Scientific Computations on Modern Parallel

Vector SystemsLeonid Oliker

Julian Borrill, Jonathan Carter, Andrew Canning, John Shalf, David SkinnerLawrence Berkeley National Laboratories

Stephane EthierPrinceton Plasma Physics Laboratory

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

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

Applications studied

LBMHD Plasma Physics 1,500 lines grid basedLattice 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, 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

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)

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)

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 boundariesExpect 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 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. 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: ScaLAPACKSteps 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(1012) to the pixel domain - maps - O(108) to the multipole domain - power spectra - O(104)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%)

0%20%40%60%80%

100%

P=

16

P=

16

P=

16

P=

16

P=

64

P=

64

P=

64

P=

64

P=

25

6

P=

25

6

P=

25

6

P=

25

6

P=

10

24

P=

10

24

Sbg ES Phx Cmb Sbg ES Phx Cmb Sbg ES Phx Cmb Sbg ES

% o

f th

eo

reti

cal

pe

ak

CalcCalc+MPICalc+MPI+I/OCalc+MPI+I/O+Rmp

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.

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% 20% 11% 9.4 4.3 4.1 1.1PARATEC 57% 33% 54% 58% 20% 8.2 3.9 1.4 3.9MADbenc

h 49% --- 19% 37% 17% 6.3 --- 3.5 1.4

Average 19.9 7.2 4.5 2.4