Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms GTC 2015 Speaker: Dr. Matteo Lulli Prof. M. Bernaschi and Prof. G. Parisi March the 19th, 2015 Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 1 / 31
Outlook Cubic stencils 1 PRNGs 2 Multi-GPU and MPI 3 Results 4 Conclusions & Perspectives 5 Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 2 / 31
Motivations Phase Transitions in disordered systems Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 3 / 31
Motivations Phase Transitions in disordered systems Equilibrium Monte Carlo analysis works well for non-disordered systems Disordered systems are very hard to equilibrate Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 3 / 31
Motivations Phase Transitions in disordered systems Equilibrium Monte Carlo analysis works well for non-disordered systems Disordered systems are very hard to equilibrate A large number of disorder realizations, samples , is required 3D Ising spin glass at most L = 40 equilibrated so far (Janus, FPGA dedicated machine) Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 3 / 31
Motivations Phase Transitions in disordered systems Equilibrium Monte Carlo analysis works well for non-disordered systems Disordered systems are very hard to equilibrate A large number of disorder realizations, samples , is required 3D Ising spin glass at most L = 40 equilibrated so far (Janus, FPGA dedicated machine) Robust out-of-equilibrium methods for the study of phase transitions would be very useful Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 3 / 31
Motivations Phase Transitions in disordered systems Equilibrium Monte Carlo analysis works well for non-disordered systems Disordered systems are very hard to equilibrate A large number of disorder realizations, samples , is required 3D Ising spin glass at most L = 40 equilibrated so far (Janus, FPGA dedicated machine) Robust out-of-equilibrium methods for the study of phase transitions would be very useful Why GPUs Even with out-of-equilibrium methods usual CPUs are not sufficiently powerful in order to obtain good estimates Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 3 / 31
Outline for section 1 Cubic stencils 1 PRNGs 2 Multi-GPU and MPI 3 Results 4 Conclusions & Perspectives 5 Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 4 / 31
Standard checkerboard pattern Nearest-neighbours based problems in spz 3D spy smx i spx smy smz Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 5 / 31
Standard checkerboard pattern Nearest-neighbours based problems in spz 3D spy Cubic lattice of linear size L = 2 n : smx Checkerboard colouring i spx Each lattice site has nearest neighbours smy of the other colour smz Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 5 / 31
Standard checkerboard pattern Nearest-neighbours based problems in spz 3D spy Cubic lattice of linear size L = 2 n : smx Checkerboard colouring i spx Each lattice site has nearest neighbours smy of the other colour smz Allocation choices Unified allocation: one array Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 5 / 31
Standard checkerboard pattern Nearest-neighbours based problems in spz 3D spy Cubic lattice of linear size L = 2 n : smx Checkerboard colouring i spx Each lattice site has nearest neighbours smy of the other colour smz Allocation choices Unified allocation: one array Separated allocation: two arrays Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 5 / 31
Standard checkerboard pattern Nearest-neighbours based problems in spz 3D spy Cubic lattice of linear size L = 2 n : smx Checkerboard colouring i spx Each lattice site has nearest neighbours smy of the other colour smz Allocation choices Unified allocation: one array Separated allocation: two arrays The parity, ( − 1) x i + y i + z i , of each lattice site has to be taken into account Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 5 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 6 7 6 0 7 0 4 4 5 5 2 2 3 3 0 0 0 0 0 1 1 0 x ′ x 0 0 z ′ = 0 z = 0 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 6 7 6 0 7 0 4 4 5 5 2 2 3 3 0 0 0 0 0 1 1 0 0 x ′ x 0 0 z ′ = 0 z = 0 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 6 7 6 0 7 0 4 4 5 5 2 2 3 3 0 0 0 0 0 1 1 0 0 2 x ′ x 0 0 z ′ = 0 z = 0 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 6 7 6 0 7 0 4 4 5 5 2 2 3 3 0 0 0 0 0 1 1 0 0 2 5 x ′ x 0 0 z ′ = 0 z = 0 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 6 7 6 0 7 0 4 4 5 5 2 2 3 3 0 0 0 0 0 1 1 0 0 2 5 7 x ′ x 0 0 z ′ = 0 z = 0 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 14 14 15 15 12 12 13 13 10 10 11 11 10 0 8 8 0 9 9 0 0 2 5 7 x ′ x 0 0 z ′ = 0 z = 1 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 14 14 15 15 12 12 13 13 10 10 11 11 10 12 0 8 0 8 9 9 0 0 2 5 7 x ′ x 0 0 z ′ = 0 z = 1 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
Sliced checkerboard pattern: definition It is always possible to remap the lattice sites Yavors’kii et al. , Heisenberg spin glass, ’snake-like’ pattern, unified Ferrero et al. , Potts 2D, separated Sliced scheme y ′ y 14 14 15 15 12 12 13 13 10 10 11 11 10 12 15 0 8 0 8 9 9 0 0 2 5 7 x ′ x 0 0 z ′ = 0 z = 1 Periodic boundary conditions are necessary Matteo Lulli - arXiv: 1114.0127 - lullimat.org Solutions for Efficient Memory Access for Cubic Lattices and Random Number Algorithms 6 / 31
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