Scientific Computations on Modern Parallel Vector Systems Leonid Oliker Julian Borrill, Jonathan Carter, Andrew Canning, John Shalf, David Skinner Lawrence Berkeley National Laboratories Stephane Ethier Princeton Plasma Physics Laboratory

Architectural Comparison Node TypeWhere CPU/ Node Clock MHz Peak GFlop Mem BW GB/s Peak byte/flop Netwk BW GB/s/P Bisect BW byte/flop MPI Latency usec Network Topology Power3NERSC Fat-tree Power4ORNL Fat-tree AltixORNL Fat-tree ESESC Crossbar X1ORNL D-torus  Custom vector architectures have High memory bandwidth relative to peak Superior interconnect: latency, point to point, and bisection bandwidth  Overall ES appears as the most balanced architecture, while Altix shows best architectural balance among superscalar architectures  A key ‘balance point’ for vector systems is the scalar:vector ratio

Applications studied LBMHD Plasma Physics 1,500 lines grid based Lattice Boltzmann approach for magneto-hydrodynamics CACTUS Astrophysics 100,000 lines grid based Solves Einstein’s equations of general relativity PARATEC Material Science 50,000 lines Fourier space/grid Density Functional Theory electronic structures codes GTC Magnetic Fusion 5,000 lines particle based Particle in cell method for gyrokinetic Vlasov-Poisson equation MADbench Cosmology 2,000 lines dense linear algebra Maximum likelihood two-point angular correlation, I/O intensive Applications chosen with potential to run at ultrascale  Computations contain abundant data parallelism ES runs require minimum parallelization and vectorization hurdles  Codes originally designed for superscalar systems  Ported onto single node of SX6, first multi-node experiments performed at ESC

Plasma Physics: LBMHD  LBMHD uses a Lattice Boltzmann method to model magneto-hydrodynamics (MHD)  Performs 2D simulation of high temperature plasma  Evolves from initial conditions and decaying to form current sheets  2D spatial grid is coupled to octagonal streaming lattice  Block distributed over 2D processor grid  Main computational components: Collision requires coefficients for local gridpoint only, no communication Stream values at gridpoints are streamed to neighbors, at cell boundaries information is exchanged via MPI Interpolation step required between spatial and stream lattices  Developed George Vahala’s group College of William and Mary, ported Jonathan Carter Current density decays of two cross- shaped structures

LBMHD: Porting Details  Collision routine rewritten: For ES loop ordering switched so gridpoint loop (~1000 iterations) is inner rather than velocity or magnetic field loops (~10 iterations) X1 compiler made this transformation automatically: multistreaming outer loop and vectorizing (via strip mining) inner loop Temporary arrays padded reduce bank conflicts  Stream routine performs well: Array shift operations, block copies, 3 rd -degree polynomial eval  Boundary value exchange MPI_Isend, MPI_Irecv pairs Further work: plan to use ES "global memory" to remove message copies (left) octagonal streaming lattice coupled with square spatial grid (right) example of diagonal streaming vector updating three spatial cells

Material Science: PARATEC  PARATEC performs first-principles quantum mechanical total energy calculation using pseudopotentials & plane wave basis set  Density Functional Theory to calc structure & electronic properties of new materials  DFT calc are one of the largest consumers of supercomputer cycles in the world Induced current and charge density in crystallized glycine  Uses all-band CG approach to obtain wavefunction of electrons  33% 3D FFT, 33% BLAS3, 33% Hand coded F90  Part of calculation in real space other in Fourier space Uses specialized 3D FFT to transform wavefunction  Computationally intensive - generally obtains high percentage of peak  Developed Andrew Canning with Louie and Cohen’s groups (UCB, LBNL)

 Transpose from Fourier to real space  3D FFT done via 3 sets of 1D FFTs and 2 transposes  Most communication in global transpose (b) to (c) little communication (d) to (e)  Many FFTs done at the same time to avoid latency issues  Only non-zero elements communicated/calculated  Much faster than vendor 3D-FFT PARATEC: Wavefunction Transpose (a)(b) (e) (c) (f) (d)

Astrophysics: CACTUS  Numerical solution of Einstein’s equations from theory of general relativity  Among most complex in physics: set of coupled nonlinear hyperbolic & elliptic systems with thousands of terms  CACTUS evolves these equations to simulate high gravitational fluxes, such as collision of two black holes  Evolves PDE’s on regular grid using finite differences  Uses ADM formulation: domain decomposed into 3D hypersurfaces for different slices of space along time dimension  Exciting new field about to be born: Gravitational Wave Astronomy - fundamentally new information about Universe  Gravitational waves: Ripples in spacetime curvature, caused by matter motion, causing distances to change.  Developed at Max Planck Institute, vectorized by John Shalf Visualization of grazing collision of two black holes Communication at boundaries Expect high parallel efficiency

Magnetic Fusion: GTC  Gyrokinetic Toroidal Code: transport of thermal energy (plasma microturbulence)  Goal magnetic fusion is burning plasma power plant producing cleaner energy  GTC solves 3D gyroaveraged gyrokinetic system w/ particle-in-cell approach (PIC)  PIC scales N instead of N 2 – particles interact w/ electromagnetic field on grid  Allows solving equation of particle motion with ODEs (instead of nonlinear PDEs)  Main computational tasks: Scatter d eposit particle charge to nearest point Solve Poisson eqn to get potential for each point Gather c alc force based on neighbors potential Move particles by solving eqn of motion Shift particles moved outside local domain 3D visualization of electrostatic potential in magnetic fusion device Developed at Princeton Plasma Physics Laboratory, vectorized by Stephane Ethier

GTC: Scatter operation  Particle charge deposited amongst nearest grid points.  Calculate force based on neighbors potential, then move particle accordingly  Several particles can contribute to same grid points, resulting in memory conflicts (dependencies) that prevent vectorization  Solution: VLEN copies of charge deposition array with reduction after main loop However, greatly increases memory footprint (8X)  Since particles are randomly localized - scatter also hinders cache reuse

Cosmology: MADbench  Microwave Anisotropy Dataset Computational Analysis Package  Optimal general algorithm for extracting key cosmological data from Cosmic Microwave Background Radiation (CMB)  CMB encodes fundamental parameters of cosmology: Universe geometry, expansion rate, number of neutrino species  Preserves full complexity of underlying scientific problem  Calculates maximum likelihood two-point angular correlation function  Recasts problem in dense linear algebra: ScaLAPACK Steps include: mat-mat, mat-vec, chol decomp, redistribution  High I/O requirement - due to out-of-core nature of calculation  Developed at NERSC/CRD by Julian Borrill

CMB analysis moves from the time domain - observations - O(10 12 ) to the pixel domain - maps - O(10 8 ) to the multipole domain - power spectra - O(10 4 ) calculating the compressed data and their reduced error bars at each step. CMB Data Analysis

MADBench: Performance Characterization  In depth analysis shows performance contribution of each component for evaluated architectures  Identifies system specific balance and opportunities for optimization  Results show that I/O has more effect on ES than Seaborg - due to ratio between I/O performance and peak ALU speed  Demonstrated IPM capabilities to measure MPI overhead on variety of architectures without the need to recompile, at a trivial runtime overhead (<1%)

Overview Tremendous potential of vector architectures: 5 codes running faster than ever before  Vector systems allows resolution not possible with scalar arch (regardless of # procs)  Opportunity to perform scientific runs at unprecedented scale  ES shows high raw and much higher sustained performance compared with X1 Non-vectorizable code segments become very expensive (8:1 or even 32:1 ratio) Evaluation codes contain sufficient regularity in computation for high vector performance GTC example code at odds with data-parallelism Important to characterize full application including I/O effects Much more difficult to evaluate codes poorly suited for vectorization  Vectors potentially at odds w/ emerging techniques (irregular, multi-physics, multi-scale)  Plan to expand scope of application domains/methods, and examine latest HPC architectures Code (P=64) % peak(P=Max avail) Speedup ES vs. Pwr3Pwr4AltixESX1Pwr3Pwr4AltixX1 LBMHD7%5%11%58%37% CACTUS6%11%7%34%6% GTC9%6%5%20%11% PARATEC57%33%54%58%20% MADbench49%---19%37%17% Average