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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 1 1 1

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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

3. The Kardar-Parisi-Zhang (KPZ) equation3
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

4. KPZ equation can describe4
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

5. Mapping of KPZ growth in 2+1 dimensions5
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, (2009)
G. Ódor, B. Liedke and K.-H. Heinig, PRE79, (2010)
Bit coded simulations on x sized systems provided high precision scaling exponent estimates and probability distributions J. Kelling and G. Ódor Phys. Rev. E 84 (2011) 5 5 5

6. Domain decomposition method6
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: 6 6 6

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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

8. 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

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

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

12. 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.

13. 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: , submitted to Phys. Rev. E

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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, (2011), J. Kelling, G. Ódor, J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) G. Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, (2014) J. Kelling, G. Ódor, S. Gemming, INES2016 Budapest conference proc., arXiv: J. Kelling, G.Ódor, S. Gemming: arXiv: , submitted to Phys. Rev. E 14 14 14