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12/6/16 1 An Overview of High Performance Computing and Benchmark Changes for the Future Jack Dongarra University of Tennessee Oak Ridge National Laboratory

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Page 1: Dongarra 12-06-2016 Seminar.pdf

12/6/16 1

An Overview of High Performance Computing and Benchmark Changes for

the Future

Jack DongarraUniversity of Tennessee

Oak Ridge National Laboratory

Page 2: Dongarra 12-06-2016 Seminar.pdf

State of Supercomputing in 2016• Pflops (> 1015 Flop/s) computing fully established

with 117 systems.• Three technology architecture possibilities or

“swim lanes” are thriving.• Commodity (e.g. Intel)• Commodity + accelerator (e.g. GPUs) (88 systems)• Lightweight cores (e.g. IBM BG, ARM, Knights Landing)

• Interest in supercomputing is now worldwide, and growing in many new markets (~50% of Top500 computers are in industry).

• Exascale (1018 Flop/s) projects exist in many countries and regions.

• Intel processors largest share, 92% followed by AMD, 1%.

2

Page 3: Dongarra 12-06-2016 Seminar.pdf

3

H. Meuer, H. Simon, E. Strohmaier, & JD

- Listing of the 500 most powerfulComputers in the World

- Yardstick: Rmax from LINPACK MPPAx=b, dense problem

- Updated twice a yearSC‘xy in the States in NovemberMeeting in Germany in June

- All data available from www.top500.org

Size

Rat

e

TPP performance

Page 4: Dongarra 12-06-2016 Seminar.pdf

Performance Development of HPC over the Last 24 Years from the Top500

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

1E+09

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 201420152016

59.7GFlop/s

400MFlop/s

1.17TFlop/s

93PFlop/s

349TFlop/s

672PFlop/s

SUM

N=1

N=500

1 Gflop/s

1 Tflop/s

100 Mflop/s

100 Gflop/s

100 Tflop/s

10 Gflop/s

10 Tflop/s

1 Pflop/s

100 Pflop/s

10 Pflop/s

1 Eflop/s

6-8 years

My iPhone & iPad 4 Gflop/s

My Laptop 70 Gflop/s

4

Page 5: Dongarra 12-06-2016 Seminar.pdf

PERFORMANCE DEVELOPMENT

1

10

100

1000

10000

100000

1000000

10000000

100000000

1E+09

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020

SUM

N=1

N=100

1 Gflop/s

1 Tflop/s100 Gflop/s

100 Tflop/s

10 Gflop/s

10 Tflop/s

1 Pflop/s

100 Pflop/s

10 Pflop/s

1 Eflop/s

N=10

TflopsAchievedASCI RedSandia NL

PflopsAchieved

RoadRunnerLos Alamos NL

EflopsAchieved?

5

Page 6: Dongarra 12-06-2016 Seminar.pdf

November 2016: The TOP 10 SystemsRank Site Computer Country Cores Rmax

[Pflops]% of Peak

Power[MW]

GFlops/Watt

1National Super

Computer Center in Wuxi

Sunway TaihuLight, SW26010(260C) + Custom China 10,649,000 93.0 74 15.4 6.04

2National Super

Computer Center in Guangzhou

Tianhe-2 NUDT, Xeon (12C) + IntelXeon Phi (57C)

+ CustomChina 3,120,000 33.9 62 17.8 1.91

3 DOE / OS Oak Ridge Nat Lab

Titan, Cray XK7, AMD (16C) + Nvidia Kepler GPU (14C) +

CustomUSA 560,640 17.6 65 8.21 2.14

4 DOE / NNSA L Livermore Nat Lab

Sequoia, BlueGene/Q (16C) + custom USA 1,572,864 17.2 85 7.89 2.18

5 DOE / OS L Berkeley Nat Lab

Cori, Cray XC40, Xeon Phi (68C) + Custom USA 622,336 14.0 50 3.94 3.55

6 Joint Center for Advanced HPC

Oakforest-PACS, Fujitsu Primergy CX1640, Xeon Phi (68C)

+ Omni-PathJapan 558,144 13.6 54 2.72 4.98

7 RIKEN AdvancedInst for Comp Sci

K computer Fujitsu SPARC64 VIIIfx (8C) + Custom Japan 705,024 10.5 93 12.7 .827

8 Swiss CSCSPiz Daint, Cray XC50, Xeon (12C) + Nvidia P100(56C) +

CustomSwiss 206,720 9.78 61 1.31 7.45

9 DOE / OS Argonne Nat Lab

Mira, BlueGene/Q (16C) + Custom USA 786,432 8.59 85 3.95 2.07

10 DOE / NNSA / Los Alamos & Sandia

Trinity, Cray XC40,Xeon (16C) + Custom USA 301,056 8.10 80 4.23 1.92

500 Internet company Inspur Intel (8C) + Nnvidia China 5440 .286 71

TaihuLight is 5.2 X Performance of TitanTaihuLight is 1.1 X Sum of All DOE Systems

Page 7: Dongarra 12-06-2016 Seminar.pdf

Recent Developments¨ US DOE planning to deploy O(100) Pflop/s systems for 2017-

2018 - $525M hardware¨ Oak Ridge Lab and Lawrence Livermore Lab to receive IBM

and Nvidia based systems¨ Argonne Lab to receive Intel based system

Ø After this Exaflops

¨ US Dept of Commerce is preventing some China groups from receiving Intel technologyØ Citing concerns about nuclear research being done with the

systems; February 2015.Ø On the blockade list:

ØNational SC Center Guangzhou, site of Tianhe-2ØNational SC Center Tianjin, site of Tianhe-1AØNational University for Defense Technology, developerØNational SC Center Changsha, location of NUDT

7

Page 8: Dongarra 12-06-2016 Seminar.pdf

Since the Dept of Commerce Action …• Expanded focus on Chinese made HW and SW

• “Anything but from the US”• Three separate developments in HPC

• Wuxi• ShenWei O(100) Pflops all Chinese, June 2016

• NUDT • Tianhe-2A O(100) Pflops will be Chinese ARM + accelerator,

2017• Sugon - CAS ICT

• AMD? new processors

• In the latest “5 Year Plan”• Govt push to build out a domestic HPC ecosystem.• Exascale system, will not use any US chips

12/6/168

Page 9: Dongarra 12-06-2016 Seminar.pdf

• China’s first homegrown many-core processor• Vendor: Shanghai High Performance IC Design Center

• Supported by National Science and Technology MajorProject (NMP): Core Electronic Devices, High-end GenericChips, and Basic Software

• 28 nm technology

• 260 Cores

• 3 Tflop/s peak

SW26010 Processor

Page 10: Dongarra 12-06-2016 Seminar.pdf

Sunway TaihuLight http://bit.ly/sunway-2016• SW26010 processor• Chinese design, fab, and ISA• 1.45 GHz• Node = 260 Cores (1 socket)

• 4 – core groups• 64 CPE, No cache, 64 KB scratchpad/CPE• 1 MPE w/32 KB L1 dcache & 256KB L2 cache

• 32 GB memory total, 136.5 GB/s• ~3 Tflop/s, (22 flops/byte)

• Cabinet = 1024 nodes• 4 supernodes=32 boards(4 cards/b(2 node/c))• ~3.14 Pflop/s

• 40 Cabinets in system• 40,960 nodes total• 125 Pflop/s total peak

• 10,649,600 cores total• 1.31 PB of primary memory (DDR3) • 93 Pflop/s for HPL, 74% peak • 0.32 Pflop/s for HPCG, 0.3% peak • 15.3 MW, water cooled

• 6.07 Gflop/s per Watt• 1.8B RMBs ~ $280M, (building, hw, apps, sw, …)

Page 11: Dongarra 12-06-2016 Seminar.pdf

GordonBellAward

11

• Since1987theGordonBellPrizeisawardedattheSCconferencetorecognizeoutstandingachievementinhigh-performancecomputing.

• Thepurposeoftheawardistotracktheprogressofparallelcomputing,withemphasisonrewardinginnovationinapplyingHPCtoapplications.

• Financialsupportofthe$10,000awardisprovidedbyGordonBell,apioneerinhigh-performanceandparallelcomputing.

• Authors‘marktheirSCpaperasapossibleGordonBellPrizecompetitor.

•GordonBellcommitteereviewsthepapersandselects6papersforthecompetition.

• PresentationsaremadeatSCandawinnerischosen.

Page 12: Dongarra 12-06-2016 Seminar.pdf

GordonBellAwardFinalistsatSC16• “ModelingDiluteSolutionsUsingFirst-PrinciplesMolecularDynamics:ComputingMorethanaMillionAtomswithOveraMillionCores,”

• Lawrence-Livermore NationalLaboratory(Calif.)

• “TowardsGreenAviationwithPythonatPetascale,”• ImperialCollegeLondon(England)

• “SimulationsofBelow-GroundDynamicsofFungi:1.184Pflops AttainedbyAutomatedGenerationandAutotuning ofTemporalBlockingCodes,”

• RIKEN(Japan),ChibaUniversity(Japan),KobeUniversity(Japan)andFujitsuLtd.(Japan)

• “Extreme-ScalePhaseFieldSimulationsofCoarseningDynamicsontheSunwayTaihulight Supercomputer,”

• ChineseAcademyofSciences, theUniversityofSouthCarolina,ColumbiaUniversity(NewYork),theNationalResearchCenterofParallelComputerEngineering andTechnology(China)andtheNationalSupercomputingCenterinWuxi(China)

• “AHighlyEffectiveGlobalSurfaceWaveNumericalSimulationwithUltra-HighResolution,”

• FirstInstituteofOceanography(China),NationalResearchCenterofParallelComputerEngineering andTechnology(China)andTsinghuaUniversity(China)

• “10M-CoreScalableFully-ImplicitSolverforNonhydrostatic AtmosphericDynamics,”

• ChineseAcademyofSciences,TsinghuaUniversity(China),theNationalResearchCenterofParallelComputerEngineeringandTechnology(China)andBeijingNormalUniversity(China)

12

Page 13: Dongarra 12-06-2016 Seminar.pdf

HPE,112,22%

SGI,28,6%Lenovo,96,

19%CrayInc.,56,11%

Sugon, 47, 9%

IBM,36,7%

Bull,Atos,20,4%Huawei,16,3%

Inspur,18,4%

Dell,13,3%

Fujitsu,11,2%NUDT, 4, 1%Others,43,9%

HPE

SGI

Lenovo

CrayInc.

Sugon

IBM

Bull,Atos

Huawei

VENDORS / SYSTEM SHARE

#ofsystems,%of500

36%oftheVendorsarefromChina

Page 14: Dongarra 12-06-2016 Seminar.pdf

Countries Share

China has 1/3 of the systems, while the number of systems in the US has fallen to the lowest point since the TOP500 list was created. 14

34

5

5

5

6

7

17

20

27

32

171

171

0 50 100 150 200

Others

India

Russia

Saudi Arabia

Italy

Poland

UK

France

Japan

Germany

China

US

Number of Systims on Top500

Page 15: Dongarra 12-06-2016 Seminar.pdf

0

1

10

100

1,000

10,000

100,000

2000

2002

2004

2006

2008

2010

2012

2014

2016

Tota

l Per

form

ance

[T

flop

/s]

US

Performance of Countries

Page 16: Dongarra 12-06-2016 Seminar.pdf

0

1

10

100

1,000

10,000

100,000

2000

2002

2004

2006

2008

2010

2012

2014

2016

Tota

l Per

form

ance

[T

flop

/s]

US

EU

Performance of Countries

Page 17: Dongarra 12-06-2016 Seminar.pdf

0

1

10

100

1,000

10,000

100,000

2000

2002

2004

2006

2008

2010

2012

2014

2016

Tota

l Per

form

ance

[T

flop

/s]

US

EU

Japan

Performance of Countries

Page 18: Dongarra 12-06-2016 Seminar.pdf

0

1

10

100

1,000

10,000

100,000

2000

2002

2004

2006

2008

2010

2012

2014

2016

Tota

l Per

form

ance

[T

flop

/s]

US

EU

Japan

China

Performance of Countries

Page 19: Dongarra 12-06-2016 Seminar.pdf

19Confessions of an Accidental Benchmarker

• Appendix B of the Linpack Users’ Guide• Designed to help users extrapolate execution time for

Linpack software package• First benchmark report from 1977;

• Cray 1 to DEC PDP-10

Began in late 70’s Time when floating point operations were expensive compared to other operations and data movement

Page 20: Dongarra 12-06-2016 Seminar.pdf

http://tiny.cc/hpcg

Many Other Benchmarks• TOP500• Green 500• Graph 500• Sustained Petascale

Performance • HPC Challenge• Perfect• ParkBench• SPEC-hpc• Big Data Top100• Livermore Loops• EuroBen

• NAS Parallel Benchmarks• Genesis• RAPS• SHOC• LAMMPS• Dhrystone • Whetstone• I/O Benchmarks• WRF• Yellowstone• Roofline• Neptune

20

Page 21: Dongarra 12-06-2016 Seminar.pdf

High Performance Linpack (HPL)• Is a widely recognized and discussed metric for ranking

high performance computing systems • When HPL gained prominence as a performance metric in

the early 1990s there was a strong correlation between its predictions of system rankings and the ranking that full-scale applications would realize.

• Computer system vendors pursued designs that would increase their HPL performance, which would in turn improve overall application performance.

• Today HPL remains valuable as a measure of historical trends, and as a stress test, especially for leadership class systems that are pushing the boundaries of current technology.

21

Page 22: Dongarra 12-06-2016 Seminar.pdf

The Problem• HPL performance of computer systems are no longer so

strongly correlated to real application performance, especially for the broad set of HPC applications governed by partial differential equations.

• Designing a system for good HPL performance can actually lead to design choices that are wrong for the real application mix, or add unnecessary components or complexity to the system.

22

Page 23: Dongarra 12-06-2016 Seminar.pdf

HPCG• High Performance Conjugate Gradients (HPCG).• Solves Ax=b, A large, sparse, b known, x computed.• An optimized implementation of PCG contains essential computational

and communication patterns that are prevalent in a variety of methods for discretization and numerical solution of PDEs

• Synthetic discretized 3D PDE (FEM, FVM, FDM).• Sparse matrix:

• 27 nonzeros/row interior. • 8 – 18 on boundary.• Symmetric positive definite.

• Patterns:• Dense and sparse computations.• Dense and sparse collectives.• Multi-scale execution of kernels via MG (truncated) V cycle.• Data-driven parallelism (unstructured sparse triangular solves).

• Strong verification (via spectral properties of PCG).

23

hpcg-benchmark.org

Page 24: Dongarra 12-06-2016 Seminar.pdf

HPCGResults,Nov2016,1-10# Site Computer Cores Rmax

PflopsHPCGPflops

HPCG/HPL

%ofPeak

1 RIKENAdvancedInstituteforComputationalScience

Kcomputer,SPARC64VIIIfx2.0GHz,Tofuinterconnect

705,024 10.5 0.603 5.7% 5.3%

2 NSCC/Guangzhou Tianhe-2NUDT,Xeon12C2.2GHz+IntelXeonPhi57C+Custom

3,120,000 33.8 0.580 1.7% 1.1%

3 JointCenter forAdvancedHPC,Japan

Oakforest-PACS– PRIMERGYCX600M1,IntelXeonPhi

557,056 24.9 0.385 2.8% 2.8%

4 NationalSupercomputingCenterinWuxi, China

SunwayTaihuLight – SunwayMPP,SW26010

10,649,600

93.0 0.3712 0.4% 0.3%

5 DOE/SC/LBNL/NERSCUSA

Cori – XC40,IntelXeonPhiCray

632,400 13.8 0.355 2.6% 1.3%

6 DOE/NNSA/LLNLUSA

Sequoia– IBMBlueGene/Q,IBM

1,572,864 17.1 0.330 1.9% 1.6%

7 DOE/SC/OakRidgeNatLab Titan- CrayXK7,Opteron627416C2.200GHz,CrayGemini interconnect,NVIDIAK20x

560,640 17.5 0.322 1.8% 1.2%

8 DOE/NNSA/LANL/SNL Trinity - CrayXC40,IntelE5-2698v3,Ariescustom

301,056 8.10 0.182 2.3% 1.6%

9 NASA/MountainView Pleiades - SGIICEX,IntelE5-2680,E5-2680V2,E5-2680V3,InfinibandFDR

243,008 5.9 0.175 2.9% 2.5%

10 DOE/SC/ArgonneNationalLaboratory

Mira- BlueGene/Q,PowerBQC16C1.60GHz,Custom

786,432 8.58 0.167 1.9% 1.7%

Page 25: Dongarra 12-06-2016 Seminar.pdf

Peak Performance - Per Core

Floating point operations per cycle per coreÊ Most of the recent computers have FMA (Fused multiple add): (i.e.

x ←x + y*z in one cycle)Ê Intel Xeon earlier models and AMD Opteron have SSE2

Ê 2 flops/cycle DP & 4 flops/cycle SP

Ê Intel Xeon Nehalem (’09) & Westmere (’10) have SSE4Ê 4 flops/cycle DP & 8 flops/cycle SP

Ê Intel Xeon Sandy Bridge(’11) & Ivy Bridge (’12) have AVX Ê 8 flops/cycle DP & 16 flops/cycle SP

Ê Intel Xeon Haswell (’13) & (Broadwell (’14)) AVX2Ê 16 flops/cycle DP & 32 flops/cycle SP

Ê Xeon Phi (per core) is at 16 flops/cycle DP & 32 flops/cycle SP

Ê Intel Xeon Skylake (server) AVX 512Ê 32 flops/cycle DP & 64 flops/cycle SP

Ê Knight’s Landing

We arehere(almost)

Page 26: Dongarra 12-06-2016 Seminar.pdf

CPU Access Latencies in Clock Cycles

In 167 cycles can do 2672 DP Flops

Cycles

Cycles

Page 27: Dongarra 12-06-2016 Seminar.pdf

Classical Analysis of Algorithms May Not be Valid

• Processors over provisioned for floating point arithmetic

• Data movement extremely expensive• Operation count is not a good indicator of the time to solve a problem.

• Algorithms that do more ops may actually take less time.

12/6/1627

Page 28: Dongarra 12-06-2016 Seminar.pdf

68 cores Intel Xeon Phi KNL, 1.3 GHzThe theoretical peak double precision is 2662 Gflop/s

Compiled with icc and using Intel MKL 2017b1 20160506

Level 1, 2 and 3 BLAS68 cores Intel Xeon Phi KNL, 1.3 GHz, Peak DP = 2662 Gflop/s

Matrix size (N), vector size (NxN)2k 4k 6k 8k 10k 12k 14k 16k 18k 20k

Gflo

p/s

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400dgemm BLAS Level 3dgemv BLAS Level 2daxpy BLAS Level 1

80.3 Gflop/s

35.1 Gflop/s

2100 Gflop/s

35x

C = C + A*B

y = y + A*x

y = �*x + y

Page 29: Dongarra 12-06-2016 Seminar.pdf

0k 4k 8k 12k 16k 20kcolumns (matrix size N ⇥ N )

0

10

20

30

40

50

spee

dup

over

eisp

ack

square, with vectors

lapack QRlapack QR (1 core)linpack QReispack (1 core)

SingularValueDecompositionLAPACKVersion1991

Level1,2,&3BLASFirst Stage 8/3 n3 Ops

Dual socket – 8 coreIntel Sandy Bridge 2.6 GHz (8 Flops per core per cycle)

QR refers to the QR algorithm for computing the eigenvalues

LAPACK QR (BLAS in ||, 16 cores)LAPACK QR (using1 core)(1991)LINPACK QR (1979)EISPACK QR (1975)

3 Generations of software compared

Page 30: Dongarra 12-06-2016 Seminar.pdf

BottleneckintheBidiagonalizationTheStandardBidiagonal Reduction:xGEBRDTwoSteps:FactorPanel&UpdateTailingMatrix

­Characteristics• Total cost 8n3/3, (reduction to bi-diagonal)• Too many Level 2 BLAS operations• 4/3 n3 from GEMV and 4/3 n3 from GEMM• Performance limited to 2* performance of GEMV• èMemory bound algorithm.

factor panel k then update è factor panel k+1

Q*A*PH

0 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 30350 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 2750 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 2250 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 2500

Requires 2 GEMVs

Page 31: Dongarra 12-06-2016 Seminar.pdf

Recent Work on 2-Stage Algorithm

­Characteristics• Stage 1:

• Fully Level 3 BLAS• Dataflow Asynchronous execution

• Stage 2:• Level “BLAS-1.5”• Asynchronous execution• Cache friendly kernel (reduced communication)

0 20 40 60

0

10

20

30

40

50

60

nz = 36000 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 1190 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 605

First stageTo band

Second stageBulge chasingTo bi-diagonal

Page 32: Dongarra 12-06-2016 Seminar.pdf

0 20 40 60

0

10

20

30

40

50

60

nz = 36000 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 1190 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 605

First stageTo band

Second stageBulge chasingTo bi-diagonal

flops ⇡n�nb

nbÂ

s=12n3

b +(nt� s)3n3b +(nt� s) 10

3 n3b+(nt� s)⇥ (nt� s)5n3

b

+

n�nbnbÂ

s=12n3

b +(nt� s�1)3n3b +(nt� s�1) 10

3 n3b+(nt� s)⇥ (nt� s�1)5n3

b

⇡ 103 n3 + 10nb

3 n2 + 2nb3 n3

⇡ 103 n3(gemm)first stage

(7)

second stage:

flops = 6⇥nb ⇥n2(gemv)second stage (8)

maximium performance bound:

Pmax = reference number of operationsminimum time tmin

=83 n3

tmin(103 n3 flops in gemm)+tmin(6nbn2 flops in gemv)

=8n3PgemmPgemv

10n3Pgemv+18nbn2Pgemm

=) 840 Pgemm Pmax 8

12 Pgemm

if Pgemm is about 20x Pgemv and 120 nb 240.

(9)

Speedup:

speedup = time of one-stagetime of two-stage

=4n3/3Pgemv + 4n3/3Pgemm10n3/3Pgemm+6nbn2/Pgemv

=) 8470 Speedup 84

15

=) 1.2 Speedup 5.6

if Pgemm is about 20x Pgemv and 120 nb 240.

(10)

3

flops ⇡n�nb

nbÂ

s=12n3

b +(nt� s)3n3b +(nt� s) 10

3 n3b+(nt� s)⇥ (nt� s)5n3

b

+

n�nbnbÂ

s=12n3

b +(nt� s�1)3n3b +(nt� s�1) 10

3 n3b+(nt� s)⇥ (nt� s�1)5n3

b

⇡ 103 n3 + 10nb

3 n2 + 2nb3 n3

⇡ 103 n3(gemm)first stage

(7)

second stage:

flops = 6⇥nb ⇥n2(gemv)second stage (8)

maximium performance bound:

Pmax = reference number of operationsminimum time tmin

=83 n3

tmin(103 n3 flops in gemm)+tmin(6nbn2 flops in gemv)

=8n3PgemmPgemv

10n3Pgemv+18nbn2Pgemm

=) 840 Pgemm Pmax 8

12 Pgemm

if Pgemm is about 20x Pgemv and 120 nb 240.

(9)

Speedup:

speedup = time of one-stagetime of two-stage

=4n3/3Pgemv + 4n3/3Pgemm10n3/3Pgemm+6nbn2/Pgemv

=) 8470 Speedup 84

15

=) 1.2 Speedup 5.6

if Pgemm is about 20x Pgemv and 120 nb 240.

(10)

3

More Flops, original did 8/3 n3

25% More flops

Recent work on developing new 2-stage algorithm

Page 33: Dongarra 12-06-2016 Seminar.pdf

Recent work on developing new 2-stage algorithm

0 20 40 60

0

10

20

30

40

50

60

nz = 36000 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 1190 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 605

First stageTo band

Second stageBulge chasingTo bi-diagonal

Analysis of BRD using 2-stages

June 10, 2015

1 Two stage analysis

f lops ⇡n�nb

nbÂ

s=1

2n3

b +(nt � s)3n3

b +(nt � s) 10

3

n3

b+(nt � s)⇥ (nt � s)5n3

b

+

n�nbnbÂ

s=1

2n3

b +(nt � s�1)3n3

b +(nt � s�1) 10

3

n3

b+(nt � s)⇥ (nt � s�1)5n3

b

⇡ 10

3

n3 + 10nb3

n2 + 2nb3

n3

⇡ 10

3

n3(gemm)first stage

(1)

f lops = 6⇥nb ⇥n2(gemv)second stage

(2)

Pmax = re f erence number o f operationsminimum time tmin

=8

3

n3

tmin(10

3

n3 f lops in gemm)+tmin(6nbn2 f lops in gemv)

=8n3PgemmPgemv

10n3Pgemv+18nbn2Pgemm

=) 8

40

Pgemm Pmax 8

12

Pgemm

if Pgemm is about 20x Pgemv and 120 nb 240.

(3)

speedup = time of one-stage

time of two-stage

=4n

3/3P

gemv

+ 4n

3/3P

gemm

10n

3/3P

gemm

+6n

b

n

2/P

gemv

=) 84

70

Speedup 84

15

=) 1.8 Speedup 7

if Pgemm is about 22x Pgemv and 120 nb 240.

(4)

1

2k 4k 6k 8k 10k 12k 14k 16k 18k 20k 22k 24k 26k

1

2

3

4

5

6

Matrix size

Spee

dup

2−stages / MKL (DGEBRD)data2

25% More flops and 1.8 – 6 times faster

16 Sandy Bridge cores 2.6 GHz

Page 34: Dongarra 12-06-2016 Seminar.pdf

Critical Issues at Peta & Exascale for Algorithm and Software Design• Synchronization-reducing algorithms

§ Break Fork-Join model

• Communication-reducing algorithms§ Use methods which have lower bound on communication

• Mixed precision methods§ 2x speed of ops and 2x speed for data movement

• Autotuning§ Today’s machines are too complicated, build “smarts” into

software to adapt to the hardware

• Fault resilient algorithms§ Implement algorithms that can recover from failures/bit flips

• Reproducibility of results§ Today we can’t guarantee this. We understand the issues,

but some of our “colleagues” have a hard time with this.

Page 35: Dongarra 12-06-2016 Seminar.pdf

Collaborators and SupportMAGMA teamhttp://icl.cs.utk.edu/magma

PLASMA teamhttp://icl.cs.utk.edu/plasma

Collaborating partnersUniversity of Tennessee, KnoxvilleLawrence Livermore National Laboratory, Livermore, CAUniversity of California, BerkeleyUniversity of Colorado, DenverINRIA, France (StarPU team)KAUST, Saudi Arabia