Scientific Computations on Modern Parallel Vector Systems

1. Scientific Computations on Modern Parallel Vector Systems Leonid Oliker Computer Staff Scientist Future Technologies Group Computational Research DivisionLawrence Berkeley National Laboratories loliker@lbl.gov http://crd.lbl.gov/~oliker/paperlinks.html

2. Overview • Superscalar cache-based architectures dominate US HPC market • Leading architectures are commodity-based SMPs due to generality and perspection 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 • Currently conducting evaluation study of DOE applications on modern parallel vector architectures: final year of three year project • 09/2003 MOU between NERSC and ES was completedVisited ES center December 8th-17th, 2003First international team to conduct performance evaluation study at ES

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: only effective if sufficient regularity discoverable in program structure • Suffers greatly even if small % of code non-vectorizable (Amdahl’s Law)

4. ES Processor Overview • 8 Gflops per CPU • 8 CPU per SMP • 8 way replicated vector pipe • 72 vec registers, 256 64-bit words • Divide Unit • 32 GB/s pipe to FPLRAM • 4-way superscalar o-o-o @ 1 Gflop • 64KB I$ & D$ • Earth Simulator: 640 nodes • ES: newly developed FPLRAM (Full Pipelined RAM) SX6: DDR-SDRAM 128/256Mb • ES: uses IN 12.3 GB/s bi-dir btw any 2 nodes, 640 nodes SX6: uses IXS 8GB/s bi-dir btw any 2 nodes, max 128 nodes

5. Earth Simulator Overview • Machine type : 640 nodes, each node is 8-way SMP vector processors (5120 total procs) • Machine Peak: 40TF/s (proc peak 8GF/s) • OS : Extended version of Super-UX: 64 bit Unix OS based on System V-R3 • Connection structure : a single stage crossbar network (1500 miles of cable), 83,000 copper cables: 7.9 TB/s aggregate switching capacity 12.3 GB/s bi-di between any two nodes • Global Barrier Counter within interconnect allows global barrier synch <3.5usec • Storage: 480 TB Disk, 1.5 PB Tape • Compilers : Fortran 90, HPF, ANSI C, C++ • Batch: similar to NQS, PBS • Parallelization: vectorization processor level OpenMP, Pthreads, MPI, HPF

6. Earth Simulator Cost Approx costs Development: $400MBuilding: $70MMaintenance: $50M/yearElectricity: $8M/year

7. ES Programming Environment • Only benchmarking size runs were submitted (no production runs) • ES not connected to Internet • Interactive, S cluster, L cluster (2 nodes, 14 nodes, 624 nodes) • No global file system • Few numerical libraries • Using >10 nodes requires minimum vectorization ratio: 95% parallelization efficiency: 50% • Examples of required parallelization ratio (as defined by Amdahl’s Law):16 nodes 99.21% ; 64 nodes 99.80% ; 256 nodes 99.95% • Lack of external access and usage hurdles inhibits scientific productivity • All codes were ported/vectorized on single node SX6 at ARSC (Oliker et al, SC2003) • Multi-node vector runs first attempted at ES center

8. Cray X1 Overview SSP: 3.2GF computational core VL = 64, dual pipes (800 MHz)2-way scalar 0.4 GF (400MHz) MSP: 12.8 GF combines 4 SSPshares 2MB data cache (unique) Node: 4 MSP w/ flat shared mem Interconnect: modified 2D torusfewer links then full crossbar butsmaller bisection bandwidth Globally addressable: procs can directly read/write to global mem Parallelization: Vectorization (SSP) Multistreaming (MSP) Shared mem (OMP, Pthreads) Inter-node (MPI2, CAF, UPC) SSP MSP Node

9. Altix3000 Overview • Itanium2@ 1.5GHz (peak 6 GF/s) 128 FP registers, 32K L1, 256K L2, 6MB L3 • Cannot store FP in values in L1 • EPIC Bundles instruction • Bundles processed in-order, instructions within bundle processed in parallel • Consists of “Cbricks” : 4 Itanium2, memory, 2 controller ASICS called SHUB • Uses high bandwidth, low latency Numalink3 interconnect (fat-tree) • Implements CCNUMA protocol in hardware • A cache miss caused data to be communicated/replicated via SHUB • Uses 64-bit Linux with single system image (256 processor / few for OS services) • Scalability to large numbers of processors ?

10. 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

11. Memory Performance Triad Mem Test: A(i) = B(i) + s*C(i) NO MachineSpecific Optimizations • For strided access, SX6 achieves 10X, 100X, 1000X improvement over X1, Pwr4, Pwr3 • For gather/scatter, SX6/X1 show similar performance, exceed scalar at higher data sizes • All machines performance can be improved via architecture specific optimizations • Example: On X1 using non-cachable & unroll(4) pragma improves strided BW by 20X

12. Developed Architectural Probe: allows stress the balance points of processor design (PMEO-04) Tunable parameters to mimic behavior of important scientific kernel Analysis using‘Architectural Probe’ Interested in developing application driven architectural probes for evaluation of emerging petascale systems

13. Sample Kernel Performance NPB FT Class B Nbody (Barnus-Hut) Interested in exploring advanced algorithmic optimizations on emerging systems - preliminary work described in CUG04

14. Applications studied Applications chosen with potential to run at ultrascale • 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 • LBMHDPlasma Physics 1,500 linesgrid based Lattice Boltzmann approach for magneto-hydrodynamics • GTCMagnetic Fusion 5,000 linesparticle based Particle in cell method for gyrokinetic Vlasov-Poisson equation • MADCAPCosmology 5,000 linesdense linear algebra Extracts key data from Cosmic Microwave Background Radiation

15. 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 • What are gravitational waves? Ripples in spacetime curvature, caused by matter motion, causing distances to change: • Developed at Max Planck Institute, vectorized by John Shalf

16. CACTUS: Performance • ES achieves fastest performance to date: 45X faster than Power3! • Vector performance related to x-dim (vector length) • Scalar performance better on smaller problem size (cache effects) • Excellent scaling on ES using fixed data size per proc (weak scaling) • X1 surprisingly poor (4X slower than ES) - low ratio scalar:vector • Note boundary radiation vectorized for X1 but not on ES giving the X1 an advantage • Unvectorized boundary, required 15-20% of runtime on ES (30+% on X1) • < 5% for the scalar version: unvectorized code can quickly dominate cost

17. 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 w/ Louie and Cohen’s groups (UCB, LBNL), A Canning

18. 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)

19. PARATEC: Performance • ES achieves fastest performance to date! Over 2Tflop/s on 1024 procs • 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 8X slower than ES • Power4 4X slower - Federation has increased speed 2X compared with Colony • Altix 1.5X slower - high memory and interconnect bandwidth, low latency switch

20. PARATEC Scaling: ES vs. Power3 • ES can run the same system about 10 times faster than the IBM SP (on any number of processors) • Main advantage of ES for these types of codes is the fast communication network • Fast processors require less fine-grain parallelism in code to get same performance as RISC machines • Vector arch allow opportunity to simulate systems not possible on scalar platforms

21. 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 proc 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

22. 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

23. 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 (up to 4.70 Gflop/P) • ES is 44X, 16X, and 7X faster than Power3, Power4, and Altix • Low CI and high memory requirement (30GB) hurt scalar performance • Altix best scalar due to: high memory bandwidth, fast interconnect

24. 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 Interested in research of CAF and UPC performance and optimization

25. 80 80 70 70 60 50 60 collision % time 40 stream 50 comm 30 % time 40 20 30 10 0 20 P3 P4 ES X1 10 0 P3 P4 ES LBMHD: Performance 8192 x 8192 Grid 256 processors 8192 x 8192 Grid 64 processors • Preliminary time breakdown shown relative to each architecture • Cray X1 has highest % spent in communication (P=64), CAF version reduced this • ES shows best memory bandwidth performance (stream) • Communication increases at higher scalability (as expected)

26. 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

27. GTC: Scatter operation • Particle charge deposited amongst nearest grid points. The particles can be anywhere inside the domain • Several particles can contribute to same grid points, resulting in memory conflicts (dependencies) that prevent vectorization • Since particles are randomly localized - scatter also hinders cache reuse • Solution: VLEN copies of charge deposition array w/ reduction after main loop

28. GTC: Porting Details • Large vector memory footprint requiried eliminate dependencies P=64 uses 42 GB on ES compared w/ 5 GB on Power3 • Relatively small mem per processor (ES=2GB, X1=4GB) limits problem size runs • GTC has second level of parallelism via OpenMP (hybrid programming). However, on ES/X1 memory footprint increased: additional8X, about 320GB • Non-vectorized “Shift” routine accounted for: 54% X1, 11% on ES • Due to high penalty of serialized sections on X1 when multistreaming • The shift routine vectorized on X1, but NOT on ES - X1 has advantage • Limited time at ES prevented vectorization of shift routine • Now shift account for only 4% of X1 runtime

29. GTC: Performance • Vectors achieve fastest per-processor performance of any tested architecture! • P=64 on X1 is 35% faster than P=1024 on Power3! • X1 9% faster than ES (but has additional code section vectorized) • Advantage of ES for PIC codes may reside in higher statistical resolution simulations.The greater speed allows for more particles per grid cell • Larger testes could not be performed at ES due to parallelization/vectorization efficiency hurdles • Low Altix performance due under investigation (random # generation)

30. GTC: Performance • With increasing processors, and fixed problem size, the vector length decreases • Limited scaling due to decreased vector efficiency rather than communications overhead. • MPI communication by itself has near perfect scaling.

31. Cosmology: MADCAP • Microwave Anisotropy Dataset Computational Analysis Package • Optimal general algorithm for extracting key cosmological data from Cosmic Microwave Background Radiation (CMB) • Anisotropies in the CMB contains early history of the Universe Temperature anisotropies inCMB measured by Boomerang • Calculates maximum likelihood two-point angular correlation function • Recasts problem in dense linear algebra: ScaLAPACKSteps include: mat-mat, matrix-inv, mat-vec, Cholesky decomp, data redistribution • Porting: ScaLAPACK plus rewrite of Legendre polynomial recursion, such that large batches are computed in inner loop • Developed at NERSC by Julian Borrill

32. MADCAP: Performance • Only partially ported due to code’s requirements of global file system • All systems sustain relatively low % peak considering MADCAP’s BLAS3 ops • Complex tradeoffs: architectural paradigm, interconnect technology, and I/O filesystem • Detailed analysis presented HiPC 2004 • Further work is required to: reduce I/O, remove system calls, and remove global file system requirements • Plan to implement new MADCAP version for next ES visit

33. 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) • ES shows much higher sustained and often higher raw 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) • Vectors potentially at odds w/ emerging techniques (sparse, irregular, multi-physics) • GTC example code at odds with data-parallelism • Much more difficult to evaluate codes poorly suited for vectorization • Return to ES in October - evaluate new codes and higher scalability studies • Potential opportunity of large-scale scientific runs (not just benchmarking)

34. Future directions Leverage evaluation suite, (unclassified) application expertise, emerging arch research • Develop application driven architectural probes for evaluation of emerging petascale systems • Research the enhancement of commodity scalar processors with vector features for increased scientific productivity (including investigation into VIVA2 with IBM) • Software-controlled scratchpad, programmable prefetch/preload • Investigate algorithmic optimizations for leading vector systems and examine an architecture-independent algorithmic analysis: • Fundamental resource requirements of key algorithms (FPU, locality, bdwith, latency-tolerance) • Explore new application areas on leading parallel systems • Evaluate codes traditionally at odds with vector architectures • Study the potential of implicit parallel programming languages: UPC and CAF • Especially codes difficult to express via MPI (portability requirement tradeoffs) • Evaluate soon-to-be-released supercomputing systems and identify classes of applications best suited for their unique architectural balance • Cray X1e, XD1 & Red-Storm, IBM Power5, Bluegene/*, Hitachi SR11000, NEC SX8,

35. Publications • L. Oliker, A. Canning, J. Carter, J. Shalf, and S. Ethier.“Scientific Computations on Modern Parallel Vector Systems”, Supercomputing 2004, to appear.Nominated for Best Paper award • J. Carter, J. Borrill, and L. Oliker. “Performance Characteristics of a Cosmology Package on Leading HPC Architectures”, International Conference on Higher Performance Computing: HIPC 2004, to appear.Nominated for Best Paper award • L. Oliker, J. Borrill, A. Canning, J. Carter, H. Shan, D. Skinner, R. Biswas, J. Djomheri, “Performance Evaluation of the SX-6 Vector Architecture”, Journal of Concurrency and Computation 2004, to appear. • L. Oliker and Rupak Biswas, “Performance Modeling and Evaluation of Ultra-Scale Systems”, Minisymposium organized at SIAM Conference on Parallel Processing for Scientific Computing: SIAMPP 2004. • L. Oliker, J. Borrill, A. Canning, J. Carter, H. Shan, D. Skinner, R. Biswas, J. Djomheri, “A Performance Evaluation of the Cray X1 for Scientific Applications”, International Meeting on High Performance Computing for Computational Science: VECPAR 2004. • H. Shan, E. Strohmaier, L. Oliker, “Optimizing Performance of Superscalar Codes For a Single Cray X1 MSP Processor”, 46th Cray User Group Conference, CUG 2004. • G. Griem, L. Oliker, J. Shalf, K. Yelick, “Identifying Performance Bottlenecks on Modern Microarchitectures using an Adaptable Probe”, Performance Modeling, Evaluation, Optimization of Parallel & Distributed Systems PMEO 2004 • L. Oliker, J. Carter, J. Shalf, D. Skinner, S. Ethier, R. Biswas, J. Djomehri, R. Van der Wijngaart. “Evaluation of Cache-based Superscalar and Cacheless Vector Architectures for Scientific Computations”, Supercomputing 2003.