1 Simulation of surface growth on GPU supercomputers Jeffrey Kelling, Dresden (HZDR), Géza Ódor, Budapest (MTA-EK-MFA), NVIDIA Professor Partrnership 2010- NIIF Supercomputig GPUday 02/06/2016 www.mfa.kfki.hu/~odor 1 1 1

2 Our Motivation was In nanotechnologies large areas of nano-patterns are needed fabricated today by expensive techniques, e.g. electron beam lithography or direct writing with electron and ion beams. Similar phenomena: sand dunes, chemical reactions … → Universality & Nonequilibrium physics Better understanding of basic surface growth phenomena is needed ! 2 2 2

The Kardar-Parisi-Zhang (KPZ) equation 3 The Kardar-Parisi-Zhang (KPZ) equation t h(x,t) = 2h(x,t) + λ ( h(x,t))2 + (x,t)‏ σ : (smoothing) surface tension coefficient λ : local growth velocity, up-down anisotropy η : roughens the surface by a zero-average, Gaussian noise field with correlator: <(x,t) (x',t')> = 2 D  (x-x')(t-t')‏ Characterization of surface growth: Interface Width: Fundamental model of non-equilibrium surface physics Surface growth power-laws: 3 3 3

KPZ equation can describe 4 KPZ equation can describe Simplest model describing surface growth Transformation : W ~ e h/2 t W(x,t) =  2 W(x,t) + (x,t) W(x,t) Directed polymers in random media Transformation: v = - h t v(x,t) =  2 v(x,t) +v(x,t)v(x,t)+ (x,t) Noisy Burgers equation : Randomly stirred fluids Magnetic flux lines in superconductors And many more … Same universal scaling in all ! W(x,t): partition function of polymer length t 4 4

Mapping of KPZ growth in 2+1 dimensions 5 Mapping of KPZ growth in 2+1 dimensions Octahedron model Cellular automaton update: Driven diffusive gas of pairs (dimers) G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 021125 (2009) G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 031112 (2010) Bit coded simulations on 131072 x 131072 sized systems provided high precision scaling exponent estimates and probability distributions J. Kelling and G. Ódor Phys. Rev. E 84 (2011) 061150 5 5 5

Domain decomposition method 6 Domain decomposition method Bit-coded representation of slopes: 2 bit/lattice site, 4x4 sized tiles in a 32-bit world. For vectorization even/odd sites are grouped and stored at consecutive memory locations Updates by logical instructions TinyMT random number generator for each thread randomly seeded J. Kelling, G. Ódor, S. Gemming, INES2016 Budapest conference proc., arXiv:1606.00310 6 6 6

7 Benchmarking Highest speed is for p=0.5, where the GPU code is memory-bound For arbitrary update probabilities the speed decreases by a factor: 4-5 Limited by the random generator A speedup of up to 14× was found for GPU over a single socket six core CPU 7 7

Physical ageing in systems without detailed balance Known & practically used since prehistoric times (metals, glasses) systematically studied in physics since ~ 1970 Discovery : ageing effects are reproducible & universal ! They occur in different systems: structural glasses, spin glasses, polymers, simple magnets, . . . Dynamical scaling, growing length scale: L(t) ~ t1/z Broken time-translation-invariance 8

Universality (in permission with Timothy Halpin Healy) Completely new RSOS, KPZ Euler, and Directed Polymer in Random Medium (DPRM) simulations: 2014 EPL 105 50001 Full agreement G.Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, 032146 (2014) 9 9

SCA correlations saturate to finite values After subtracting the constant height scaling is the same but slope scaling is different than for RSA!

RSOS simulations with different step sizes in 2+1 dimensions Multi-surface parallel implementation: Multiples of 128 realizations simultaneously: vectorized data-parallel workload Double tiling domain decomposition, with random origin moving between two lattice sweeps Sustained performance of ~ 1010 deposition/sec.

RSOS Simulation results Universal growth exponents for all (N=1,3,5,7) models ruling out: = 0.25, and = 0.4 Corrections to scaling is the smallest for N=1 models High slopes are not needed to describe the long wavelength KPZ scaling J. Kelling, G.Ódor, S.Gemming: arXiv:1605.02620, submitted to Phys. Rev. E

14 Conclusions & outlook Extremely large (217 x 217) fast simulation of 2+1 d KPZ class surface model simulations on GPUs Stochastic Cellular automaton simulations of the underlying binary lattice gas using checkerboard updates and bit vectorization → max x16 speedup to I7 (6coreCPUs) Determination of universal aging behavior → Aim: full dynamical functional form !!! RSOS surface growth simulations with different step sizes confirmation of KPZ universality and correction to scaling analysis Vectorized multi-surface simulation + double tiling decomposition → 1010 upd/sec Acknowledgements: OTKA, W2/W3, WH-KO-606, NIIF HPC Debrecen2, Budapest2, ZIH TU Dresden, Recent related publications: J. Kelling and G. Ódor, Phys. Rev. E 84, 061150 (2011), J. Kelling, G. Ódor, J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) 175-187 G. Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, 032146 (2014) J. Kelling, G. Ódor, S. Gemming, INES2016 Budapest conference proc., arXiv:1606.00310 J. Kelling, G.Ódor, S. Gemming: arXiv:1605.02620, submitted to Phys. Rev. E 14 14 14