT, LANL), Rensselaer, Mississippi State U. 2
computer architectures + algorithms “Nature” “Nature” ? ? 3
Discrete-event systems ! Cellular communication networks (call arrivals) ! Internet traffic routing/queueing systems § •Dynamics is asynchronous •Updates in the local “configuration” are discrete events in continuous time (Poisson arrivals) ⇒ discrete-event simulation Modeling the evolution of spatially extended interacting systems: updates in “local” configuration as discrete-events ! Magnetization dynamics in condensed matter (Ising model with single-spin flip Glauber dynamic) ! Spatial epidemic models (contact process) 4
Parallelization for asynchronous dynamics The paradoxical task: ! (algorithmically) parallelize (physically) non-parallel dynamics Difficulties: ! Discrete events (updates) are not synchronized by a global clock ! Traditional algorithms appear inherently serial (e.g., Glauber attempt one site/spin update at a time) " However, these algorithms are not inherently serial (B.D. Lubachevsky ’87) 5
Parallel discrete-event simulation •Spatial decomposition on lattice/grid (for systems with short-range interactions only local synchronization between subsystems) •Changes/updates: independent Poisson arrivals " Each subsystem/block of sites, carried by a processing element (PE) must must have its own local simulated time, { τ i } (“virtual time”) " Synchronization scheme " PEs must concurrently advance their own Poisson streams, without violating causality 6
Two approaches d =1 τ i (site index) i " Optimistic (or speculative) ! PEs assume no causality violations ! Rollbacks to previous states once causality violation is found (extensive state saving or reverse simulation) ! Rollbacks can cascade (“avalanches”) " Conservative ! PE “idles” if causality is not guaranteed ! utilization, 〈 u 〉 : fraction of non-idling PEs 7
Basic conservative approach “Worst-case” analysis: •One-site-per PE, N PE = L d • t =0,1,2, ¢ parallel steps • τ i ( t ) fluctuating time horizon •Local time increments are iid exponential random variables τ ≤ τ •Advance only if min{ nn } i (nn: nearest neighbors) " Scalability modeling ! utilization (efficiency) 〈 u ( t ) 〉 (fraction of non-idling PEs) density of local minima ! width (spread) of time surface: N 1 PE ∑ 2 = τ − τ 2 w ( t ) [ ( t ) ( t )] i N 8 i = 1 PE
Coarse graining for the stochastic time surface evolution G. K., Toroczkai, Novotny, Rikvold, ‘00 ( ) ( ) τ + − τ = Θ τ − τ Θ τ − τ η ( t 1 ) ( t ) ( t ) ( t ) ( t ) ( t ) ( t ) i i i − 1 i i + 1 i i • Θ (…) is the Heaviside step-function • η i ( t ) iid exponential random numbers M 2 Kardar-Parisi-Zhang 2 ∂ τ ∂ τ ∂ τ = − λ + η ( x , t ) equation t 2 ∂ ∂ x x 2 ∂ τ 1 Steady state ( d =1): ∫ τ ∝ − P [ ( x )] exp dx Edwards-Wilkinson ∂ 2 D x Hamiltonian " Random-walk profile: short-range correlated local slopes 9
" Universality/roughness " Utilization/efficiency 2 β << t , if t t const . 2 × 〈 〉 w ( t ) ~ 〈 〉 ≅ 〈 〉 + u L u L α 2 >> L , if t t ∞ × L z = α / β t ~ L , z σ = 〈 2 〉 − 〈 〉 2 1 / 2 u u ~ 1 / L L L L β ≈ α ≈ 0 . 33 , 0 . 5 ( d =1) exact KPZ: β =1/3 α =1/2 〈 〉 ≈ u 0 . 2464 ∞ 10
Higher- d simulations (one site per PE) d = N L PE d =1 〈 u 〉 ≈ 0 . 246 ∞ d =2 〈 u 〉 ≈ 0 . 12 ∞ d =3 〈 〉 ≈ u 0 . 075 ∞ 11
Implications for scalability z t >> L Simulation reaches steady state for (arbitrary d ) const . 〈 〉 ≅ 〈 〉 ∞ + u L u " Simulation phase: scalable 2 ( 1 − α ) L 〈 u 〉 ∞ asymptotic average growth rate (simulation speed or utilization ) is non-zero Krug and Meakin, ‘90 " Measurement (data management) phase: not scalable measurement at τ meas : α 2 2 〈 〉 w ~ L (e.g., simple averages) L w " But CAN be made scalable by considering complex underlying communication topologies among PEs 12
Actual implementation l × l blocks N PE =( L / l ) 2 1. Local time increment: ∆τ =-ln( r ), r ✌ U(0,1) 2. If chosen site is on the boundary, PE must wait until τ≤ min{ τ nn } 13
Application: metastability and dynamic phase transition in spatially extended bistable systems 〈 τ 〉 ( T , H ) { i s } metastable lifetime t 1 / 2 { Q } half-period of i the oscillating field ≈ τ 〈 〉 > τ 〈 〉 < τ 〈 〉 t t t 1 / 2 1 / 2 1 / 2 1 ∫ = Q s ( t ) dt i i period-averaged spin 2 t 14 1 / 2
Application: metastability and hysteresis Kinetic Ising model 2 L ∑ ∑ = ± = − − s 1 J>0 H J s s H ( t ) s i i j i < > = i , j i 1 L × L lattice with periodic boundary conditions # Single-spin-flip Glauber dynamics # Periodic square-wave field of amplitude H # Half-period: t 1/2 Magnetization: m ( t ) = ( 1/L 2 ) Σ i s i ( t ) T<T c H → − H t= τ : m=0 t=0: m=1 escape from metastable well: Lifetime: 〈 τ 〉 = 〈 τ ( T,H ) 〉 15
Hysteresis and dynamic response ∑ ∑ = − − H J s s H ( t ) s i j i < > i , j i $ Periodic square-wave field of amplitude H o $ Half-period t 1/2 ; Θ = t 1/2 / < τ (T,H o )> Θ >>1 symmetric limit cycle Θ <<1 asymmetric limit cycle 16
Dynamic Phase Transition (DPT) 1 t ∫ = Q m ( t ) dt Θ = 1 / 2 2 t 〈 τ 〉 ( T , H ) 1 / 2 4 th order cumulant order parameter fluctuations # Θ >> Θ c : |Q| ≈ 0 symmetric dynamic phase # Θ << Θ c : |Q| ≈ 1 symmetry-broken dynamic phase # Θ = Θ c ¿ 1 ( t 1/2 ¿ 〈 τ 〉 ) large fluctuations in Q → DPT finite-size scaling evidence for a Sides et.al., PRL’98, PRE’99 } continuous (dynamic) phase transition 17 G.K. et.al., PRE’01
Large-scale finite-size analysis of the dynamic phase transition : Absence of the Tri-critical Point 1 ∫ period-averaged magnetization = Q m ( t ) dt 2 t ( dynamic order parameter) 1 / 2 18
Summary and outlook $ The tools and machinery of non-equilibrium statistical physics (coarse-graining, finite-size scaling, universality, etc.) can be applied to scalability modeling and algorithm engineering $ Conservative schemes can be made scalable $ Optimistic schemes: rollbacks (avalanches in virtual time): Self-organized criticality ??? $ Non-Poisson asynchrony (e.g., in “fat-tail” internet traffic) $ Applications: metastability, nucleation, and dynamic phase transition in spatially extended bistable systems 19
Recommend
More recommend
