Combinative preconditioning based on Relaxed Nested Factorization and Tangential Filtering preconditioner

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Factorization and Tangential Filtering preconditioner

Pawan Kumar, Laura Grigori, Frédéric Nataf, Qiang Niu

To cite this version:

Pawan Kumar, Laura Grigori, Frédéric Nataf, Qiang Niu. Combinative preconditioning based on

Relaxed Nested Factorization and Tangential Filtering preconditioner. [Research Report] RR-6955,

INRIA. 2009. �inria-00392881�

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Thème NUM

Combinative preconditioning based on Relaxed

Nested Factorization and Tangential Filtering

preconditioner

Pawan Kumar — Laura Grigori — Frederic Nataf — Qiang Niu

N° 6955

Juin 2009

Centre de recherche INRIA Saclay – Île-de-France

pre onditioner PawanKumar

∗

, Laura Grigori

†

, Frederi Nataf

‡

, Qiang Niu

§

ThèmeNUMSystèmesnumériques Équipe-ProjetGrand-large

Rapportdere her he n°6955Juin200924pages

Abstra t: Theproblemofsolvingblo ktridiagonallinearsystemsarisingfrom thedis retizationofPDEis onsidered. Thenestedfa torizationpre onditioner introdu edby[J.R. Appleyardand I.M.Cheshire,NestedFa torization,SPE 12264,presentedattheSeventhSPESymposiumonReservoirSimulation,San Fran is o,1983℄isanee tivepre onditionerfor ertain lassofproblemsanda similarmethodisimplementedin S hlumerger'sE lipseoilreservoirsimulator. In this paper, a relaxed version of Nested Fa torization pre onditioner is proposed as a repla ementto ILU(0). Indeed, the proposed pre onditioner is SPDand leadsto astable splittingiftheinput matrixisS.P.D.. ForILU(0), equivalent properties hold ifthe inputmatrix isa M-matrix. Moreoverit has no storage ost. Ee tive multipli ative/additive pre onditioning is a hieved in ombination with Tangential ltering pre onditioner with the lter ve tor hosen as ve tor of

ones

. Numeri al tests are arried out with both additive and multipli ative ombinations. With thissetup thenewpre onditionerisas robustasthe ombinationofILU(0)withtangentiallteringpre onditioner. Key-words: pre onditioner,linearsystem,frequen ylteringde omposition, GMRES,nestedfa torization,in ompleteLU,eigenvalues

∗

INRIA Sa lay - Ilede Fran e, Laboratoire de Re her he enInformatique, Universite Paris-Sud11,Fran e(Email:pawan.kumarlri.f r)

†

INRIA Sa lay - Ilede Fran e, Laboratoire de Re her he enInformatique, Universite Paris-Sud11,Fran e(Email:laura.grigoriinr ia.f r)

‡

Laboratoire J. L. Lions, CNRS UMR7598, Universite Paris 6, Fran e; Email: natafann.jussieu.fr

§

S hoolofMathemati alS ien es,XiamenUniversity,Xiamen,361005,P.R.China;The workofthisauthorwas performedduringhisvisittoINRIA,fundedbyChina S holarship Coun il;Email:kangniugmail. om

Résumé : Dans e papier nous présentonsune version de fa torisation em-boîtéave relaxation. Cettefa torisationestproposée ommerempla ementde ILU(0). Lepré onditionneurproposéest SPDsi lamatri eoriginaleest SPD. Pour ILU(0)des propriété équivalentes existent si la matri e originaleest une M-matri e. Un pré onditionnemente a eest obtenu en ombinaisonave le ltragetangentielave leve teurdeltragede

ones

. Destestsnumériquessont présentés pour des ombinaisonsadditifset multipli atifs. Ils montrentque le pré onditionneurest aussirobustequela ombinaisonobtenueparILU(0)ave leltragetangentiel.

Mots- lés : pré onditionnement, fa torisation emboîtée, fa torisation LU in omplète

1 INTRODUCTION

Several appli ations are modelled by non-linear partial dierential equations, examples in lude oil reservoir simulations and uid ow through porous me-dia. TheseequationsareusuallysolvedbytheNewton'smethod orxedpoint method,whereatea hstepapre onditionediterativemethodisgenerallyused forsolvingalargesparselinearsystem

Ax = b.

(1)

Thelinearsystems hangeat ea hstep, andsolvingthem onstitutethemost omputationallyintensivepartofthesimulation. Therefore,ane ient pre on-ditionerwith fastsetuptime playsanimportantrole in theoverallsimulation pro ess. Also, sin ethe systemsare verylarge, the pre onditionershould not be too demanding in terms of memory requirements. This problem has al-readybeenextensivelystudied,see [7,16, 18℄forthedes riptionof algorithms and their omparisons. Inparti ular, one lassof methods whi h haveproved parti ularly su essful are the algebrai multigrid methods, see for example, [17, 19, 20, 24℄. However,for ertain problems the algebrai multigrid meth-ods involverelatively high setup ost, whi h leadsus to look into some other alternativesinthiswork.

Withfastsetuptimeandverymodeststoragerequirement,theNested Fa -torization (NF) pre onditioner introdu ed in [1℄ is a powerful pre onditioner; it has been found [1, 15℄ that for ertain lass of problems it performsbetter thanthewidelyusedILU(0)orModiedILU[18℄. NFtakesasinputamatrix whi h hasanestedblo ktridiagonal stru ture. Matri es arisingfrom the dis- retization ofP.D.E anbepermuted to thisform. Themethod of NFdiers fromILU(0)orModiedILU(0)inthatthepre onditioningmatrixinNFisnot formed stri tly from upper and lowertriangular fa tors. Instead, blo k lower andupperfa torsare onstru tedusingapro edurewhi haddsonedimension atatimetothepre onditioningmatrixhavingthediagonalmatrixonthelowest level.

TheNFpre onditionerhassomeimportantproperties. If

B

N F

istheNF pre- onditioner,then

colsum(B

N F

− A) = 0

(alsoknownaszero olsumproperty), asa onsequen ethesumoftheresidualsinsu essiveKryloviterationsremain zero,providedasuitableinitialsolutionisused[1℄. Thisproperty anprovidea veryuseful he konthe orre tnessoftheimplementation. Further,quoting[1℄, the fa torizationpro edure onservesmaterial exa tlyforea h phaseat ea h lineariteration,anda ommodatesnon-neighbour onne tions(arisingfromthe treatmentofthefaults, ompletingthe ir leinthree-dimensional oning stud-ies, numeri al aquifers, dual porosity/permeability systemset .) in anatural way. Moreover,foruidowproblemsitisprovedin[2℄thatthelowerandthe uppertriangularfa torsofNFarenonsingular. Duetothesedesirablequalities, the NF pre onditioner is of parti ular interest in the oil reservoirindustry; a methodsimilartoNFisimplementedinS hlumerger'sE lipseoilreservoir sim-ulator [12℄. However,aswewill see later in thetables, theNF pre onditioner anfailonsomeproblems.

The main result of this paper is the introdu tion of a new pre onditioner whi h orrespondstoarelaxedversionofNF,namelyRNF(0,0). Comparedto

ˆ If

A

isS.P.D.then, RNF(0,0) isS.P.D.

RNF(0,0) leadstoastablesplitting ˆ nosetup ost

ˆ nostoragerequirement

Thenew pre onditioner is parti ularly useful when multipli ate/additive pre- onditioningisa hievedin ombinationwithtangentiallteringpre onditioner [4℄. With this setup the new pre onditioner is as robust as the ombination ofILU(0)with tangentialltering pre onditioner. Notably bothmultipli ative andadditive ombinationaretried. Goodresultsareobtainedwiththeadditive ombinationwhi hmoreoverhasanadvantageovermultipli ative ombination onparallelsystems;ea hofthepre onditionersolvesinvolvedwiththeadditive ase anbedonesimultaneouslyontwodierentpro essors. Inorder tomake a omprehensive study, we onsider a general relaxed version of NF, namely RNF(

α

,

β

).

This arti le is organized as follows. Notations play an important role in explainingthemethodof NF;we arefullyintrodu ethem inthenextse tion. Inse tion3,NF isexplainedand laterin these tion RNFis introdu ed. The method of LFTFD is dis ussed briey in se tion 4. In se tion 5, we present onvergen eresultsforthe ombinationinvolvingRNF(0,0)andLFTFDwhi h shows that the ombination is onvergent. In se tion 6 we will omment on numeri alresults. Andnally,se tion7 on ludesthepaper.

2 NOTATIONS

Throughout the paper, matrix

A

is onsidered to be arising from the nite dieren e or nite volume dis retization of two or three dimensional partial dierentialequations. Forexample,using7-pointformulaforthreedimensional problemsforan

nx×ny ×nz

grid,where

nx

denotesthenumberofpointsonthe line,

ny

denotesthenumberoflinesonea hplane,and

nz

denotesthenumber ofplanes,weobtainan(

nx × ny × ny) × (nx × ny × nz)

matrix. Let

nxy

denote

nx × ny

,thenumberofunknownsonea hplane,

nyz

denote

ny × nz

,thetotal numberoflinesonallplanes,and

nxyz

denote

nx × ny × nz

,thetotalnumber ofunknowns. Then theresulting matrixhasa nestedtridiagonalstru ture as follows:

A =













b

D

1

U

b

3

1

b

L

1

3

D

b

2

. . . . . . . . .

U

b

nz−1

3

b

L

nz−1

3

D

b

nz













.

(2)

Herethediagonalblo ks

D

b

i

s

aretheblo ks orrespondingtotheunknowns ofthe

i

th

plane. Theblo ks

L

b

i

3

and

U

b

i

3

arediagonal matri esofsize

nxy

,and they orrespondto the onne tions betweenthe

i

th

and

(i + 1)

th

plane. F ur-ther, thediagonal blo ks

D

b

i

s

are themselvesblo ktridiagonal, i.e. they have

thefollowingstru ture,

b

D

i

=















D

_{(i−1)∗ny+1}

U

(i−1)∗ny+1

2

L

(i−1)∗ny+1

2

D

(i−1)∗ny+2

. . . . . . . . .

U

i∗ny−1

2

L

i∗ny−1

2

D

i∗ny















where the blo ks

L

i

2

and

U

i

2

are diagonal matri es of size

nx

ea h and they orrespondto the onne tions between thelines. The diagonal blo ks

D

i

are themselvestridiagonalmatri es,

D

i

=













˜

D

_{(i−1)∗nx+1}

U

˜

1

(i−1)∗nx+1

˜

L

(i−1)∗nx+1

₁

D

˜

_{(i−1)∗nx+2}

. . . . . . . . .

U

˜

i∗nx−1

˜

L

i∗nx−1

1

D

˜

i∗nx













with thes alars

L

˜

(i−1)∗nx+j

1

and

U

˜

(i−1)∗nx+j

1

being the orresponding onne -tions between the ells. We observe that this matrix has nested tridiagonal stru ture. For any matrix

K

,

Diag(K)

refersto the stri t diagonal of

K

; for anyve tor

v

,

Diag(v)

refersto thediagonalmatrix formedfrom theve tor

v

. Foranydiagonalmatrix

M

of size

nxyz

,the

i

th

entryof

M

is denotedby

M

˜

i

, the

i

th

lineblo kof

M

isdenotedby

M

i

,andthe

i

th

planeblo kof

M

isdenoted by

M

c

i

. Belowwelistthestru turesofthematri es

L

1

,

U

1

,

L

2

,

U

2

,

L

3

,and

U

3

tobereferredtolater.

Diag(A) =











˜

D

1

˜

D

2

. . .

˜

D

nxyz











,

M =











˜

M

1

˜

M

2

. . .

˜

M

nxyz











L

1

=











0

˜

L

1

1

0

. . . . . .

˜

L

nxyz−1

1

0











,

U

1

=













0

U

˜

1

1

0

. . . . . .

U

˜

1

nxyz−1

0













L

2

=



















0

L

1

2

. . . . . . . . . . . . . . .

L

nyz−1

2

0



















,

U

2

=



















0 U

1

2

. . . . . . . . . . . . . . .

U

nyz−1

2

0



















L

3

=



















0

b

L

1

3

. . . . . . . . . . . . . . .

b

L

nz−1

3

0



















,

U

3

=



















0

U

b

1

3

. . . . . . . . . . . . . . .

U

b

nz−1

3

0



















3 RELAXED NESTED FACTORIZATION

Inthisse tion,weintrodu eRNFforthree-dimensional ase,thetwo-dimensional ase will be onsidered a spe ial ase of three-dimensional ase with just one plane. Werst introdu easpe ial versionof thepre onditioner for whi h we provethatitisS.P.Dwhentheinputmatrix

A

isS.P.D.Later, weintrodu ea moregeneralversionwhi hwillprovesigni antforthetwo-dimensional prob-lems.

3.1 A spe ial ase : RNF(0,0)

Wedene theRNF(0,0)pre onditionerasfollows:

B

RN F

(0,0)

=

(P + L

3

)(I + P

−1

U

3

)

P

=

(T + L

2

)(I + T

−1

U

2

)

T

=

(M + L

1

)(I + M

−1

U

1

)

M

=

diag(A) .

3.2 The general ase : RNF(

α

,

β

)

Thegeneral version of relaxed nestedfa torization pre onditioner denoted by

B

RN F

(α,β)

=

(P + L

3

)(I + P

−1

U

3

),

P

=

(T + L

2

)(I + T

−1

U

2

),

T

=

(M + L

1

)(I + M

−1

U

1

),

M

=

diag(A) − α L

1

M

−1

U

1

−

β colsum(L

2

T

−1

U

2

)

−

β colsum(L

3

P

−1

U

3

),

where

colsum(K) = Diag(1K)

for any square matrix

K

, and

1

denotes the ve tor

[1, 1, ..., 1]

. Note that with

α = 1

and

β = 1

wegetthe lassi alNested Fa torizationpre onditionerasdes ribedin [1℄.

ForRNF(1,1)weobservethat

B

RN F(1,1)

− A = L

2

T

−1

U

2

− colsum(L

2

T

−1

U

2

) + L

3

P

−1

U

3

− colsum(L

3

P

−1

U

3

)

sothatthepre onditionersatises

1

_B

_{RN F}

_(1,1)

_{= 1A,}

(3)

i.e.,a olumnsum onstraintasin MILU[18℄.

Thematrix

M

is omputedbeforetheiterationbegins. Finding

M

requires writing down the above expression for

M

. We noti e that

M

is a diagonal matrix, sin e

Diag(A)

,

L

1

M

−1

_U

1

,

colsum(L

2

T

−1

_U

2

)

, and

colsum(L

3

P

−1

_U

3

)

arediagonalmatri es. Thematri es

T

and

P

areblo kdiagonalmatri eswith squarediagonalblo ksofsizes

nx

and

nxy

respe tively. Wehave

L

1

M

−1

U

1

=

0

B

B

B

@

0

˜

L

1

1

M

˜

−1

1

U

˜

1

1

. . .

˜

L

nxyz−1

1

M

˜

−1

nxyz−1

U

˜

nxyz−1

1

1

C

C

C

A

,

T

=

0

B

B

B

@

T

1

T

2

. . .

T

nyz

1

C

C

C

A

,

colsum(L

2

T

−1

U

2

)

=

0

B

B

B

B

@

0

colsum(L

1

2

T

−1

1

U

1

2

)

. . .

colsum(L

nyz−1

2

T

_nyz−1

−1

U

nyz−1

2

)

1

C

C

C

C

A

,

P

=

0

B

@

P

1

. . .

P

nz

1

C

A , and

colsum(L

3

P

−1

U

3

)

=

0

B

B

B

@

0

colsum(b

L

1

3

P

−1

1

U

b

1

3

)

. . .

colsum(b

L

nz−1

3

P

−1

nz−1

U

b

nz−1

3

)

1

C

C

C

A

.

3.3 Constru tion of RNF(

α, β

) pre onditioner

Adetailedpseudo odeforthe onstru tionof

RN F (α, β)

pre onditioneris pro-videdinAlgorithm(1), webrieyoutlinethe onstru tioninthis se tion. The entries of

M

are al ulated in a single sweep through the grid. Reading the entries rowbyrowfortheexpressionof

M

,we andetermine

M

.

Therstrowgives

M

˜

1

=

D

˜

1

. Forthese ond ellonwardsuptothelast ell ofrstlinethereare ontributionsfromtheterm

L

1

M

−1

_U

1

,andwehave

˜

M

i

= ˜

D

i

− ˜

L

1

i−1

U

˜

1

i−1

/ ˜

M

i−1

, i = 2, . . . , nx.

Whentheupdatefromtherstlineisnished,ea helementof

M

forthese ond linealsodependsonthepreviouslinethroughtheterm

colsum(L

2

T

−1

_U

2

)

,and ea h element of

M

for se ond plane onwards depends on the previous plane through the term

colsum(L

3

P

−1

_U

₃

₎

. The

(j + 1)

th

line blo k of the blo k diagonalmatrix

colsum(L

2

T

−1

_U

₂

₎

, anbe omputedasfollows:

colsum(L

j

₂

T

−1

j

U

j

2

) = Diag(

1

L

j

2

T

j

−1

U

j

2

) = Diag((U

j

2

)

T

T

j

−T

(L

j

2

)

T

1

).

Asimilar al ulation anbeperformedforthe omputationoftheplaneblo ks of

colsum(L

3

P

−1

_U

3

)

.

3.4 Solution pro edure

Inthisse tionwewill seehowthenestedexpressionfor

B

isusedto solvethe equation

B

RN F

(α,β)

u

=

v

requiredin apre onditioned Kryloviterativemethod. At theoutermostlevel, wesolve

(P + L

3

)(I + P

−1

U

3

)u = v

doingthe forwardsweepsof theform

q = P

−1

i

(v − b

L

i−1

3

q)

, and theba kward sweepsofthe form

u = u − P

−1

i

U

b

3

i

q.

These equationsare solvedfor oneplane at a time. The solution involves solving equations of the form

P

i

w = x

for ea hplane. Tosolvethisequationweusethefa t thatea hplaneblo kofthe pre onditionerhasanin ompleteblo kfa torizationrepresentedby

P = (T + L

2

)(I + T

−1

U

2

),

andwesolve

P

i

w = x

usingtheforwardsweepsoftheform

r = T

−1

i

(x − L

i−1

2

r)

, andtheba kwardsweepsoftheform

w = w − T

−1

i

U

i

2

r

. Wenoti ethatweneed to solveequationsof theform

T

i

y = z

orresponding toea hline blo kofthe pre onditioner,andforthisweusethefollowingfa toredformof

T

T = (M + L

1

)(I + M

−1

U

1

),

andwesolve

T

i

y = z

usingtheforwardsweepsoftheform

s = ˜

M

−1

i

(z − ˜

L

i−1

1

s)

, andtheba kwardsweepsoftheform

y = y − ˜

M

−1

i

U

˜

1

i

s

.

Algorithm1PSEUDOCODETOFIND

M

FORRNF(

α

,

β

) INPUT:

Diag(A)

,

L

1

,

L

2

,

L

3

,

U

1

,

U

2

,

U

3

OUTPUT:

M

for

i

=1tonz(Numberofplanes)do

if

i 6= 1

and

β 6= 0

(Nottherstplane)then Solve

P

T

i−1

b

x

=

(c

L

3

i−1

)

T

_.1

c

M

i

=

M

c

i

-

β(c

U

3

i−1

)

T

_b

_x

endif UPDATE-FROM-LINES

( c

M

i

)

endfor UPDATE-FROM-LINES(

M

c

i

)

for

j

=1tony(Numberoflinesin urrentplane)do if

j 6= 1

and

β 6= 0

(Nottherstline)then

Solve

T

T

j−1

x

=

(L

2

j−1

)

T

_.1

M

_{(i−1)∗ny+j}

=

M

(i−1)∗ny+j

-

β(U

j−1

2

)

T

x

endif UPDATE-FROM-CELLS

(M

(i−1)∗ny+j

)

endfor UPDATE-FROM-CELLS

(M

i

)

˜

M

_{(i−1)∗nx+1}

=

D

˜

(i−1)∗nx+1

(Update fromrst ellof urrentline) for

k

=2to nx(Update fromrestofthe ells)do

˜

M

_{(i−1)∗nx+k}

=

M

˜

(i−1)∗nx+k

-

α ˜

L

(i−1)∗nx+k−1

1

M

˜

_{(i−1)∗nx+k−1}

−1

U

˜

1

(i−1)∗nx+k−1

endfor

4 LOW FREQUENCYTANGENTIAL FIL TER-ING DECOMPOSITION

In[4℄,aLowFrequen yTangentialFilteringDe omposition(LFTFD) pre ondi-tionerwasdeveloped. If

t

isanyve torand

B

LF

istheLFTFDpre onditioner, thenthepre onditionersatisestherightlteringproperty

At = B

LF

t .

(4)

Webriey des ribethede ompositionpro edure. Thepre onditioner

B

LF

isdenedas follows

B

LF

= (Q + L

3

)(I + Q

−1

U

3

)

,

where

Q

is ablo k diagonalmatrixwith diagonal blo ks

Q

i

of size

nxy

ea h, theblo ks

Q

i

s

aredened asfollows

Q

i

=

(

b

D

1

, i = 1,

b

D

i

− b

L

i−1

3

(2β

i−1

− β

i−1

Q

i−1

β

i−1

) b

U

3

i−1

, i = 2 to nz

where

β

i

isgivenby

β

i

= Diag((Q

−1

i

U

b

i

t

i+1

)./( b

U

i

t

i+1

))

(5)

Here./isapointwiseve tordivision,

Diag(v)

isthediagonalmatrix onstru ted fromtheve tor

v

and

t = [t

1

, ..., t

nz

]

T

isthelterve tor. Divisionbyzerobeing undened,itisassumedthatno omponentof

t

iszero.

Thispre onditionerseemstoberobustenoughwhen ombinedwithILU(0) usingmultipli ativepre onditionersimilar to(2) forawide lass problemsfor whi h it was tested. For more on this we refer the reader to [4℄. For other pre onditioners satisfying the ltering property (4), the reader is referred to [9,10,21, 22,23℄.

5 ANALYSISOFTHEMULTIPLICATIVE

PRE-CONDITIONERSINVOLVINGRNF(0,0),RNF(1,0), AND LFTFD

Foranymatrix

K

, let

K ≻ 0

denote that thematrix

K

is symmetri positive denite;let

K > 0

denotethatea hentryof

K

ispositive,andlet

λ

i

(K)

denote anarbitraryeigenvalueof

K

.

For onvenien e,wedenotethepre onditionersRNF(0,0)by

B

RN F

0

,RNF(1,0) by

B

RN F

1

,andLFTFDby

B

LF

. WhentheresultsapplytobothRNF(0,0)and RNF(1,0),wedenote thepre onditioneras

B

RN F

; further,let

B

c

denote

B

−1

c

= B

−1

RN F

+ B

LF

−1

− B

−1

LF

AB

RN F

−1

.

(6) TheFollowingtheoremshowsthatthemultipli ativepre onditionerobtained usingformula(6)satisestherightlteringproperty(4).

Lemma5.1 If

B

LF

satises the right ltering property (4) on ve tor

t

, then thepre onditioner

B

c

1

obtainedby ombining

B

LF

and

B

RN F

(α,β)

usingformula (6)satises theright ltering property (or onstraint)on thesame ve tor,i.e.

B

c1

t = At

.

Proof: FromEqn. (6)wehave

I − B

−1

c1

A = (I − B

−1

RN F

(α,β)

A)(I − B

LF

−1

A)

Fromthis identitywehave,

(I − B

−1

c1

A)t

= 0 ( since (I − B

LF

−1

A)t = 0)

Hen e thelemma.

InLemma 5.2, we provethat the xed point iterationinvolvingRNF(0,0) pre onditioneris onvergent. Lemma5.3 anbefoundin[4℄. Inourlastresult Theorem 5.6, we prove that the xed point iteration involving

B

c1

pre ondi-tioneris onvergent. Lemma5.2 If

A ≻ 0

, then

B

RN F0

≻ 0

,

B

RN F

1

≻ 0

,

λ

i

(B

−1

RN F

0

A) ∈ (0, 1]

, and

λ

i

(B

−1

RN F

1

A) ∈ (0, 1]

.

Proof: TheRNF(0,0)pre onditionerisdenedasfollows:

B

RN F

0

=

(P

RN F0

+ L

3

)P

RN F0

−1

(P

RN F0

+ L

T

3

),

P

RN F

0

=

(T

RN F

0

+ L

2

)T

_{RN F}

−1

₀

(T

RN F

0

+ L

T

2

),

T

RN F

0

=

(M

RN F0

+ L

1

)M

RN F0

−1

(M

RN F

0

+ L

T

1

),

M

RN F

0

=

Diag(A).

It is easy to see that sin e

A ≻ 0

,

M

RN F

0

= Diag(A) > 0

, so

T

RN F

0

≻ 0

,

P

RN F

0

≻ 0

, and hen e

B

RN F0

≻ 0

. Sin e

B

−1

RN F

0

A

is similar to asymmetri matrix

B

−

1

2

RN F

0

AB

−

1

2

RN F

0

, all the eigenvalues of

B

−1

RN F

0

A

are real. Also, sin e

B

RN F0

,

T

RN F

0

, and

P

RN F

0

arepositivedeniteand

B

RN F0

= A + L

1

M

_{RN F}

−1

0

L

T

1

+ L

2

T

_{RN F}

−1

0

L

T

2

+ L

3

P

RN F0

−1

L

T

3

,

wehave

(B

RN F

0

x, x)

= (Ax, x) + (M

RN F

−1

0

L

T

1

x, L

T

1

x) + (T

RN F

−1

0

L

T

2

x, L

T

2

x) + (P

RN F

−1

0

L

T

3

x, L

T

3

x),

≥ (Ax, x),

> 0, ∀x 6= 0,

and from this we have

λ

i

(B

−1

RN F

0

A) ∈ (0, 1]

. The

B

RN F

1

pre onditioner has similarhierar hi alrepresentationas

B

RN F

0

, itisdened asfollows:

B

RN F

1

=

(P

RN F1

+ L

3

)P

RN F1

−1

(P

RN F1

+ L

T

3

),

P

RN F

1

=

(T

RN F

1

+ L

2

)T

RN F

−1

1

(T

RN F

1

+ L

T

2

),

Sin e

A ≻ 0

,wehave

T

RN F

1

= Diag(A) + L

1

+ L

T

1

≻ 0

,

P

RN F

1

≻ 0

,andhen e

B

RN F

1

≻ 0

. Also

B

RN F

1

= A + L

2

T

RN F

−1

1

L

2

T

+ L

3

P

RN F

−1

1

L

T

3

,

(B

RN F

1

x, x)

≥ (Ax, x),

> 0, ∀x 6= 0,

andfromthiswehave

λ

i

(B

−1

RN F

1

A) ∈ (0, 1]

. Hen etheproof.

Lemma5.3 Let

A ≻ 0

,then

B

LF

≻ 0

,

B

LF

− A = N ≻ 0

, and

λ

i

(B

−1

LF

A) ∈

(0, 1]

.

Proof: Fortheproof ofthe fa t that

B

LF

≻ 0

and

B

LF

− A = N ≻ 0

see Lemma2.2 in [4℄. Nowbythe similarargumentasin thepreviouslemma, we have

(B

LF

x, x) = (Ax, x) + (N x, x) ≥ (Ax, x) > 0, ∀x 6= 0,

andfromthisitfollowsthat

λ

i

(B

−1

LF

A) ∈ (0, 1]

.

If

A ≻ 0

, then the inner produ t

( , )

A

dened by

(u, v)

A

= u

T

_Av

is a well dened inner produ t, and it indu es the energy norm

k k

A

dened by

kvk

A

= (v, v)

A

1

2

foranyve tor

v

. Amatrix

K

is alledA-selfadjointif

(Ku, v)

A

= (u, Kv)

A

.

orequivalentlyif,

A

−1

_K

T

_{A = K}

(7) Lemma5.4 IfAissymmetri ,thenthematri es

I − B

−1

RN F

A

,

I − B

−1

LF

A

,and

I − B

−1

c

A

areA-selfadjoint.

Proof: Using riteria for self-adjointness dened by (7), it an be easily provedthat

I −B

−1

RN F

A

and

I −B

−1

LF

A

areA-selfadjoint. Toprovethat

I −B

−1

c

A

isself-adjoint,wehave

A

−1

_{(I − B}

−1

c

A)

T

A =

A

−1

(I − B

−1

LF

A)

T

_{(I − B}

−1

RN F

A)

T

_A,

=

A

−1

_{(I − AB}

−1

RN F

− AB

LF

−1

+ AB

LF

−1

AB

−1

RN F

)A,

=

I − B

−1

c

A.

Hen ethetheorem.

Foranymatrix

K

, let

ρ(K) = max

i

(|λ

i

(K)|)

denote thespe tralradiusof

K

.

Lemma5.5 [Ashby, Holst, Manteuel, and Saylor℄ [5 ℄ If

A ≻ 0

and

K

is A-selfadjoint,then

kKk

A

= ρ(K)

.

Theorem5.6 If

A ≻ 0

, then the xed point iteration with the orresponding error propagationmatrix

(I − B

−1

c

A)

is onvergent,i.e.,

ρ(I − B

−1

c

A) < 1.

Proof: UsingTheorem(5.4)andLemma(5.5),wehave

ρ(I − B

−1

c

A) =

kI − B

c

−1

Ak

A

,

=

k(I − B

−1

RN F

A)(I − B

LF

−1

A)k

A

,

≤

k(I − B

−1

RN F

A)k

A

k(I − B

−1

LF

A)k

A

,

=

ρ(I − B

−1

RN F

A)ρ(I − B

LF

−1

A),

<

1 (U sing Lemma 5.2 and Lemma 5.3).

Hen e theproof.

6 NUMERICAL EXPERIMENTS

Inthisse tionwepresentnumeri alresultsforNF,RNF,andits ombinations with LFTFD.Weshall ompareea hof thesepre onditioners,and outlinethe advantages and disadvantages of ea h of them. First, we will ompare the onvergen eofRNF(

α

,

β

)withother lassi alpre onditioners,andlaterweshall observethespe trumbehavior. Se ondly,weshall ompareseveral ombinative pre onditionings involving RNF(

α

,

β

). Thirdly, we shall ompare the results betweenadditiveandmultipli ativepre onditionings,andthennally weshall dis usstheresultsof ombinativepre onditioningswiththelterve tor hosen asRitzve tor.

6.1 Sour e of matri es

We onsidertheboundaryvalueproblemasin [4℄

div(a(x)u) − div(κ(x)∇u) = f in Ω,

u = 0 on ∂Ω

D

,

∂u

∂n

= 0 on ∂Ω

N

,

(8) where

Ω = [0, 1]

n

(

n = 2

,or3),

∂Ω

N

= ∂Ω \ ∂Ω

D

.

Theve toreld

a

andthe tensor

κ

arethegiven oe ientsofthepartialdierentialoperator. In2D ase, wehave

∂Ω

D

= [0, 1] × {0, 1}

,andin 3D ase,wehave

∂Ω

D

= [0, 1] × {0, 1} ×

[0, 1]

.

Thefollowingve asesare onsidered:

Case 4.1: Adve tion-diusion problem with a rotating velo ityin two dimen-sions:

Thetensor

κ

isidentity, andthe velo ityis

a

= (2π(x

2

− 0.5), 2π(x

1

− 0.5))

T

. Wetestproblemsforthetwo-dimensional ase.

Case 4.2: Non-homogeneous problems with large jumps in the oe ients in twodimensions:

The oe ient

a

iszero. Thetensor

κ

isisotropi anddis ontinuous. Itjumps from the onstantvalue

10

3

in the ring

1

2

√

2

≤ |x − c| ≤

1

2

,

c = (

1

2

,

1

2

)

T

, to 1 outside. Wetestproblemsforthetwo-dimensional ase.

Case 4.3: Skys raper problems:

Thetensor

κ

isisotropi anddis ontinuous. Thedomain ontainsmanyzonesof highpermeabilitywhi hareisolatedfromea hother. Let

[x]

denotetheinteger valueof

x

. Fortwo-dimensional ase,wedene

κ(x)

asfollows:



andforthree-dimensional ase

κ(x)

is denedasfollows:

κ(x) =



10

3

_{∗ ([10 ∗ x}

2

] + 1),

if [10 ∗ x

i

] = 0 mod(2) , i = 1, 2, 3,

1,

otherwise.

Case 4.4: Conve tive skys raperproblems:

ThesamewiththeSkys raperproblemsex eptthatthevelo ityeldis hanged tobe

a

= (1000, 1000, 1000)

T

. Bothtwoandthreedimensional aseare onsid-ered.

Case 4.5: Anisotropi layers:

The domain is made of 10 anisotropi layers with jumps of up to four or-ders of magnitude, and an anisotropy ratio of up to

10

3

in ea h layer. For three-dimensional ase, the ube is divided into 10 layers parallel to

z = 0

, of size 0.1, in whi h the oe ients are onstant. The oe ient

κ

x

in the

i

th layer is given by

v(i)

, the latter being the

i

th omponent of the ve tor

v = [α, β, α, β, α, β, γ, α, α]

, where

α = 1

,

β = 10

2

and

γ = 10

4

. We have

κ

y

= 10κ

x

and

κ

z

= 1000κ

x

. Thevelo ityeldiszero.

We onsiderproblemsonuniformgridwith

n×n

nodesforthetwo-dimensional ase, where we hoose

n = 100, 200, 300

, and

400

; whereas, for the three-dimensional ase we onsider the uniform grid with

n × n × n

nodes, and we hoose

n = 20, 30

,and

40

.

6.2 Comments on the numeri al tests

All ourtest resultswere obtainedwith GMRES(20) in double pre ision arith-meti withinitial solutionve torasve torofallzeros. Theknownsolutionis arandomve tor. Thestopping riteriais thede rease ofthe relativeresidual below

10

−12

. Themaximumnumberof iterationsallowedis

200

. Themethod of NF is implemented in Fortran 90, whereas, the method of LFTFD is im-plemented in MATLAB, for this reason insteadof CPU time, weprovidethe op ount (see Table 1) for a

nx × nx × nx

uniform grid forall the methods onsidered.

Intable1,weintrodu eshort notationsforthemethodsand thetest ases onsidered.

To ompareRNF(1,1)andRNF(0,0)withother lassi almethods,wepresent resultsinTable(3). Wenoti ethat RNF(1,1)takessmallernumberofstepsto onverge as ompared to the lassi alpre onditioners like ILU and MILU.In mostof thetest ases, we observethat wheneverMILU onverged within

200

steps,RNF(1,1) onvergedaswell. Formatri eswithdis ontinuous oe ients like skys rapperand onve tiveskys rapperproblems, RNF(1,1),ILU(0), and MILU fail to onverge within the maximum number of iterations. The on-ditioning behaviorofRNF(0,0) seemsto be similar to ILU(0) in terms ofthe numberofstepsrequiredfor onvergen e;theproblems forwhi hILU(0) on-verges, RNF(0,0) onverges as well, and for those for whi h ILU(0) does not onverge, RNF(0,0)doesnot onvergeeither. Theseobservationssuggestthat RNF(0,0)alone anrepla eILU(0)for ertainproblems;theadvantageofdoing thisisthat forRNF(0,0),no ostof onstru tionisrequired,and thereare no storagerequirements(see Table2). On the other hand, omparing RNF(0,0) andRNF(1,0)(seetable3andtable4),wendthatRNF(1,0) onvergesfaster when omparedtoRNF(0,0). So,RNF(1,0) analsorepla eILU(0)forthetest ases onsidered.

Table1: Shortnotationsforthemethodsandthetest ases onsidered. METHODS STANDS FOR

ILU In ompleteLUwithzeroll-in MILU ModiedILUwith olsum onstraint

TFILU Combinativepre onditioningwithLFTFDandILU RNF(

α

,

β

) Relaxednestedfa torizationwithparameters

α

and

β

. TFRNF(

α

,

β

) Combinativepre onditioningwithRNF(

α

,

β

)andLFTFD TF(r,l)RNF(1,1) Combinativepre onditioningwithRNF(1,1)and

LFTFDwithlterve torasthelargestRitzve tor TF(r,s)RNF(0,0) Combinativepre onditioningwithRNF(0,0)and

LFTFDwithlterve torasthesmallestRitzve tor MATRICES STANDS FOR

2DNHm 2-dimensionalnon-homogeneousproblemofsize mbym 2DADm 2-dimensionaladve tiondiusionproblemofsize mbym. 2DSKYm 2-dimensionalskys rapperproblemofsizembym. 2DCONSKYm 2-dimensional onve tiveskys rapperofsize mbym. 3DSKYm 3-dimensionalskys rapperproblem ofsizembym bym. 3DCONSKYm 3-dimensional onve tiveskys rapperofsizembymbym. 3DANIm 3-dimensionalanisotropi problemofsizem bymbym.

Table2: Cost of onstru tion, ost of applying, and thestorage requirements forseveralmethods(nnz(K) =no. ofnon-zerosin matrixK).

Method Constru tion Toapply Storage RNF(

α

,

β

)

≈ 720 nx

7

_{≈ 18 nx}

3

_nx

3

RNF(1,0)

≈ 4nx

3

_{≈ 18 nx}

3

_nx

3

RNF(0,0)

0

≈ 18 nx

3

Null LFTFD

≈

2

3

nx

7

_≈

2

3

nx

7

nnz(

A

)-2nnz(

L

3

) ILU(0)

≈ 7nx

3

_{≈ 14nx}

3

nnz(

A

)

Tostudythespe trumofRNF(

α

,

α

)pre onditionedmatrixfordierent val-uesoftheparameter

α

,wedisplaythespe trumfor

30×30

skys rapperproblem in theFigure(1). Thespe trum ofTFRNF(0,0) pre onditionedmatrixis dis-playedintheFigure(2). Asweobserveinthegure,theparameterhasinuen e onthespe trum;with

α = 0

,theeigenvaluesofRNF(0,0)liesbetweenzeroand one,whi hisfavorableforthemultipli ativepre onditioningasseeninse tion 5.

Forafair omparison,intables3and4,thenumberofiterations orrespond-ingto multipli ative ombinationaredoubled. Ontheotherhand,thenumber ofiterationsfortheadditive asearethea tualnumberofiterations,keepingin mindthat ea hof thepre onditionersolves anbeperformedon twodierent pro essorsonparallelar hite tures.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−1

0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−1

0

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−1

0

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−1

0

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−1

0

1

0

500

1000

1500

2000

2500

−2

0

2

x 10

−12

Figure1: Spe trumplotforRNF(

α

,

α

)pre onditioned2D30

×

30skys rapper problem: (toptobottom)

α

=0,0.2,0.4,0.6,0.8, and1

Comparingthe ombinative pre onditioning TFILU and TFRNF(1,

α

), we ndthat the ombinationTFRNF(1,

α

)performsmu hbetter omparedtothe ombination TFILU or other lassi al methods (see table 3 and 4) when pa-rameter is hosen lose to

1

. We noti e that the signi ant number of steps an be saved for two-dimensional problems, but for three-dimensional prob-lems dependen e on the parameter is almost negligible. This dependen e of iterationnumberontheparameterforea htypeofmatri esareplottedin Fig-ure (4) for two-dimensional ase and in Figure (5) for three-dimensional ase

0

1

2

3

4

5

6

7

x 10

4

−50

0

50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.05

0

0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−0.02

0

0.02

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

−0.02

0

0.02

Figure 2: Spe trum plot for 2D 60

×

60 skys rapper problem: (top to bot-tom) Spe trum of A followed by spe trum of RNF(0,0), LFTFD, and RNF mult.LFTFD pre onditionedmatrix

forRNF(

1

,

α

),RNF(

α

,

α

), forbothadditiveandmultipli ative ombinationsof TFRNF(1,

α

),andforadditiveandmultipli ative ombinationofTFRNF(

α

,

α

). In these plots, the a tualnumber of iterationsare presented. From these g-uresweobservethat for three-dimensionalproblems,the ombinations involv-ingRNF(0,0)andRNF(1,0)remaingoodparameterindependent hoi esasnot mu hisgainedwithanyvalueofparameters. Betweenthe ombinations involv-ing RNF(0,0) and RNF(1,0), we observe that TFRNF(1,0) performs slightly betterthanTFRNF(0,0). OntheotherhandinTFRNF(0,0),RNF(0,0)hasno ostof onstru tionandhasnostoragerequirement.

As mentioned before, there are two hoi es of ombinative pre ondition-ing, rst is amultipli ativeapproa h, and se ond is an additiveapproa h. In termsoftotalnumberofiterations,themultipli ativeapproa hperformsbetter ompared to additive approa h. On parallel ar hite ture, ea h pre onditioner solvefortheadditive ase anbedoneontwodierentpro essors,whi hisan advantageoverthemultipli ative ase.

InFigure(3),wepresent onvergen e urvesforRNF(1,1),TFILU,TFRNF(0,0), ILU(0), and MILUforonematrixea hfrom the test ases. Fromthese plots, weobserve that the onvergen e behavior of the ombination TFRNF(0,0) is

idewithea hother;thereasonforsu hsimilaritymaybeduetothefa tthat RNF(0,0)in somesense behaveasILU(0) foralltheproblems onsidered(see Table3).

Othermeaningfulattemptinvolves ombiningRNF(1,1)withLFTFDwhere thelterve tor hosenisthelargestRitzve tor. Inourtest asesweobserve that the reasonfor thefailure of RNF(1,1) is thepresen e of someverylarge eigenvaluesin thespe trum of the pre onditioned matrix, see Figure (1). To deate one of these largest eigenvalues, we hose the lter ve tor to be the largestRitzve torobtainedafter20stepsofArnoldiiteration[6℄withRNF(1,1) pre onditioned matrix. We observethat in Table (4), where the total ost is shown, this ombination is not robust enough, the reason beingthe presen e ofmany otherlarge eigenvaluesin theRNF(1,1)pre onditioned matrix whi h ause di ulties in the pre onditioned GMRES method. On the other hand RNF(0,0)pre onditionedmatrixhasfewsmalleigenvalues,seeFigure(2). For thisreasonwe hoosethelterve tor orrespondingto thesmallestRitzvalue inmagnitude. This ombination,i.e.,TF(r,s)RNF(0,0)isnotrobusteither,and thereasonbeingthepresen eofothersmalleigenvaluesinthespe trumofthe pre onditionedmatrix.

However, in our tests we found that for the ombinative pre onditioning, the hoi eofthelterve torasve torofallonesis quiterobust. Moreover,it eliminatesthe ostofformingthelterve tor.

0

50

100

150

200

250

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 2dNH200200

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

(a) 2D

200

× 200

non-homogeneous problem

0

50

100

150

200

250

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 2dAD200200

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

(b) 2D

200

× 200

adve tion diusion problem

0

50

100

150

200

250

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 2dCONSKY200200

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

( ) 2D

200

×200

onve tiveskys rapper problem

0

50

100

150

200

250

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 3dSKY303030

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

(d) 3D

30

×30×30

skys rapperproblem

0

50

100

150

200

250

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 3dCONSKY303030

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

(e) 3D

30

×30×30

onve tive skys rap-perproblem

0

10

20

30

40

50

60

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Convergence curves for the matrix 3dANI303030

Number of iterations

Norm of relative residual

RNF(1,1)

ILU(0)+LFTFD

RNF(0,0)+LFTFD

ILU(0)

MILU

(f) 3D

30

× 30× 30

anisotropi problem

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of alpha Vs no. of iters. for 2dNH200200

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

(a) 2D

200

× 200

non-homogeneous problem.

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of Alpha Vs No. of iters. for 2dNH200200

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(b) 2D

200

× 200

non-homogeneous problem.

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of itertions

Plot of alpha Vs no. of iters. for 2dAD200200

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

( ) 2D

200

× 200

adve tion-diusion problem.

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of Alpha Vs No. of iters. for 2dAD200200

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(d) 2D

200

× 200

adve tion-diusion problem.

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of alpha Vs no. of iters. for 2dSKY200200

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

(e) 2D

200

× 200

skys rapperproblem.

0

0.2

0.4

0.6

0.8

1

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of alpha Vs no. of iters. for 2dSKY200200

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(f) 2D

200

×200

skys rapperproblem.

0

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dSKY202020

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

(a)3D

20

×20×20

skys rapperproblem.

0

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

120

140

160

180

200

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dSKY202020

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(b)3D

20

× 20 × 20

skys rapper prob-lem.

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70

80

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dANI202020

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

( )3D

20

× 20× 20

anisotropi problem.

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

70

80

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dANI202020

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(d)3D

20

× 20 × 20

anisotropi prob-lem.

0

0.2

0.4

0.6

0.8

1

10

20

30

40

50

60

70

80

90

100

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dCONSKY202020

RNF(1,alpha) mult. LFTFD

RNF(1,alpha) add. LFTFD

RNF(1,alpha)

(e)3D

20

× 20 × 20

onve tive skys rap-perproblem.

0

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

120

alpha

number of iterations

Plot of alpha Vs no. of iters. for 3dCONSKY202020

RNF(alpha,alpha) mult. LFTFD

RNF(alpha,alpha) add. LFTFD

RNF(alpha,alpha)

(f)3D

20

×20×20

onve tive skys rap-perproblem.

Pawan Kumar , L aur a Grigori , F r e deri Nataf , Qiang Niu sol.)

Meth./Mat. ILU MILU TFILU RNF(0,0) TFRNF(0,0) RNF(1,1)

Mult. Add. Mult. Add.

2dNH100 - 72(e-12) 56(e-11) 39(e-11) - 52(e-11) 38(e-11) 37(e-11) 2dNH200 - 109(e-11) 78(e-10) 53(e-10) - 72(e-11) 53(e-11) 54(e-11) 2dNH300 - 141(e-11) 96(e-10) 66(e-10) - 88(e-10) 65(e-10) 67(e-10) 2dNH400 - 169(e-11) 110(e-10) 75(e-10) - 104(e-10) 74(e-10) 78(e-10) 2dAD100 - 65(e-11) 56(e-11) 39(e-11) - 52(e-11) 38(e-11) 36(e-11) 2dAD200 - 95(e-11) 80(e-10) 54(e-10) - 74(e-10) 53(e-10) 53(e-11) 2dAD300 - 121(e-11) 96(e-10) 67(e-10) - 88(e-10) 65(e-10) 67(e-11) 2dAD400 - 146(e-11) 110(e-10) 75(e-10) - 104(e-10) 74(e-10) 79(e-11) 2dS100 - - 56(e-08) 36(e-08) - 50(e-07) 37(e-08) -2dS200 - - 82(e-07) 54(e-07) - 74(e-07) 55(e-07)

-2dS300 - - 100(e-06) 68(e-07) - 96(e-06) 69(e-06) 110(e-09) 2dS400 - - 124(e-06) 82(e-06) - 116(e-06) 86(e-06)

-2dCS100 - - 50(e-09) 39(e-09) - 50(e-09) 41(e-09) -2dCS200 - - 66(e-08) 59(e-09) - 76(e-08) 67(e-08) -2dCS300 - 92(e-10) 78(e-08) 73(e-08) - 98(e-08) 80(e-08) 63(e-08) 2dCS400 - - 120(e-08) 101(e-07) - 138(e-08) 120(e-08) -3dCS15 113(e-11) 99(e-12) 22(e-11) 28(e-11) 92(e-10) 34(e-11) 30(e-10) 47(e-11) 3dCS20 88(e-10) 77(e-11) 20(e-10) 23(e-10) 122(e-10) 32(e-10) 29(e-10) 15(e-11) 3dCS30 169(e-09) - 32(e-10) 27(e-10) 162(e-10) 32(e-10) 29(e-10) 138(e-10) 3dCS40 169(e-09) - 28(e-10) 25(e-10) 191(e-09) 34(e-10) 30(e-10) 24(e-10) 3dS20 196(e-07) - 24(e-09) 20(e-09) - 24(e-09) 20(e-09) 16(e-09) 3dS30 - - 30(e-08) 22(e-08) - 26(e-08) 22(e-08) -3dS40 - - 32(e-08) 23(e-08) - 28(e-08) 23(e-08) 26(e-08) 3dANI20 29(e-08) 43(e-08) 22(e-09) 15(e-08) 72(e-08) 26(e-08) 21(e-08) 20(e-08) 3dANI30 51(e-07) 58(e-08) 24(e-08) 17(e-08) 104(e-07) 28(e-08) 23(e-08) 23(e-08) 3dANI40 55(e-07) 72(e-08) 26(e-08) 18(e-07) 132(e-07) 30(e-08) 23(e-08) 23(e-08)

Combinative pr e onditioning b ase d on RNF and T angential Filtering Meth./Mat. RNF(1,0) TFRNF(1,0) TFRNF(.9,.9) TFRNF(1,0.9) TF(r,l)RNF(1,1) TF(r,s)RNF(0,0)

Mult. Add. Mult. Add. Mult. Add. Mult. Mult.

2dNH100 168(e-10) 48(e-11) 35(e-11) 36(e-11) 24(e-11) 34(e-11) 21(e-10) 52(e-11) 52(e-11) 2dNH200 - 70(e-11) 48(e-10) 50(e-10) 34(e-11) 48(e-10) 29(e-10) 78(e-11) -2dNH300 - 84(e-10) 59(e-10) 62(e-10) 40(e-10) 58(e-10) 36(e-10) 104(e-10) -2dNH400 - 96(e-10) 69(e-10) 72(e-10) 46(e-10) 70(e-10) 42(e-10) 122(e-10) -2dAD100 168(e-10) 48(e-11) 35(e-11) 36(e-11) 24(e-11) 32(e-11) 21(e-10) 52(e-11) 52(e-11) 2dAD200 - 70(e-11) 49(e-10) 50(e-10) 33(e-10) 48(e-10) 30(e-10) 78(e-11) 394(e-09) 2dAD300 - 84(e-10) 59(e-10) 64(e-10) 40(e-10) 60(e-10) 37(e-10) 100(e-10)

-2dAD400 - 96(e-10) 67(e-10) 70(e-10) 45(e-10) 70(e-10) 42(e-10) 128(e-10) -2dS100 - 44(e-07) 33(e-08) 42(e-07) 22(e-07) 42(e-07) 23(e-07) - -2dS200 - 68(e-07) 51(e-07) 64(e-07) 32(e-07) 60(e-07) 30(e-07) - -2dS300 - 84(e-06) 60(e-07) 58(e-06) 38(e-06) 56(e-07) 34(e-06) 124(e-09) -2dS400 - 102(e-06) 72(e-06) 94(e-06) 47(e-06) 92(e-06) 46(e-06) -

-2dCS100 - 46(e-09) 39(e-09) 38(e-09) 31(e-09) 38(e-09) 32(e-09) - 390(e-09) 2dCS200 - 70(e-08) 57(e-08) 52(e-09) 39(e-08) 52(e-08) 39(e-09) -

-2dCS300 - 80(e-08) 73(e-08) 56(e-08) 43(e-08) 48(e-08) 39(e-08) 108(e-09) -2dCS400 - 124(e-07) 109(e-07) 74(e-08) 59(e-08) 72(e-08) 58(e-08) -

-3dCS15 92(e-10) 30(e-11) 28(e-10) 44(e-10) 36(e-11) 48(e-11) 37(e-11) 76(e-11) 154(e-10) 3dCS20 97(e-10) 30(e-11) 27(e-10) 28(e-10) 27(e-11) 28(e-10) 26(e-11) 30(e-11) 302(e-10) 3dCS30 168(e-09) 28(e-10) 26(e-10) 44(e-10) 29(e-10) 44(e-10) 29(e-10) - 120(e-09) 3dCS40 - 28(e-10) 27(e-10) 26(e-10) 24(e-09) 24(e-09) 24(e-10) 78(e-10) 170(e-09) 3dS20 - 24(e-09) 20(e-09) 26(e-09) 20(e-09) 26(e-09) 20(e-09) 28(e-09) 278(e-08) 3dS30 - 26(e-08) 22(e-08) 50(e-09) 25(e-09) 52(e-09) 26(e-08) - 188(e-08) 3dS40 - 28(e-08) 22(e-08) 30(e-08) 21(e-08) 32(e-09) 21(e-08) 46(e-09)

-3dANI20 57(e-08) 26(e-09) 19(e-08) 18(e-08) 14(e-08) 18(e-08) 13(e-08) 28(e-09) 84(e-09) 3dANI30 75(e-07) 28(e-08) 21(e-08) 20(e-08) 15(e-08) 20(e-08) 14(e-08) 30(e-08) 72(e-06) 3dANI40 106(e-07) 28(e-08) 22(e-08) 20(e-07) 15(e-07) 20(e-08) 14(e-08) 32(e-08) 84(e-08)

RR

n

°

7 CONCLUSIONANDFUTUREDIRECTIONS Wehaveintrodu edarelaxedversionofNF,andanee tive ombinative pre- onditioningisa hievedin ombinationwithtangentiallteringpre onditioners [4℄. Compared to ILU(0), it hasseveral important properties. If

A

is S.P.D. thenRNF(0,0)is S.P.D.and leadstoastable splitting. There is nosetup ost nor storage requirement. The new pre onditioner is parti ularly useful when multipli ate/additivepre onditioningis a hievedin ombinationwith tangen-tial lteringpre onditioner [4℄. With this setup the newpre onditioner is as robustasthe ombinationofILU(0)withtangentiallteringpre onditioner.

A futurework onsistsin studyingothermodiedversionofNF, similarto onesuggestedbyGustason[13℄,[14℄forMILU,wherethediagonalisperturbed byanadditivetermoforder

O(h

2

₎

,whi hseemstoimprovethe onditioningof modiedin ompletefa torizationmethodsfor ertain lassofproblems.

Referen es

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[2℄ J. R. Appleyard, Proof that simple Colsum modied ILU fa toriza-tion of matri es arising in Fluid folw problems are not singular, by John Appleyard, Polyhedron Software Ltd, May 2004. Available online : http://www.polyhedron. om/ olsum-proof.

[3℄ J.R.AppleyardandI.M.Cheshire,Stri tlyTriangular L.Uformulationof NestedFa torization, by JohnAppleyard, PolyhedronSoftwareLtd, May 2004.Availableonline: http://www.polyhedron. om/stri tlu.

[4℄ Y. A hdouandF.Nataf,Low frequen y tangentialltering de omposition, Numer.LinearAlgebraAppl.,14,(2007),pp.129-147.

[5℄ S. F. Ashby, M. J. Holst,T. A. Manteuel, and P. E. SaylorThe role of inner produ t in stopping riteria for onjugate gradient iterations, BIT., 41, (2001),pp.26-52.

[6℄ Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst Tem-platesfortheSolutionofAlgebrai EigenvalueProblems: APra ti alGuide ,SIAM,Philadelphia,2000.

[7℄ M.Benzi,A.M.Tuma,A omparativestudyofsparseapproximateinverse pre onditioners,Appl.Numer.Math.,30,(1999),pp.305-340.

[8℄ D.BraessTowardsalgebrai multigridfor ellipti problems ofse ondorder, Computing,55(4), (1995),pp.379-393.

[9℄ A.Buzdin,Tangentialde omposition,Computing.,61,(1998),pp.257-276. [10℄ A. Buzdin and G. Wittum, Two-frequen y de omposition, Numer. Math.,

[11℄ C.Wagner andG. Wittum,Adaptive ltering,Numer.Math.,78, (2004), pp.305-328.

[12℄ P. A. Crumpton, P. A. Fjerstad, and J. Berge, Parallel omputing Using ECLIPSE Parallel S hlumberger, E lipse, Te hni al Des ription 2003A, Chapter38: ParallelOption.2003.

[13℄ GustafssonI,A lass ofrstorderfa torizationmethods,BIT18,pp. 142-56,1978.

[14℄ GustafssonI,Modiedin omplete holesky(MIC)methods,InD.Evans, ed-itor,Pre onditioningMethods-TheoryandAppli ations,pp.265-293. Gor-donandBrea h,NewYork,1983.

[15℄ N. N. Kuznetsova,O. V. Diyankov, S. S. Kotegov, S. V. Koshelev, I.V. Krasnogorov,V.Y.Pravilnikov,andS.Y.Maliassov,Thefamilyof nested fa torizations, Russian Journal of Numeri al Analysis and Mathemati al Modelling.22(4)(2007),pp.393–412.

[16℄ G. Meurant Computer Solution of Large Linear Systems North-Holland PublishingCo. Amsterdam,1999.

[17℄ J. W. Ruge, K. Stüben Algebrai multigrid. Multigrid Methods, Frontiers ofAppliedMathemati s,vol.3.SIAMPhiladelphia,PA, 1987;73–130. [18℄ Y. Saad, Iterative Methods for Sparse Linear Systems, PWS publishing

ompany,Boston,MA,1996.

[19℄ K. Stüben, A review of algebrai multigrid, J. Comput. Appl. Math. and applied mathemati s, 128(1-2), (2001), pp. 281-309, Numeri al analysis 2000,vol.VII,Partialdierentialequations.

[20℄ U. Trottenberg, C. W. Oosterlee, and A. S hüller, Multigrid, A ademi PressIn .SanDiego,CA,(2001).With ontributionsbyBrandtA,Oswald P,andStübenK.

[21℄ C. Wagner, Tangential frequen y ltering de ompositions for symmetri matri es,Numer.Math.,78,(1997),pp.119-142.

[22℄ C.Wagner,Tangentialfrequen yltering de ompositions for unsymmetri matri es Numer.Math.,78,(1997),pp.143-163.

[23℄ C.Wagner andG. Wittum,Adaptive ltering,Numer.Math.,78, (1997), pp.305-382.

[24℄ P. Van¥k, M. Brezina, and J. Mandel Convergen e of algebrai multigrid basedonsmoothedaggregation,Numeris heMathematik,88(3),(2001),pp. 559-579.

[25℄ L.Grigori,F.Nataf,andQ.Niu,Twosides tangentialltering de omposi-tion,INRIATR6554.

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