Scientific Computations on Modern Parallel Vector Systems

1. Scientific Computations on Modern Parallel Vector Systems Leonid Oliker, Jonathan Carter, Andrew Canning, John Shalf Lawrence Berkeley National Laboratories Stephane Ethier Princeton Plasma Physics Laboratory http://crd.lbl.gov/~oliker

2. Overview • Superscalar cache-based architectures dominate HPC market • Leading architectures are commodity-based SMPs due to generality and perception of cost effectiveness • Growing gap between peak & sustained performance is well known in scientific computing • Modern parallel vectors may bridge gap this for many important applications • In April 2002, the Earth Simulator (ES) became operational: Peak ES performance > all DOE and DOD systems combined Demonstrated high sustained performance on demanding scientific apps • Conducting evaluation study of scientific applications on modern vector systems • 09/2003 MOU between ES and NERSC was completedFirst visit to ES center: December 8th-17th, 2003 (ES remote access not available)First international team to conduct performance evaluation study at ES • Examining best mapping between demanding applications and leading HPC systems - one size does not fit all

3. Vector Paradigm • High memory bandwidth • Allows systems to effectively feed ALUs (high byte to flop ratio) • Flexible memory addressing modes • Supports fine grained strided and irregular data access • Vector Registers • Hide memory latency via deep pipelining of memory load/stores • Vector ISA • Single instruction specifies large number of identical operations • Vector architectures allow for: • Reduced control complexity • Efficiently utilize large number of computational resources • Potential for automatic discovery of parallelism However: most effective if sufficient regularity discoverable in program structure • Suffers even if small % of code non-vectorizable (Amdahl’s Law)

4. Architectural Comparison • 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

5. Applications studied LBMHDPlasma Physics 1,500 linesgrid based Lattice Boltzmann approach for magneto-hydrodynamics CACTUS Astrophysics 100,000 lines grid based Solves Einstein’s equations of general relativity PARATECMaterial Science50,000 linesFourier space/grid Density Functional Theory electronic structures codes GTCMagnetic Fusion 5,000 linesparticle based Particle in cell method for gyrokinetic Vlasov-Poisson equation • 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

6. 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 Current density decays of two cross-shaped structures • 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

7. LBMHD: Porting Details (left) octagonal streaming lattice coupled with square spatial grid (right) example of diagonal streaming vector updating three spatial cells • 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, 3rd-degree polynomial eval • Boundary value exchange • MPI_Isend, MPI_Irecv pairs • Further work: plan to use ES "global memory" to remove message copies

8. LBMHD: Performance • ES achieves highest performance to date: over 3.3 Tflops for P=1024 • X1 comparable absolute speed up to P=64 (lower % peak) • But performs 1.5X slower at P=256 (decreased scalability) • CAF improved X1 to slightly exceed ES at P=64 (up to 4.70 Gflop/P) • ES is 44X, 16X, and 7X faster than Power3, Power4, and Altix • Low CI (1.5) and high memory requirement (30GB) hurt scalar performance • Altix best scalar due to: high memory bandwidth, fast interconnect

9. LBMHD on X1 MPI vs CAF • X1 well-suited for one-sided parallel languages (globally addressable mem) • MPI hinders this feature and requires scalar tag matching • CAF allows much simpler coding of boundary exchange (array subscripting): • feq(ista-1,jsta:jend,1) = feq(iend,jsta:jend,1)[iprev,myrankj] • MPI requires non-contiguous data copies into buffer, unpacked at destination • Since communication about 10% of LBMHD, only slight improvements • However, for P=64 on 40962 performance degrades. Tradeoffs: • CAF reduced total message volume 3X (eliminates user and system buffer copy) • But CAF used more numerous and smaller sized message

10. 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 Visualization of grazing collision of two black holes Communication at boundariesExpect high parallel efficiency • 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

11. CACTUS: Performance • ES achieves fastest performance to date: 45X faster than Power3! • Vector performance related to x-dim (vector length) • Excellent scaling on ES using fixed data size per proc (weak scaling) • Scalar performance better on smaller problem size (cache effects) • X1 surprisingly poor (4X slower than ES) - low ratio scalar:vector • Unvectorized boundary, required 15% of runtime on ES and 30+% on X1 • < 5% for the scalar version: unvectorized code can quickly dominate cost • Poor superscalar performance despite high computational intensity • Register spilling due to large number of loop variables • Prefetch engines inhibited due to multi-layer ghost zones calculations

12. 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 chargedensity 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)

13. PARATEC:Wavefunction Transpose (a) (b) • 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 timeto avoid latency issues • Only non-zero elements communicated/calculated • Much faster than vendor 3D-FFT (c) (d) (e) (f)

14. PARATEC: Performance • ES achieves fastest performance to date! Over 2Tflop/s on 1024 procs • Main advantage for this type of code is fast interconnect system • X1 3.5X slower than ES (although peak is 50% higher) • Non-vectorizable code can be much more expensive on X1 (32:1 vs 8:1) • Lower bisection bandwidth to computation ratio • Limited scalability due to increasing cost of global transpose and reduced vector length • Plan to run larger problem size next ES visit • Scalar architectures generally perform well due to high computational intensity • Power3, Power4, Alitx are 8X, 4X, 1.5X slower than ES • Vector arch allow opportunity to simulate systems not possible on scalar platforms

15. 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 N2 – particles interact w/ electromagnetic field on grid • Allows solving equation of particle motion with ODEs (instead of nonlinear PDEs) • Main computational tasks: • Scatter deposit particle charge to nearest point • Solve Poisson eqn to get potential for each point • Gather calc 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

16. 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

17. GTC: Performance • ES achieves fastest performance of any tested architecture! • First time code achieved 20% of peak - compared with less 10% on superscalar systems • Vector hybrid (OpenMP) parallelism not possible due to increased memory requirements • P=64 on ES is 1.6X faster than P=1024 on Power3! • Reduced scalability due to decreasing vector length, not MPI performance • Non-vectorizable code portions expensive on X1 • Before vectorization shift routine accounted for 11% of ES and 54% of X1 overhead • Larger tests could not be performed at ES due to parallelization/vectorization hurdles • Currently developing new version with increased particle decomposition • Advantage of ES for PIC codes may reside in higher statistical resolution simulations • Greater speed allow more particles per cell

18. Overview Tremendous potential of vector architectures: 4 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 • Limited X1 specific optimization - optimal programming approach still unclear (CAF, etc) • 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 • 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

19. Second ES visit • Evaluate high-concurrency PARATEC performance using large-scale Quantum Dot simulation • Evaluate CACTUS performance using updated vectorization of radiation boundary condition • Evaluate MADCAP performance using a newly optimized version, without global file systems requirements and improved I/O behavior • Examine 3D version of LBMHD, and explore optimization strategies • Evaluate GTC performance using updated vectorization of shift routine as well as new particle decomposition approach designed to increase concurrency • Evaluate performance of FVCAM3 (Finite Volume atmospheric model), at high concurrencies and resolution (1x1.25 , 0.5 x 0.625, 0.25 x 0.375) Papers available athttp://crd.lbl.gov/~oliker