Scientific Computations on Modern Parallel

Vector Systems

Leonid Oliker

http://crd.lbl.gov/~oliker

Andrew Canning, Jonathan Carter, John ShalfLawrence Berkeley National Laboratories

Stephane Ethier Princeton Plasma Physics Laboratory

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: second 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

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)

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

Earth Simulator OverviewMachine 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-R3Connection 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 nodesGlobal Barrier Counter within interconnect allows global barrier synch <3.5usecStorage: 480 TB Disk, 1.5 PB TapeCompilers : Fortran 90, HPF, ANSI C, C++Batch: similar to NQS, PBSParallelization: vectorization processor level OpenMP, Pthreads, MPI, HPF

Earth Simulator Cost

Approx costs

Development: $400MBuilding: $70MMaintenance: $50M/yearElectricity: $8M/year

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%

All codes were ported/vectorized on single node SX6 at ARSC (Oliker et al, SC2003)

Multi-node vector runs first attempted at ES center

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) Node

SSP MSP

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 ?

Architectural Comparison

Node Type Where CPU/

NodeClockMHz

PeakGFlop

Mem BW GB/s

Peak byte/fl

op

NetwkBW

GB/s/P

BisectBW

byte/flop

MPI Latenc

yusec

NetworkTopolog

y

Power3 NERSC 16 375 1.5 1.0 0. 47 0.13 0.087 16.3 Fat-tree

Power4 ORNL 32 1300 5.2 2.3 0.44 0.13 0.025 7.0 Fat-tree

Altix ORNL 2 1500 6.0 6.4 1.1 0.40 0.067 2.8 Fat-treeES ESC 8 500 8.0 32.0 4.0 1.5 0.19 5.6 CrossbarX1 ORNL 4 800 12.8 34.1 2.7 6.3 0.088 7.3 2D-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

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

1

10

100

1000

10000

100000

0 100 200 300 400 500

Stride

Triad (MB/s)Power3Power4SX-6X1

1

10

100

1000

10000

1003007001.5K4.5K10K25K65K150K375K900K2.2M5.5M13M

Data Size (Bytes)

Triad Gath/Scat (MB/s)

Power3 Power4 SX-6 X1

Analysis using‘Architectural Probe’

Developed Architectural Probe: allows stress the balance points of processor design (PMEO-04) Tunable parameters to mimic behavior of important scientific kernel

Gather/Scatter expensive on commodity cache-based systems

Power4 can is only 1.6% (1 in 64)Itanium2: much less sensitive at 25% (1 in 4)

Huge amount of computation may be required to hide overhead of irregular

data access

Itanium2 requires CI of about 9 flops/wordPower4 requires CI of almost 75!

What % of memory access can be random before performance decreases by

half?

How much computational intensity is required to hide the penalty of all random

access?Reducing performance by 50%

1.6%

25%

6.3%

0.8%

0%

1%

10%

100%

Itanium 2 Opteron Power3 Power4

% Indirection

CI required to hide indirection

9.3

149.3

18.7

74.7

0

50

100

150

200

Itanium 2 Opteron Power3 Power4Computational Intensity

(CI)

Sample Kernel Performance

NPB FT Class B Nbody (Barnus-Hut)

FFT computationally intensive with data parallel operations

Well suited for vectorization: 17X, 4X faster than Power3/4

Fixed cost interprocessor communication hurts scalability

Nbody requires irregular, unstructured data access, control flow and communication

Poorly suited for vectorization:2X and 5X slower than Power/4

Vector architectures not general purpose systems

0.0

0.5

1.0

1.5

2.0

1 2 4 8 16 32 64Processors

GFlops/s

Power3Power4SX-6X1

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 4 8 16 32 64Processors

GFlops/s

Power3Power4SX-6X1

Applications studied

Applications chosen with potential to run at ultrascale

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

LBMHD Plasma Physics 1,500 lines grid basedLattice Boltzmann approach for magneto-hydrodynamics

GTC Magnetic Fusion 5,000 lines particle based Particle in cell method for gyrokinetic Vlasov-Poisson equation

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

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

Visualization of grazing collision of two black holes

Communication at boundariesExpect high parallel efficiency

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

ProblemSize P

Power 3 Power 4 Altix ES X1

Gflops/P

%peak

Gflops/P

%peak

Gflops/P

%peak

Gflops/P %peak Gflops/

P%pea

k80X80x8

0per

processor

16 0.31 21% 0.58 11% 0.89 15% 1.5 18% 0.54 4%64 0.22 14% 0.50 10% 0.70 12% 1.4 17% 0.43 3%256 0.22 14% 0.48 9% --- --- 1.4 17% 0.41 3%

250x80x80

perprocessor

16 0.10 6% 0.56 11% 0.51 9% 2.8 35% 0.81 6%64 0.08 6% --- --- 0.42 7% 2.7 34% 0.72 6%256 0.07 5% --- --- --- --- 2.7 34% 0.68 5%

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

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

Induced current and chargedensity in crystallized glycine

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

PARATEC:Wavefunction Transpose(a) (b)

(e)

(c)

(f)

(d)

PARATEC: Performance

DataSize P

Power 3 Power4 Altix ES X1

Gflops/P%peak

Gflops/P %peak Gflops/

P %peakGflops/P

%peak

Gflops/P

%peak

432Atom

32 0.95 63% 2.0 39% 3.7 62% 4.7 60% 3.0 24%64 0.85 57% 1.7 33% 3.2 54% 4.7 59% 2.6 20%128 0.74 49% 1.5 29% --- --- 4.7 59% 1.9 15%256 0.57 38% 1.1 21% --- --- 4.2 52% --- ---512 0.41 28% --- --- --- --- 3.4 42% --- ---

686Atom

128 4.9 62% 3.0 24%256 4.6 57% 1.3 10%

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

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 platforms10

100

1000

10000

32 64 128 256 512 1024

Processors

GFlops

309 QD - Ideal

309 QD - Pwr3

432 Si - Pwr3

432 Si - ES

686 Si - ES

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

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, 3rd-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

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

DataSize P

Power 3 Power4 Altix ES X1

Gflops/P

%peak

Gflops/P

%peak

Gflops/P

%peak

Gflops/P

%peak

Gflops/P %peak

4096 x

4096

16 0.11 7% 0.28 5% 0.60 10% 4.6 58% 4.3 34%64 0.14 9% 0.30 6% 0.62 10% 4.3 54% 4.4 34%

256 0.14 9% 0.28 5% --- --- 3.2 40% --- ---

8192x

8192

64 0.11 7% 0.27 5% 0.65 11% 4.6 58% 4.5 35%256 0.12 8% 0.28 5% --- --- 4.3 53% 2.7 21%1024 0.11 7% --- --- --- --- 3.3 41% --- ---

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

DataSize P

X1-MPI X1-CAF

Gflops/P %peak Gflops/

P%peak

4096216 4.32 34% 4.55 36%64 4.35 34% 4.26 33%

8192264 4.48 35% 4.70 37%256 2.70 21% 2.91 23%

LBMHD: Performance

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)

–0–10–20–30–40–50–60–70–80

–P3 –P4 –ES –X1

– % t

ime

–collision

–stream

–comm

8192 x 8192 Grid 64 processors

–0–10–20–30–40–50–60–70–80

– % t

ime

–P3 –P4 –ES

8192 x 8192 Grid 256 processors

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

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

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: additional 8X, 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

GTC: Performance

Number

Particles

P

Power 3 Power4 Altix ES X1

Gflops/P

%peak

Gflops/P %peak Gflops/

P%pea

kGflops/

P %peak Gflops/P

%peak

10/cell

20M

32 0.13 9% 0.29 5% 0.29 5% 0.96 12% 1.00 8%64 0.13 9% 0.32 5% 0.26 4% 0.84 10% 0.80 6%

100/cell

200M

32 0.13 9% 0.29 5% 0.33 6% 1.34 17% 1.50 12%64 0.13 9% 0.29 5% 0.31 5% 1.25 16% 1.36 11%

1024 0.06 4%

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)

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.

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)

Code(P=64) % peak (P=Max avail) Speedup ES

vs.

Pwr3 Pwr4 Altix ES X1 Pwr3 Pwr4 Altix X1LBMHD 7% 5% 11% 58% 37% 30.6 15.3 7.2 1.5CACTUS 6% 11% 7% 34% 6% 45.0 5.1 6.4 4.0

GTC 9% 6% 5% 16% 11% 9.4 4.3 4.1 0.9PARATE

C 57% 33% 54% 58% 20% 8.2 3.9 1.4 3.9

Future directions

Develop application driven architectural probes for evaluation of emerging petascale systems

Research the enhancement of commodity scalar processors with vector components for increased scientific productivity (including investigation into VIVA2)

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

Examples includes sparse linear-algebra, molecular dynamics, and combustion Evaluate more difficult codes traditionally at odds with vector architectures Interested in examining AMR codes - key component for future multi-scale simulations

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, Hitachi SR11000, NEC SX8, Cray Red-Storm, IBM Power5, Bluegene/*

Publications

L. Oliker, A. Canning, J. Carter, J. Shalf, and S. Ethier.“Scientific Computations on Modern Parallel Vector Systems”, Supercomputing 2004, to appear.

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.

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.