ParILUT – A New Parallel Threshold ILUhanzt/talks/ParILUT.pdf · 2 05/07/2018 Hartwig Anzt:...

KIT – The Research University in the Helmholtz Association

Hartwig AnztSteinbuch Centre for Computing (SCC)

www.kit.edu

ParILUT – ANewParallel Threshold ILU05/07/2018Kolloquiumsvortrag inderFakultätfürInformatik

Steinbuch Centre for Computing2 05/07/2018 Hartwig Anzt:ParILUT – Anew parallelthreshold ILU

Motivation

Wearelookingforafactorization-basedpreconditionersuchthat .isagoodapproximationwithmoderatenonzerocount(e.g.).

• Whereshouldthesenonzeroelementsbelocated?• Howcanwecomputethepreconditioner inahighlyparallelfashion?

A ⇡ L · Unnz(L+ U) = nnz(A)

S(A) = {(i, j) 2 N2 : Aij 6= 0}


ExactLUFactorization

• Decomposesystemmatrixintoproduct .• BasedonGaussianelimination.• Triangularsolvestosolveasystem:

• De-Factostandardforsolvingdenseproblems.• Whataboutsparse?Oftensignificantfill-in…

A = L · U

Ax = b

IncompleteLUFactorization(ILU)

• Focusedonrestrictingfill-intoaspecificsparsitypattern.

• ForILU(0),isthesparsitypatternof.• Workswellformanyproblems.• Isthisthebestwecangetfornonzerocount?

• Fill-ininthresholdILU(ILUT)basesonthesignificanceofelements(e.g.magnitude).

• Oftenbetterpreconditionersthanlevel-basedILU.

• Difficulttoparallelize.

S

S

A

S




Motivation

Ly = b ) y ) Ux = y ) x





A = L · U

Ax = b







S

S

A

S




Motivation

Ly = b ) y ) Ux = y ) x





A = L · U

Ax = b







S

S

A

S




Motivation

Ly = b ) y ) Ux = y ) x

L 2 Rn⇥n

U 2 Rn⇥n

Lij = Uij = 0 8(i, j) /2 S

lower(unit-)triangular,sparse.

uppertriangular,sparse.

R = L� U, Rij = 0 8(i, j) 2 S.

.





A = L · U

Ax = b







S

S

A

S




Motivation

Ly = b ) y ) Ux = y ) x


Motivation



Rethinktheoverallstrategy!

• Useaparalleliterativeprocesstogeneratefactors.

• Thepreconditionershouldhaveamoderatenumberofnonzeroelements,butwedon’tcaretoomuchabout intermediatedata.

1. Selectasetofnonzero locations.2. Computevaluesinthose locationssuchthat isa“good”approximation.

3. Maybechangesomelocationsinfavoroflocationsthatresult inabetterpreconditioner.4. Repeatuntilthepreconditionerqualitystagnates.

A ⇡ L · U



Considerations

• Thisisanoptimizationproblem…

1. Selectasetofnonzero locations.2. Computevalues inthoselocationssuchthat

isa“good” approximation.3. Maybechangesomelocationsinfavorof

locationsthatresultinabetterpreconditioner.4. Repeatuntilthepreconditionerqualitystagnates.

A ⇡ L · U

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

AR = L U⇥ILUresidual -


Considerations





A ⇡ L · U

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA


Considerations





A ⇡ L · U

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA


Considerations





A ⇡ L · U

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA


Considerations





A ⇡ L · U

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA=

Residual:


Considerations1. Selectasetofnonzero locations.2. Computevalues inthoselocationssuchthat



A ⇡ L · U

nnz(A� L · U)nnz(L+ U)

• Thisisanoptimizationproblemwithequationsandvariables.

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA=

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

SSparsitypattern





A ⇡ L · U


• Wemaywanttocomputethevaluesinsuchthat,theapproximationbeingexactinthe locationsincludedin,butnotoutside!

L,U

=

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

R = A� L · U = 0|S

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA

nnz(L+ U)equationsvariablesnnz(L+ U)

S


SSparsitypattern





A ⇡ L · U



• ThisistheunderlyingideaofEdmondChow’sparallelILUalgorithm1:

L,U

1ChowandPatel.“Fine-grainedParallelIncompleteLUFactorization”. In:SIAMJ.onSci.Comp.(2015).

R = A� L · U = 0|SS


L · U = A|S ) F (lij , uij) =

(1

ujj

⇣aij �

Pj�1k=1 likukj

⌘, i > j

aij �Pi�1

k=1 likukj , i j





A ⇡ L · U




• Convergesintheasymptoticsensetowards incompletefactorssuchthat

L,U


R = A� L · U = 0|SS


R = A� L · U = 0|SL,U

L · U = A|S ) F (lij , uij) =

(1

ujj

⇣aij �

Pj�1k=1 likukj

⌘, i > j

aij �Pi�1

k=1 likukj , i j





A ⇡ L · U




• Wemaynotneedhighaccuracyhere,becausewemaychangethepatternagain…

• Onesinglefixed-pointsweep.

L,U R = A� L · U = 0|SS



L · U = A|S ) F (lij , uij) =

(1

ujj

⇣aij �

Pj�1k=1 likukj

⌘, i > j

aij �Pi�1

k=1 likukj , i j


Considerations1. Selectasetofnonzero locations.2. Computevaluesinthose locationssuchthat

isa“good”approximation.3. Maybechangesomelocationsinfavorof


A ⇡ L · U

• Comparingsparsitypatternsextremelydifficult.• MaybeusetheILUresidualasconvergencecheck.

=

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA


Considerations

A ⇡ L · U

1. Selectasetofnonzero locations.2. Computevaluesinthose locationssuchthat

isa“good”approximation.3. Maybechangesomelocationsinfavorof


A• Thesparsitypatternofmightbeagoodinitialstartfornonzero locations.


Considerations

A ⇡ L · U


isa“good”approximation.3. Maybechangesomelocations infavorof


A

1Saad.“IterativeMethodsforSparseLinearSystems,2nd Edition”. (2003).

• Thesparsitypatternofmightbeagoodinitialstartfornonzero locations.• Then,theapproximationwillbeexactforalllocations

andnonzero inlocations1.S1 = (S(A) [ S(L0 · U0)) \ S(L0 + U0)S0 = S(L0 + U0)

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ?

? ?? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ?

??

? ??

1

CCCCCCA=

0

BBBBBB@

? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ?

? ? ? ?? ? ? ? ? ?

1

CCCCCCA


Considerations

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ? ? ?

? ? ?? ? ? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ? ?

? ? ? ?? ?

? ??

1

CCCCCCA=

0

BBBBBB@

? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?

1

CCCCCCA

A ⇡ L · U




A

S1 = (S(A) [ S(L0 · U0)) \ S(L0 + U0)


andnonzero inlocations1.

• Addingalltheselocations(level-fill!)mightbegoodidea…

S0 = S(L0 + U0)


Considerations

A ⇡ L · U

A

S1 = (S(A) [ S(L0 · U0)) \ S(L0 + U0)


andnonzero inlocations1.

• Addingalltheselocations(level-fill!)mightbegoodidea, butaddingthesewillagaingeneratenewnonzeroresiduals

S0 = S(L0 + U0)

S2 = (S(A) [ S(L1 · U1)) \ S(L1 + U1)

0

BBBBBB@

? ? ? ? ?? ? ? ? ?? ? ?? ?

? ? ?? ? ? ?

1

CCCCCCA�

0

BBBBBB@

?? ?? ? ?? ? ? ?

? ? ?? ? ? ? ? ?

1

CCCCCCA⇥

0

BBBBBB@

? ? ? ? ?? ? ? ? ?

? ? ? ?? ?

? ??

1

CCCCCCA=

0

BBBBBB@

? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?

1

CCCCCCA





Considerations

A ⇡ L · U




• Atsomepointweshouldremovesomelocationsagain,e.g.thesmallestelements,andstartoverlookingatlocations…R = A� Lk · Uk


Interleavingfixed-pointsweepsapproximatingvalueswithpattern-changingsymbolicroutines.

ParILUT


ParILUT :Parallelisminsidetheblocks

Interleavingfixed-pointsweepsapproximatingvalueswithpattern-changingsymbolicroutines.Parallelisminsidethebuildingblocks.


Parallelisminsidetheblocks:Fixed-pointsweeps

Fixed-pointsweepsapproximatevaluesinILUfactorsandresidual1:

• Inherentlyparalleloperation.• Elementscanbeupdatedasynchronously.

• Wecanexpect100%parallelefficiencyifnumberofcores<numberofelements

• Residualnormisaglobalreduction.


bilinearfixed-pointiterationcanbeparallelizedbyelements

F (lij , uij) =

(1

ujj

⇣aij �

Pj�1k=1 likukj

⌘, i > j

aij �Pi�1

k=1 likukj , i j


Parallelisminsidetheblocks:Candidatesearch

S⇤ = (S(A) [ S(L · U)) \ S(L+ U)

sparsematrixproduct

sparsematrixsum

• Combinationofsparsematrixproductandsparsematrixsums.

• BuildingblocksavailableinSparseBLAS.• Blockscanbecombinedintoonekernelfor

higher(memory)efficiency.• Kernelcanbeparallelizedbyrows.

• Costheavilydependentonsparsitypattern.

• Kernelperformanceboundbymemorybandwidth.

sparsematrixsum

Identifylocationsthataresymbolicallynonzero:


Parallelisminsidetheblocks:Selectingthresholds

Athresholdseparatingthesmallestelementsisneededforremovinginsignificantlocationsandkeepingsparsity.

• Standardapproach:sort/selectionalgorithms• Highcomputationalcost• Memory-intensive• Hardtoparallelize

• Thresholdsdonotneedtobeexact:• Inaccuratethresholdsresultinafewadditional/lesselements.• Wecanusesamplingtogetreasonableapproximations.• Multiplesampling-basedselectionrunsallow togenerate

thresholdsofreasonablequalityinparallel.

• Isthisappropriateformany-corearchitectureswith5Kthreadsexecutingsimultaneously?


Thresholdselectiononparallelarchitectures

• SelectionSort (comparisons,elementswaps)• QuickSelect (averagecomparisons,elementswaps)• Floyd-Rivest algorithm()• IntroSelect (worstcasecomparisons,elementswaps)

JohnD.McCaplin (TACC)

Selectionalgorithmstraditionallybasedonre-arrangingelementsinmemory.

O(n)O(n2)O(n) O(n)

O(n) O(n)n+min(k, n� k) +O(

pn)

• Computepower(#FLOPs)growsmuchfasterthanmemorybandwidth.

• Data-rearrangingselectionalgorithmsbecomeinefficient.

”Operations arefree,memoryaccessiswhatcounts.”

64bitread,1FLOP


Thresholdselectiononparallelarchitectures:StreamSelect


• Primarygoal:reducethememorytraffic.• Accountforhighcorecounts,eachhavingasetofregisters.• Assume“nice”distributionofvalues(~uniform).• Acceptsomeinaccuracyinthegeneratedthreshold(approximation).

1. Findthelargestandsmallestelementstogetthedatarange.

2. Generateafinegridofthresholds,distribute themtothecores.3. Streamalldataonesingletime.4. Eachcorehandlesasetofthresholds andcountshowmany

elements arelarger/smaller.5. Selectthethresholdwiththeelementcountclosesttothe

targetvalue.

NVIDIAV100“Volta”7.8TFLOP/sDP16GBRAM@900GB/s

80Multiprocessors,eachwith64FP32cores[oversubscribewiththreadstohidelatency]








targetvalue.


80Multiprocessors,eachwith64FP32cores[oversubscribewiththreadstohidelatency]

Run5120threads(bindtocores)eachafewthresholds:

32 thresholds gives mesh granularity of 6.1035e-06

0 5 10 15 20 25 30 35

Thresholds per thread

0

10

20

30

40

Ru

ntim

e

LinearStreamSelect


0 5 10 15 20 25 30 35

Thresholds per thread

0

10

20

30

40

Ru

ntim

e

LinearStreamSelect







targetvalue.


OnaV100GPU:80Multiprocessorseachwith32FP64cores

64FP32cores[oversubscribewiththreadstohidelatency]

Run5120threads(bindtocores)eachafewthresholds:

32 thresholds gives mesh granularity of 6.1035e-06

• Setsizem,subsetsizen*• Foruniformdistribution: quality~meshgranularity.

• Runtimeincreaseslinearwithsetsize.• Runtimeindependentofsubset size.

103

104

105

106

107

108

Set size m

10-4

10-2

100

102

Runtim

e

103 104 105 106 107 108

Subset size n*

10-10

10-5

100

Devi

atio

n

(n-n*) / m


Isthisafuture-orientedalgorithm?





Bulk-Synchronous Algorithm!



DependenciesInterleavingfixed-pointsweepsapproximatingvalueswithpattern-changingsymbolicroutines.Parallelisminsidethebuildingblocks.



Strongdependency –wecannotstartbeforefinished.



Strongdependency –wecannotstartbeforefinished.Weakdependency – ifwestartbefore:+/- fewnonzeros.



Strongdependency –wecannotstartbeforefinished.Weakdependency – ifwestartbefore:+/- fewnonzeros.Wecanalreadystartwiththenextiteration.



Excellentcandidateforhybridhardware?Asynchronous execution?

GPU?

Strongdependency –wecannotstartbeforefinished.Weakdependency – ifwestartbefore:+/- fewnonzeros.Wecanalreadystartwiththenextiteration.



ParILUT – ANewParallelThreshold ILU

• HybridParILUT versionutilizingGPUandCPU,overlappingcommunication&computation.

• Asynchronous versionrelaxingdependencies.

• Useadifferentsparsity-patterngenerator:• Randomized?• Machinelearningtechniques?

• Increasingfill-intowards“full”factorization.

• ParILUT routinesavailableinMAGMA-sparse– theywillbeinGinkgo!

Helmholtz Impuls und VernetzungsfondVH-NG-1241

Nextsteps:

ThisresearchisincooperationwithEdmondChow(GaTech)andJackDongarra (UniversityofTennessee).

Slidesavailable:


Scalability

0 10 20 30 40 50 60 70

Number of Threads

0

10

20

30

40

50

60

70

Sp

ee

du

p

CSC CSRCandidatesResidualsILU-normCSR CSCAddSweep1Select2RmRemoveSweep2

10 20 30 40 50 60

Number of Threads

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Runtim

e fra

ctio

n

CSC CSRCandidatesResidualsILU-normSelectAddSweepsRemove

thermal2matrixfromSuiteSparse,RCMordering,8el/row.

IntelXeon Phi 7250“Knights Landing”[email protected],16GBMCDRAM@490GB/s

• Buildingblocksscalewith15%- 100%parallelefficiency.• Transpositionandsortarethebottlenecks.• Overallspeedup~35xwhenusing68KNLcores.



Scalability

10 20 30 40 50 60

Number of Threads

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Runtim

e fra

ctio

n

CSC CSRCandidatesResidualsILU-normSelectAddSweepsRemove

0 10 20 30 40 50 60 70

Number of Threads

0

10

20

30

40

50

60

70

Sp

ee

du

p

CSC CSRCandidatesResidualsILU-normCSR CSCAddSweep1Select2RmRemoveSweep2

topopt120matrixfromtopologyoptimization,67el/row.

• Buildingblocksscalewith15%- 100%parallelefficiency.• Dominatedbycandidatesearch.• Overallspeedup~52xwhenusing68KNLcores.



Performance

Matrix Origin Rows Nonzeros Ratio SuperLU ParILUT ParICT

ani7 2DAnisotropicDiffusion 203,841 1,407,811 6.91 10.48s 0.45s 23.34 0.30s 35.16

apache2 SuiteSparseMatrixCollect. 715,176 4,817,870 6.74 62.27s 1.24s 50.22 0.65s 95.37

cage11 SuiteSparseMatrixCollect. 39,082 559,722 14.32 60.89s 0.54s 112.56 --

jacobianMat9 Fun3DFluid FlowProblem 90,708 5,047,042 55.64 153.84s 7.26s 21.19 --

thermal2 ThermalProblem(Suite Sp.) 1,228,045 8,580,313 6.99 91.83s 1.23s 74.66 0.68s 134.25

tmt_sym SuiteSparseMatrixCollect. 726,713 5,080,961 6.97 53.42s 0.70s 76.21 0.41s 131.25

topopt120 Geometry Optimization 132,300 8,802,544 66.53 44.22s 14.40s 3.07 8.24s 5.37

torso2 SuiteSparseMatrixCollect. 115,967 1,033,473 8.91 10.78s 0.27s 39.92 --

venkat01 SuiteSparseMatrixCollect. 62,424 1,717,792 27.52 8.53s 0.74s 11.54 --

Runtimeof5ParILUT /ParICT stepsandspeedup overSuperLU ILUT*.

*WethankSherryLiandMeiyue Shaofortechnicalhelpingeneratingtheperformancenumbers.


HowaboutGPUs?

• Fine-grainedparallelism• Highbandwidth forcoalescentreads• Nodeepcachehierarchy

• Weneedtooversubscribecoresforhidinglatency



HowaboutGPUs?


• Weneedtooversubscribecoresforhidinglatency

Trans

Cand

ResSor

t

Trans

Add

Sweep1

Thres

Remv

Sweep2

-0.1

-0.05

0

0.05

0.1

NV

IDIA

P100 G

PU

Inte

l Hasw

ell

20c

thermal2matrixfromSuiteSparse,RCMordering,8el/row.

Runtime[s]



HowaboutGPUs?


• Weneedtooversubscribecoresforhidinglatencytopopt120matrixfromtopologyoptimization,67el/row.

Trans

Can

dRes

Sort

Trans

Add

Swee

p1

Thres

Rem

v

Swee

p2-1

-0.5

0

0.5

1

NV

IDIA

P100 G

PU

Inte

l Hasw

ell

20c

Runtime[s]



• Top-levelsolveriterationsasqualitymetric.• Fewsweepsgivea“better”preconditionerthanILU(0).• BetterthanILUT?

ParILUT quality

0 2 4 6 8 10

Number of ParICT steps (2 sweeps per step)

0

10

20

30

40

50

60

70

80

CG

Ite

ratio

ns

IC(0)ICTParICT

Anisotropicfluidflowproblemn:741,nz:4,951



ParILUT quality

0 2 4 6 8 10


0

10

20

30

40

50

60

70

80

CG

Ite

ratio

ns

IC(0)ICTParICT

0 500 1000 1500

ILU(0) Pattern discrepancy ILUT

0

1

2

3

4

5

6

7

8

9

10

Nu

mb

er

of

Pa

rIC

T s

tep

s (2

sw

ee

ps

pe

r st

ep

)





ParILUT quality

• Patternstagnatesafterfewsweeps.• Pattern“morelike”ILUTthanILU(0).

0 2 4 6 8 10


0

10

20

30

40

50

60

70

80

CG

Ite

ratio

ns

IC(0)ICTParICT

0 500 1000 1500

ILU(0) Pattern discrepancy ILUT

0

1

2

3

4

5

6

7

8

9

10

Nu

mb

er

of

Pa

rIC

T s

tep

s (2

sw

ee

ps

pe

r st

ep

)

ParICT


Testmatrices


Convergence:GMRESiterations


Convergence:CGiterations


Preconditioning

Ax = b

Weiterativelysolve alinearsystemoftheformWherenonsingularand.

Theconvergenceratetypicallydependsontheconditioning ofthe linearsystem,whichistheratiobetweenthelargestandsmallesteigenvalue.

Withwecantransform thelinearsystemintoasystemwithalowerconditionnumber:

Ifwenowapplythe iterativesolvertothepreconditionedsystem, weusuallygetfasterconvergence.

A 2 Rn⇥nb, x 2 Rn

MAx = Mb (left preconditioned)

AMy = b, x = My (right preconditioned)

cond2(A) =

�max

�min

=

1�

min

1�

max

= cond2(A�1

)

MAx = Mb

M ⇡ A�1


Preconditioning

Assume ,then: .Solutionrightavailable,butcomputing

isexpensive…

M = A�1x = MAx = Mb

M = A�1

Ax = b





A 2 Rn⇥nb, x 2 Rn



cond2(A) =

�max

�min

=

1�

min

1�

max

= cond2(A�1

)

MAx = Mb

M ⇡ A�1


Preconditioning


isexpensive…

Thepreconditionedsystemisrarelyformedexplicitely,instedisappliedimplicitely:


M = A�1

Ax = b





A 2 Rn⇥nb, x 2 Rn



cond2(A) =

�max

�min

=

1�

min

1�

max

= cond2(A�1

)

MAx = Mb

M ⇡ A�1

M zk+1 = Mrk+1

MA


Preconditioning


isexpensive…


Insteadofformingthepreconditioner explicitlyandapplyingas,IncompleteFactorizationPreconditioner(ILU)trytofindanapproximatefactorization.


M = A�1

zk+1 = Mrk+1

M ⇡ A�1

A ⇡ L · U

Ax = b





A 2 Rn⇥nb, x 2 Rn



cond2(A) =

�max

�min

=

1�

min

1�

max

= cond2(A�1

)

MAx = Mb

M ⇡ A�1

M zk+1 = Mrk+1

MA


Preconditioning


isexpensive…


Insteadofformingthepreconditioner explicitlyandapplyingas,IncompleteFactorizationPreconditioner(ILU)trytofindanapproximatefactorization.

Intheapplicationphase,thepreconditioner isonlygivenimplicitly,requiringtwotriangularsolves:


M = A�1

zk+1 = Mrk+1

M�1zk+1 = rk+1

LUzk+1| {z }=:y

= rk+1

)Ly = rk+1, Uzk+1 = y

zk+1 = Mrk+1

M ⇡ A�1

Ax = b





A 2 Rn⇥nb, x 2 Rn



cond2(A) =

�max

�min

=

1�

min

1�

max

= cond2(A�1

)

MAx = Mb

M ⇡ A�1

MAM zk+1 = Mrk+1

A ⇡ L · U

ParILUT – A New Parallel Threshold ILUhanzt/talks/ParILUT.pdf · 2 05/07/2018 Hartwig Anzt:...

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