An efficient truncated SVD of large matrices based on the low-rank approximation for inverse geophysical problems

HAL Id: hal-01414769

https://hal.archives-ouvertes.fr/hal-01414769

Submitted on 12 Dec 2016

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

the low-rank approximation for inverse geophysical

problems

Sergey Solovyev, Sébastien Tordeux

To cite this version:

Sergey Solovyev, Sébastien Tordeux. An efficient truncated SVD of large matrices based on the

low-rank approximation for inverse geophysical problems . Sibirskie Elektronnye Matematičeskie Izvestiâ,

Russian Academy of Sciences, 2015, 12, pp.592-609. �10.17377/semi.2015.12.048�. �hal-01414769�

Math-Net.Ru

All Russian mathematical portal

S. A. Solovyev, S. Tordeux, An efficient truncated SVD of large matrices based on

the low-rank approximation for inverse geophysical problems, Sib. `

Elektron. Mat.

Izv.

_{, 2015, Volume 12, 592–609}

DOI: http://dx.doi.org/10.17377/semi.2015.12.048

Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms

of use

http://www.mathnet.ru/eng/agreement

Download details:

IP: 77.141.166.117

ÑÈÁÈÑÊÈÅ ÝËÅÊÒÎÍÍÛÅ

ÌÀÒÅÌÀÒÈ×ÅÑÊÈÅ ÈÇÂÅÑÒÈß

Siberian Ele troni Mathemati alReports

http://semr.math.ns .ru

Òîì12,ñòð.592609(2015) ÓÄÊ519.61

DOI10.17377/semi.2015.12. 04 8 MSC 65F15

AN EFFICIENT TRUNCATED SVD OF LARGE MATRICES

BASED ON THE LOW-RANK APPROXIMATION FOR INVERSE GEOPHYSICAL PROBLEMS

S.A.SOLOVYEVANDS.TORDEUX

Abstra t. In this paper, we propose a new algorithm to ompute atrun ated singularvalue de omposition (T-SVD)of theBorn matrix basedonalow-rankarithmeti .Thisalgorithmis testedinthe ontext ofa ousti media.Theoreti alba kgroundtothelow-rankSVDmethod ispresented:theBornmatrixofana ousti problem anbe approxima-tedbyalow-rankapproximationderivedthankstoakernelindependent multipoleexpansion.Thenewalgorithmto omputeT-SVD approxima-tion onsistsoffoursteps,andtheyaredes ribedindetail.Thelargest singularvaluesandtheirleftandrightsingularve tors anbe approxima-tednumeri allywithoutperforminganyoperationwiththefull matrix. Thelow-rankapproximationis omputedduetoadynami panelstrategy of rossapproximation(CA)te hnique.

Attheendofthepaper,wepresentanumeri alexperimenttoillustrate thee ien yandpre isionofthealgorithmproposed.

Keywords: Born matrix, SVD algorithm, ross approximation (CA), low-rankapproximation,high-performan e omputing,parallel omputa-tions.

1. Introdu tion

Computingtheexa tSVDortheT-SVDinfull-arithmeti isaprettyexpensive task. A standard way to ompute the SVD onsists of the two steps: rst, the matrixis redu edto abidiagonalform,then theSVDof thebidiagonal matrixis

SolovyevS.A.,TordeuxS.,Aneffi ientTrun atedSVDoflargematri esbased onthelow-rankapproximationforinversegeophysi alproblems.

2015SolovyevS.A.,TordeuxS.

TheworkissupportedbyRFBR(grants14-01-31340,14-05-31222,14-05-00049,14-05-93090, 15-29-06922).

omputed. Usually, therst stepisdoneby theHouseholder ree tionsand osts

(4/3)n

2

_(3m−n)

arithmeti operations(FLOPS).These ondstep anbeperformed by an iterative renement algorithm with stoping riteron equal to the ma hine pre ision, it takes about

O(n)

iterations, ea h osts

O(n)

FLOPS [3℄. Another versionof the se ond step is a QR algorithm for the omputation of eigenvalues and takes

O(n

2

₎

FLOPS [4℄,[10℄. There are various modi ations of them, using adivide-and- onquermethod, pre onditioned andJa obiplane rotationmethods. TheseonesareimplementedinLAPACKroutines.To omparewithowndeveloped algorithmweusetheLAPACKfun tionalityfromIntelMKL.However,itisalmost impossibleto omputesu hade ompositionforlarge-s aleproblemsusingarobust arithmeti .

The obje tive of this paper is designing an e ient algorithm to ompute an approximationoftheT-SVDbasedonalow-rankarithmeti .

Morepre isely,in thispaperweaimatlookingformatri es

U

k

= {¯u

1

, . . . ¯

u

k

} ∈

C

I×k

,

V

k

= {¯v

1

, . . . ¯

v

k

} ∈ C

J×k

and

D

k

= diag{d

i

}

k

i=1

∈ R

k×k

whi h approximate

U

k

, V

k

, D

k

oftheT-SVD(28)inthefollowingsense:

I Thedieren ebetweenapproximatesingularvaluesandexa tsingularvalues issmallerthanasmallparameter

η

1

(1)

d

i

− ¯

d

i

< η

1

,

1 < i ≤ k

and

d

i

< δ, k < i ≤ n

II Theanglesbetweentheapproximatedandexa tleftandrightsingularspa es aresmallerthanasmallparameter

η

2

(2)

∠(U

k

, U

k

) < η

2

and

∠(V

k

, V

k

) < η

2

where

∠(A, B) = arccos(σ)

withthesmallestsingularvalue

σ

of

A

∗

_B

.

The major appli ation of the algorithm proposed is the geophysi al inverse problem. It onsists in determining the physi al hara teristi s of a propagation medium byinterpreting themeasureddata by re eiversfor dierentsour es.One of themain features of thisproblem is thehuge numberof re eivers,sour esand parametersofthemodelwhi hisunderstudy.Amongalldierentte hniques that exist,theFullWaveformInversion(FWI)isoneofthemost ostly.It onsistsinan interativeNewton-typepro edure whi h requires the omputation oftheso- alled BornMatrixasso iatedwiththefullwaveequation.

Itisknownthatbe auseofitspoor onditioningthisverylarges aleproblemis di ulttosolve:some( ombinations)ofthesour es,re eiversormodelparameters are very important to take into a ount whereas some other ( ombinations) do not have so strong impa t. The numeri al method proposed in this paper needs omputingtheSingularValueDe omposition [14,7℄ oftheBornmatrixto redu e the omplexityofthisproblembyidentifyingthemostimportantmodelparameters, sour esandre eivers.

This paper will is omposed as follows: In Se tion 2, we briey re all what is a low-rank approximation of a matrix and some algorithms to ompute these approximations.Se tion 3is the" ore"ofthis paper.Wedes ribeanalgorithm to omputeanumeri alapproximationoftheTrun atedSingularValuede omposition basedonalow-rankarithmeti .InSe tion4,somenumeri alresultsarepresented. Theseresultsillustratethee ien yanda ura yofthemethod. Therearethree appendi es:inAppendixA,wedes ribeindetailtheSingularValueDe omposition andtheTrun atedSingularValueDe omposition.AppendixBis on ernedwitha theoreti alresultabouttheBorna ousti matrix.Were all thattheBornmatrix

asso iated with homogeneousa ousti media an beapproximatedbya low-rank approximationundersuitableassumptions.Finally,AppendixCbrieyre alls lassi- alresultsabouttheChebyshevtensorialinterpolation.

2. The low rank approximationof amatrix

Inwhatfollows,wewillmakeanintensiveuseofthelow-rankapproximation.

2.1. Denitionofthe

ε

-rankofamatrixandlow-rankapproximation. Let

A ∈ C

I×J

beamatrixwith

I

rowsand

J

olumns with

J ≤ I

. Thematrix

A

has

ε

-rank

k

if

k

isthesmallestintegersu hthatthereexistsamatrix

A

k

∈ C

I×J

with therank

k

satisfying

(3)

kA − A

k

k

kAk

< ε.

When

k · k

istheeu lideanmatrixnorm

k · k

2

,the

ε

-rankofamatrix

A

isexpli itly givenbythenumberofsingularvalueswhi harelargerthan

ε

.Moreover,thematrix

A

k

an bededu edfrom theT-SVDofthematrix

A

(4)

A

k

= U

k

D

k

V

∗

k

with

U

k

∈ C

I×k

_{, D}

k

∈ R

k×k

+

and

V

k

∈ C

J×k

_.

Inreality,the omputation oftheT-SVDof alarge-sizematrixisveryexpensive, many authors have proposed other algorithms to ompute non-optimal (in the senseofthematrixeu lideannorm)low-rankapproximationofamatrix

A

.These methods onsist in looking for a matrix

A

k

as a produ t of the two matri es

B

k

∈ C

I×k

and

C

k

∈ C

J×k

,whi hminimize

k

,satisfying (5)

kA − B

k

C

_k

T

k

kAk

< ε.

The norm

kAk

2

is given by the largest singular value of the matrix

A

and its omputation is pretty expensive. So, we prefer to use

kAk = max

i≤I,j≤J

|A

i,j

|

norm.

Thetwo ommonte hniques toobtainthis fa torizationare theQRfa torization withpivotingofthematrixandtheCrossApproximation(CA) te hniquewhi his similar to the in omplete LUfa torization with pivoting. To determine

ε

-rankof amatrixand obtainlow-rankapproximation,therank-revealingQRmodi ation with pivoting is used (RRQR-piv) [8℄. The approximate numberof oating-point operationsforrealavorsis

(2/3)n

2

_{(3m − n), m ≥ n}

.TheRRQR-pivalgorithmis slow(howeverfasterthanthe omputationoftheT-SVDofamatrix

A

)algorithm andis almost optimalin termsoftherank

k

,whereastheCA algorithmis rapid, but an giverisetonon-optimal

k

.

Letsusbriey des ribetheRRQR-pivalgorithm anddes ribein detailvarious modi ationsoftheCAapproa h.

Crossapproximation.The rossapproximationalgorithm[11℄takesthe follow-ingform.

•

Initialization:

(6)

R

0

= A ∈ C

I×J

_,

_{n = 0}

•

Whilestopping- riteria

(R

n

) > εkAk

Step 1. Chooseapivot

(i

⋆

, j

⋆

)

in

R

n

Step 2. Denetwove tor olumns

b

n+1

∈ C

I

and

c

n+1

∈ C

J

(7)

(b

n+1

)

i

= (R

n

)

i,j

⋆

and

(c

n+1

)

j

=

(R

n

)

i

⋆

,j

(R

n

)

i⋆,j⋆

Step 3. In rement

n

:

n = n+1

.Denethematri es

B

n

∈ C

I

×n

and

C

n

∈ C

J×n

and (8)

B

n

= [b

1

, b

2

, · · · , b

n

]

and

C

n

= [c

1

, c

2

, · · · , c

n

]

Step 3. Update (9)

R

n

= A − B

n

C

T

n

.

Asaresult,weobtainbothmatri es

B

k

and

C

k

thatareinvolvedin thelow-rank approximationof

A

.

Likeforthein ompleteLUfa torizationalgorithm,wehavethefollowingoptions forthisalgorithm:

I Totalpivoting:Thestopping riterionisthefollowing:

(10) stopping- riterion

(R

n

) = max

i≤I,j≤J

(|(R

n

)

i,j

|)

Atea hiteration, thepivot is hosenbymaximizing

|(R

n

)

i,j

|

overwhole the matrix

(11)

|(R

n

)

i⋆,j⋆

| = max

i≤I, j≤J

|(R

n

)

i,j

|

II Dynami panelstrategy: The algorithm is more omplex to des ribe. This orresponds to a partial pivoting. First, as for the total pivoting we dene

i

_

and

j



su hthat

(12)

|(R

n

)

i



,j



| = max

_{i≤I, j≤J}

|(R

n

)

i,j

|

Wethendeneapanel

J



⊂ [1, J]

ofwidth

2K + 1 ∈ [1, J]

: (13)

J



=















[1, 2K + 1]

if

j



≤ K,

[j

_

− K, j



+ K]

if

K < j



≤ J − K,

[J − 2K, J]

if

j



> J − K.

Aslongas amaximumof

|R

n

|

overthis panelislargerthan

ε

(14)

max

i≤I, j∈J

_

|(R

n

)

i,j

| > ε,

thepivot(

i

⋆

, j

⋆

)willbe hoseninto thispanelof olumns

(15)

|(R

n

)

i⋆,j⋆

| =

max

i≤I, j∈J

_

|(R

n

)

i,j

|.

When (14)is notfullled anymore,another panel is onsidered in the same wayuntil

(16)

max

i≤I, j≤J

|(R

n

)

i,j

| < ε.

III Crosspivoting: The pivot is hosen in the following way: Pi k by hazard a non-zero olumn

(R

n

)

·,j△

of

R

n

, with

j

△

∈ J

. Dene the integer

i

⋆

∈ I

by lookingforamaximumof

|R

n

|

inthis olumn

(17)

|(R

n

)

i⋆,j△

| = max

Denetheinteger

j

⋆

∈ J

bylookingforamaximumof

|R

n

|

in thisrow

(18)

|(R

n

)

i

⋆

,j

⋆

| = max

j∈J

|(R

n

)

i

⋆

,j

|

Thestopping riterionisthenthefollowing:

(19)

|(R

n

)

i

⋆

,j

⋆

| > ε

Remark 1. The sear h for a maximum and the update of the matrix

R

n

is the performan e of the bottle-ne k of the CAalgorithm. The total pivoting strategy is mu hslower thanthe two other strategies be auseof

•

the sear h for the maximumismade overthefullmatrix.

•

atea h iterationthe matrixshouldbefully updated

Thedynami alpanelstrategyismoree ientsin e

•

thesear hforthemaximumismadeoverasmallsubsetofthefullmatrix.

•

onlythepanelofthematrixneedsto beupdatedatea hiteration

Sin e the panel is formed of agroup of olumns, it is important to optimize the a esstomemoryinstoringthematrixintheRAM olumn-by- olumn.Ifamatrix isstoredrows-by-rows,thepanelshould be onstru tedofagroupofrows.

The rosspartial pivotingstrategyisalso verye ientsin e

•

thesear hforthemaximumismadeovera rosswhi hisasmallsubsetof thefullmatrix.

•

the matrix

R

n

does not need not to be updated but only needs to be evaluatedforasmallnumberofindi esatea hiteration.

3. DESCRIPTIONOF THE ALGORITHM

Thealgorithm anbede omposedintofoursteps.

The rst step onsists in de omposing verti ally the matrix A into blo ks

A

i

,

A=

A

6

A

5

A

4

A

3

A

2

A

1

(Figure1),andinperformingalow-rankapproximationofea hblo k

A

i

∈ C

m

i×J

, (Figure2).

A

i

≃ B

i

C

i

T

with

B

i

∈ C

m

i×ki

and

C

i

∈ C

J×ki

To omputealow-rankapproximationofblo kswehavethethreeoptions: i)T-SVDinthefullarithmeti ;

ii)RRQR-pivalgorithm;

iii)CrossApproximation(CA) te hnique.

Remark2. The algorithmwill onlybee ientif the integer

k

i

islessthan

J

. In pra ti e, this numberissmall.

Remark3. Thea ura yofthelow-rankapproximationofthematrix

A

is hara te-rizedby asmallparameter

ε

andby the stopping riterion.ForSVD- ompression, QR-pivandCAalgorithm, ittakes theform

(20)







kA

i

− B

i

C

i

T

k

2

≤ ε

for T-SVD andRRQR algorithm,

kA

i

− B

i

C

i

T

k

∞

≤ ε

for CAalgorithm, with

k · k

2

the eu lideanmatrixnormand

kAk

∞

= max

i≤I,j≤J

|A

i,j

|

.

A

i

=

B

i

C

i

T

Fig.2. Low-rankapproximationof

A

Theresultoftherststepisdepi tedinFigure3.Inthispi tureandinthenext ones,the"plotted"partsofmatri esmeandensenon-zeroblo ks.The"white"blo ks meanzeroll-in.

At these ond step, weorthogonalizethe matri es

B

and

C

. More pre isely, weperformaQRde ompositionofthematri es

B

i

and

C

B

i

= e

B

i

R

i

and C

T

= L e

C

T

,

with

B

e

i

∈ R

m

i×ki

,

C ∈ R

e

J×k

beingorthogonal,

R

i

∈ R

k

i×ki

theuppertriangular and

L ∈ R

k×k

 the lowertriangular, where

k =

P

k

i

. The matri es

B

e

i

and

R

i

are olle tedintotheorthogonalmatrix

B

e

andintotheuppertriangularmatrix

R

, (seeFigure4).This resultin thattheLow-rankapproximationof

A

should be (21)

A ≃ e

B (R L) e

C

T

_,

with

B ∈ R

e

I×k

,

C ∈ R

e

J×k

and

RL ∈ R

k×k

beingfull.

Atthethird step,arobustT-SVD withthe a ura y

δ

of theprodu t

RL

is performed (22)

RL = U

RL

D

RL

V

∗

RL

with

kU

RL

D

RL

V

∗

RL

− RLk < δ.

TheresultofthethirdstepispresentedinFigure 5.

Remark 4. Whenthe matrix

RL

is mu hsmaller thanthe initial matrix

A

(this is a wide-spread in the pra ti e ase), the omputation of the T-SVD in the full arithmeti of the produ t

RL

isless expensive thanthe omputationofthe T-SVD of thematrix

A

.

A= =

A

6

A

5

A

4

A

3

A

2

A

1

B

i

B

C

t

i

C

t

Fig.3. Low-rankapproximationofthematrix

A

A= =

A

6

A

5

A

4

A

3

A

2

A

1

e

B

e

B

i

R

R

i

L

C

e

T

Fig.4. Theresultofthese ondstep

Atthefourthstep,we onstru tthenalmatri esby omputingtheprodu ts

U = e

BU

RL

,

V

∗

= V

∗

RL

C

˜

T

and

D = D

RL

. Asaresult,thematri es

U

and

V

have orthogonal olumns.

Ourstatementisthatthede omposition

U D V

∗

approximatestheexa tT-SVD of

A

in thesense of (1)and(2).

A= =

A

6

A

5

A

4

A

3

A

2

A

1

e

B

e

B

i

U

RL

D

RL

V

RL

∗

C

e

T

Fig.5. Theresultofthethird step

4. NUMERICALEXPERIMENTS

Thenumeri alexperimentsareaimedat demonstratingtheperforman eofour algorithmin termsof omputationtimeandpre ision.

WehavenotdevelopedourownlinearalgebralibrarybuthaveusedtheLAPACK andBLASfun tionsofIntelMKL.Theperforman ewasmeasuredonIntelCore i7-3770KCPU3.5GHz,(IvyBridge).Wehaveavoidedtheimpa tofOMP paralleliza-tionofallMKLfun tionsbyswit hingothreadingbysetting OMP_NUM_TRE-ADS=1.

The Born matrix

A

asso iated with a 2D elasti ity verti ally inhomogeneous (layered)isotropi medium,onesour e,1450re eiversand10dierentfrequen ies, hasbeen onsidered.Thetarget domain, ontaining120x20 points,is apart of a hugereal modelwhi h wewould liketoimage. Detailsofthismodelaredes ribed in [13℄. Thefullmatrix

A

has29,000lines and7,200 olumns. At thepreliminary step,thematrix

A

isseparatedinto

p = 10

blo ks.

4.1. Computationaltime. Inthersttest,thea ura y

ε

ofthelow-rank appro-ximation is

10

−6

and the threshold

δ

of the ropped exa t SVD

U

k

D

k

V

∗

k

and of

the ropped low-rank SVD

U D V

∗

is

10

−6

. Dierent options for the rst step aretested: the omputationaltimefortheSVD,the QRand theCA ompression methods are omputed. These omputationaltimes should be ompared to 970s, i.e.thetimeofarobustSVDbythe

gesvd

omputationaloptimizedroutineofIntel MKL.

Theperforman eresultsshowtheadvantageofusingpartialpivotingintheCA overthe rosspivoting and lassi altotalpivoting. Additionally, theperforman e results onrm the high a elerationof theCA approa hin omparisonwith the SVDandtheQR-pivte hniques.

Table 1. Comparison of performan e algorithms of Low-rank SVD-basedonSVD/QR/CAapproa hes

Steps, des ription SVD RRQR CA CA CA

\

Approa hes Total Cross Partial

pivot. pivot. pivot. 1-st,GetSVD/QR/CAofall

A

i

605s

255s

90s

44s

18s

2-nd,MakeQRof

B

andT-QRof

C

:

18s

20s

19s

20s

25s

3-rd, PerformSVDof

RL

:

16s

13s

13s

13s

15s

4-th,Gathering

U

,

D

,

V

7s

6s

6s

6s

6s

Totaltime

651s

294s

130s

84s

66s

Finally,wepointoutthattherststepofthemethodproposediseasily paralleli-zable: sin e the ompression of all blo ks an be done by dierent pro essors as opposedtothelastthreesteps.

4.2. Pre ision. Inthese ondtest ourattentionwillbefo ussedontwodierent errorindi ators.Onlytheresultsforthemoste ient ase(CrossApproximation withdynami panelpartialpivoting)willbepresented.

First,weareinterestedin theerrorresulted fromapproximatesingularvalues. Denotingby

I

ε

the

ε

-rankofthematrix

A

,thefollowingquantities

(23)



















max

i≤Iε

|d

i

− d

i

|

|d

1

|

(

absoluteerror

)

max

i≤Iε

|d

i

− d

i

|

|d

i

|

(

relativeerror

)

have been plotted with respe t to

ε

, the parameter relating to the a ura y of theCrossApproximationparameterandwithrespe tto

δ

,i.e. thenalthreshold parameter.TheresultsarereportedinFigure6fortheabsoluteerrorandinFigure 7fortherelativeerror.It seemsthattheabsoluteerrordoesmostlydependon

ε

.On ontrary,therelativeerrorissu ientlyrelatedtotheratio

ε/δ

.Se ond,we quantifythequalityoftheapproximationoftheleftsingularspa e.Theresultsfor therightsingularvaluesare quitesimilarand arenotpresented.We omputethe angle (in degrees) between the subspa egenerated by the olumns of

U

and the subspa egenerated bythe olumnsof

U

k

(24)

∠(U , U

k

) = arccos(σ)

180

π

withthesmallestsingularvalue

σ

ofthematrix

U

∗

U

k

.

Numeri al measurements (Figure 8) show that the angle

α = ∠(U , U

k

)

, for any threshold

δ

, an be de reased via improving the a ura y

ε

of the low-rank approximation.Thenumeri alresultsrevealthattheerrordependsmostlyon

ε/δ

liketherelativeerrorforthesingularvalues(Figure7).

4.3. Evolution of

ε

-rank. Inthelast test weinvestigatetheranksof trun ated matri es.TheyarepresentedinTables2and3fordierentlow-rankapproximations. TheSVDapproa histhemostoptimalintermsof

ε

-rank,whereastheCAte hnique istheworst(therankaftertherststepintheTables).For

ε = 10

−6

and

δ = 10

−6

, the nal ranks asso iated with dierent ompressions are slightly dierent. For

Fig.6. TheCApartialpivoting:Dependen eofthesingularvalue absoluteerrorona ura y

ε

andthreshold

δ

.

Fig.7. TheCApartialpivoting:Dependen eofthesingularvalue relativeerror.

ε = 10

−9

and

δ = 10

−6

,the nalranksare equal.This willbethe asewhenthe ompression

ε

ismu hsmallerthan

δ

.Thenalrankofalow-rankSVDshould be ompared to a rankobtained by trun ated full arithmeti SVD, i.e.

1984

. Inthe ontextofseismi imaging,asmallmistoftherankwill nothave,inouropinion, alargeimpa tonthesolutionoftheinverseproblem.

5. CONCLUSION

We have presented an algorithm to ompute the trun ated SVD of the Born matrix. This method is based on a low-rank arithmeti and the CA te hnique. Toperform the low-rank approximation, we have proposed adynami panel CA algorithm. Thisapproa his similarto thepanel lo alpivoting

LU

de omposition

Fig.8. TheCApartialpivoting:Dependen eoftheanglebetween subspa esspanned onsingularve torsofthe roppedexa tSVD andLow-rankSVD.

Table 2. Therankof matri esat intermediatesteps,

ε = δ = 10

−6

Matrixrank SVD RRQR CA CA CA

\

Approa hes Total Cross Partial pivot. pivot. pivot. Afterthe1-ststep 2366 2424 2444 2454 2795 Afterthe2-ndstep 2041 2041 2022 1961 2013 Afterthe3-thstep

1963

1963

1953

1936

1950

Table 3. Therank of matri esat intermediatesteps,

ε = 10

−9

,

δ = 10

−6

Matrixrank SVD RRQR CA CA CA

\

Approa hes Total Cross Partial pivot. pivot. pivot. Afterthe1-ststep 2963 3030 3052 3052 3565 Afterthe2-ndstep 2619 2619 2586 2541 2581 Afterthe3-thstep 1964 1964 1964 1964 1964

te hnique [12℄. The algorithm proposed is an alternative to a very popular and e ientrandomizedSVDapproa hproposedbyRokhlin[6℄.Themainadvantages are:(i)the

ε

-rankofamatrixhasnottobeknowninadvan e,(ii)the omputation of a redu ed matrix is less expensive (this has been onrmed by preliminary numeri altests whi harenotin ludedin thispaper).

Forarepresentative ongurationwehave omparedtheresultsgeneratedbythe proposed trun ated SVDalgorithm to the resultsobtainedbyan exa tSVD.We haveobservedthatthemethodisa urateandthea elerationofthe omputation has in reasedby the fa tor 10 onone-thread systems. Thealgorithm has agood

opportunityforparallelizationbothonsharedmemorysystems(usingOMP paralle-lization)andondistributedones (MPIparallelization).

Appendix A. The Singular Value De omposition andthe Trun ated Singular Value De omposition

TheSVDof amatrix

A ∈ C

I×J

with

J ≤ I

isafa torizationoftheform

(25)

A = U DV

∗

_,

where thematri es

U

and

V

ontaintheleftand rightsingularve tors

u

i

and

v

i

; thematrix

D

isdiagonaland ontainsthesingularvalues

d

i

(26)

U = {u

1

, · · · , u

J

} ∈ C

I×J

with

U

∗

_{U = I ∈ C}

J×J

V = {v

1

, · · · , v

J

} ∈ C

J×J

,

with

V

∗

_{V = I ∈ C}

J×J

D = diag{d

i

}

J

_i=1

∈ R

J×J

₊

.

Thesingularvalues

d

i

areallpositiveandordered (27)

d

1

≥ d

2

≥ · · · ≥ d

J

≥ 0.

The T-SVDof the matrix

A

is obtainedfrom the SVD byremoving thesingular values

d

k+1

, d

k+2

, . . .

whi harelowerthanasmallparameter

δ

(28)

A

k

= U

k

D

k

V

∗

k

,

(29)

U

k

= {u

1

, · · · , u

k

} ∈ C

I×k

with

U

∗

k

U

k

= I ∈ C

k×k

V

k

= {v

1

, · · · , v

k

} ∈ C

J×k

with

V

∗

k

V

k

= I ∈ C

k×k

D

k

= diag{d

i

}

k

_i=1

∈ C

k×k

.

Thematrix

A

k

isanapproximationofthematrix

A

inthefollowingsense (30)

kA − A

k

k

2

≤ δ

withtheEu lideanmatrixnorm

k · k

2

:

(31)

kAk

2

= sup

x6=0

kAxk

2

kxk

2

Appendix B. Some theoreti al resultsinvolving the a ousti Born matrix

Letus showon asimpleexample thatthe Bornmatrixof ana ousti problem an be approximated by a low-rank approximation derived thanks to a kernel independentmultipoleexpansion.

Themodelparameters.The onsideredpropagationdomain onsistsof unbo-undedthree-dimensional(3D)a ousti mediagovernedbytheHelmholtzequation withvaryingphysi al hara teristi s

µ(y)

(the squareofthewave-number).

The model is parametrized on a regular grid with the spatial step

δ

y

∈ R

+

omposedof

J

ells

K

j

⊂ R

3

(Figure9),

(32)

K

j

= [j

1

δ

y

, (j

1

+ 1)δ

y

] × [j

2

δ

y

, (j

2

+ 1)δ

y

] × [j

3

δ

y

, (j

3

+ 1)δ

y

]

withtheinteger

j ∈ [1, J]

relatedtotheintegers

j

1

∈ [0, J

1

− 1]

,

j

2

∈ [0, J

2

− 1]

and

j

3

∈ [0, J

3

− 1]

bytherelation

δ

y

δ

y

δ

y

the ell

K

j

Fig.9. Dis retizationofthemodelparameterviaa3Dgrid

Thefun tion

µ

is hosento be onstant outsidetheregulargridand pie ewise onstantontheregulargridwiththevalue

µ

j

∈ R

+

on

K

j

(34)

µ(y) = µ

j

if

y

∈ K

j

and

µ(y) = µ

0

else

.

Themodelparameters

µ

j

,

1 ≤ j ≤ J

,are olle tedinto ave tor

µ

∈ R

J

.

The dataareobtainedthanksto

I

1

experiments,ea h orrespondingtoasour e lo ated at a point

x

1

i1

∈ R

3

, with

0 ≤ i

1

≤ I

1

− 1

. For every experiment,

I

2

measurementsarerealizedbyre eiverslo atedat apoint

x

2

i2

∈ R

3

, with

0 ≤ i

2

≤

I

2

− 1

. This givesrise to a set of data omposed of

I = I

1

I

2

measurements. To organizethesedata,everyre eiversour e oupleisindexedbyaninteger

i ∈ [1, I]

givenby

i = i

1

I

1

+ i

2

+ 1.

The full wave inverse problemtakestheform: Given

f

∈ C

I

,nd

µ

∈ R

J

+

su hthat (35)

F

i

(µ) = f

i

for

1 ≤ i ≤ I

with

F

i

(µ) = u

i1

(µ; x

i2

)

where

x

7→ u

i1

(µ, x)

is denedoverall

R

3

asthe outgoingsolutionofthedire t a ousti problemforthe

i

th

1

sour e: (36)

△u

i1

(µ; x) + µ(x) u

i1

(µ; x) = −δ

x

1

i1

with

δ

x

1

i1

theDira deltafun tion lo atedatthepoint

x

1

i1

.

MostofthealgorithmsthathavebeenproposedintheliteratureareoftheNewton type [9, 1℄. They require the omputation of the Born Matrix

A ∈ C

I×J

whi h ontainsthepartial derivativewithrespe tto

µ

j

ofthenonlinearform

F

i

.

(37)

A

i,j

=

∂F

i

∂µ

j

(µ)

for

1 ≤ i ≤ I

and

1 ≤ j ≤ J.

Thematrix

A

an beexpressedas

(38)

A

i,j

= u

j

i1

(x

2

i2

)

withrespe t tothefun tion

u

i1

whi h isthepartial derivativeof

u

i1

with respe t to

µ

j

(39)

u

j

i1

(µ; x) =

∂u

i1

∂µ

j

(µ; x) = lim

h→0

u

i1

(µ + he

j

; x) − u

i1

(µ; x)

h

.

Deriving (36) with respe t to

µ

j

, we obtain that the fun tion

u

j

i1

is the unique outgoingsolutionof (40)

△u

j

i1

(µ; x) + µ(x) u

j

i1

(µ; x) = −1

K

j

(x)u

i1

(µ; x)

withthe hara teristi fun tion

1

K

j

asso iatedto

K

j

(41)

1

K

j

(x) = 1

for

x ∈ K

j

and

1

K

j

(x) = 0

for

x /

∈ K

j

.

This Born matrix an then be related to the Green fun tion asso iated with the a ousti media.Thisfun tion,whi hdependson

x

∈ R

3

and

y

∈ R

3

,issymmetri

G(µ; x, y)

and isdened forevery

y

∈ R

3

as

G(µ; x, y) = G

µ;y

(x)

with

G

µ;y

the outgoingsolutionof

(42)

∆G

µ;y

(x) + µ(x) G

µ;y

(x) = −δ

y

(x)

on

R

3

with theDira generalizedfun tion

δ

y

at

x

= y

. It followsthat thefun tion

u

i1

, whi hsolves(36),isexpli itlygivenby

(43)

u

i1

(µ; x) = G(µ, x, x

1

i1

).

Its partial derivative

u

j

i1

, whi h solves (40) with respe t to

µ

is given by the representationformula (44)

u

j

i1

(x) =

Z

R

3

G(µ; x, y)1

K

j

(x)u

i1

(µ; y)dy.

Takingintoa ount(43),weobtain

(45)

u

j

i1

(x) =

Z

K

j

G(µ; x, y)G(µ; y, x

1

i1

)dy.

ThisleadstothefollowingsimpleformulafortheBornmatrix

(46)

A

i,j

= u

j

i1

(x

2

i2

) =

Z

K

j

G(µ; x

2

i2

, y) G(µ; y, x

1

11

)dy.

For large problems (a large numberof sour es, re eiversand model parameters), the omputation of this matrix an be very expensive and an be a hieved only thanksto highperforman e omputing.However,it anbeeasilyevaluatedinthe aseofhomogeneousmedia(i.e.

µ(y) = µ

0

)

(47)

G(µ; x, y) =

e

ikr

4πr

with

k =

√

_µ

0

and

r = |x − y|.

Pra ti ally, thisparti ular hoi e, anbeseenas theinitial guessforthe a ousti mediaunderstudy.

Remark: In pra ti e, the number of rowsof the Born matrix is mu h larger thanthenumberof olumns,i.e.

I >> J

.

Let us prove that the Born Matrix dened in (46) an be approximated by a low-rank matrix under suitable assumptions on the lo ation of re eivers and sour es.Most oftheargumentsthat arepresent inthis Se tion arerather similar to those developed in the multipole theory to solve the dire t problem [5℄. The

main ingredientto obtain alow-rankapproximationof the Born matrix

A

is the so alled kernel-independent fast multipole method [2℄. This method furnishes a tensorialapproximationoftheGreen matrixunder thefollowingassumptions,see Figure10:

i)Thesour es

x

1

i1

arein ludedin a3Ddimensionalbox

B

1

(48)

B

1

= x

1

0

+ [−d

1

, d

1

]

3

with

x

1

0

∈ R

3

,the enteroftheboxand

d

1

> 0

thesizeofthebox. ii)There eivers

x

2

i2

arein ludedin a3Dbox

B

2

(49)

B

2

= x

2

0

+ [−d

2

, d

2

]

3

with

x

2

0

∈ R

3

the enteroftheboxand

d

2

thesize ofthebox. iii)Theree tors

K

j

arein ludedin a3Dbox

B

3

(50)

B

3

= y

0

+ [−d

3

, d

3

]

3

with

y

0

∈ R

3

beingits enterand

d

3

thesize ofthebox.

iv)Thediametersofthese threeboxesaresmallerorequalto awavelength

(51)

d

1

< λ,

d

2

< λ

and

d

3

< λ.

v)Thedistan e

D

1

betweentheboxes

B

1

and

B

3

andthedistan e

D

2

betweenthe boxes

B

2

and

B

3

arelargerthansomewavelengths.

(52)

D

1

>> λ

and

D

2

>> λ.

In thebox

B

ℓ

,

1 ≤ ℓ ≤ 3

, we an approximate thefun tion

f

ℓ

: B

ℓ

−→ C

in the

Lo ationofthere eivers

B

2

d

2

< λ

B

1

Lo ationof thesour es

d

1

< λ

B

3

Lo ationoftheree tors

d

3

< λ

D

2

>> λ

D

1

>>

_λ

Fig.10. Assumptionsonthesour es,re eiversandree tors

followingway (53)

f

ℓ

(x) =

M

X

m=1

f

ℓ

(X

ℓ

m

) p

ℓ

m

(x) + ε

ℓ

(x)

∀f

ℓ

∈ C

∞

(B

ℓ

),

where for

1 ≤ m ≤ M

and

ℓ ∈ [1, 3]

,

X

ℓ

m

∈ R

3

are the interpolation points the

p

ℓ

m

: B

ℓ

−→ R

are interpolating fun tions. The residual

ε

ℓ

interpolatingfun tioniswell hosen(seeAppendixCforapossible hoi e)andthe interpolatedfun tionisregular.

Inthe ontextoftheBorn matrix,theseinterpolation fun tions an beused to deneatensorialapproximationoftheGreenfun tion

(54)























G

k

(x, y)

≃

M

X

m=1

M

X

n=1

G

k

(X

1

m

, X

3

n

)p

1

m

(x)p

3

n

(y),

x

∈ B

1

,

y

∈ B

3

,

G

k

(x, y)

≃

M

X

m=1

M

X

n=1

G

k

(X

2

m

, X

3

n

)p

2

m

(x)p

3

n

(y),

x

∈ B

2

,

y

∈ B

3

.

ThisbringsaboutresidualinthefollowingapproximationoftheBornmatrix

(55)

A

i,j

≃

M

X

m1=1

M

X

m2=1

M

X

n1=1

M

X

n2=1

G

k

(X

1

m1

, Y

n1

)G

k

(X

2

m2

, Y

n2

)

p

1

_m1

(x

1

_i1

)p

2

_m2

(x

i2

)

Z

K

j

p

3

_n1

(y)p

3

_n2

(y)dy.

Rearrangingthelatter,wededu ethat theBorn matrix

A

takestheform

(56)

A = A

1

_A

2

_A

3

_,

with

A

1

_{∈ C}

I×M

2

,

A

2

_{∈ C}

M

2

×M

2

and

A

3

_{∈ C}

M

2

×J

(57)























A

1

i,m

= p

1

m1

(x

1

i1

)p

2

m2

(x

i2

)

A

2

m,n

= G

k

(X

1

m1

, Y

n1

)G

k

(X

2

m2

, Y

n2

)

A

3

n,j

=

Z

K

j

p

3

n1

(y)p

3

n2

(y)dy

wheretheintegers

m

,

n ∈ [1, M

2

_]

and

i ∈ [1, I]

arerelatedtotheintegers

m

1

,

m

2

,

n

1

,

n

2

,

i

1

and

i

2

bytherelations

(58)

m = (m

1

−1 ) M + m

2

,

n = (n

1

−1 ) M + n

2

and

i = i

1

I

1

+ i

2

+1

Equation(56)revealsthat theBorn matrix

A

anbeapproximatedbyalow-rank approximationwhen

M

2

ismu hsmallerthan

I

and

J

.Thisisthe ase,whenthe numberofre eivers,sour esandree torsisverylarge.

Wehaveprovedthat the Bornmatrixasso iatedwith ahomogeneousa ousti medium admits a low-rank approximation under very restri tive assumptions on thelo ation ofthere eivers,sour esandree tors.These resultsthat usesimilar argumentstothefastmultipolemethod anbeextended.Whentheseassumptions arenotfullled,alow-rankapproximation analsobeobtained.Itrelieson elabo-ratedargumentsoftheFastMultipoleMethod.Thiswillnotbepresentedheredue toits omplexity.

For elasti media, it is also possibleto showthat the Born matrix asso iated withahomogeneousmediaadmitsalow-rankapproximation.Thereader anrefer to[2℄forthefastmultipole methodforelasti media.

Appendix C. The tensorial Chebyshev interpolation

We would like to briey re all lassi al results about the Chebyshev tensorial interpolationofafun tion

f

inthebox

(59)

B = x

0

+ [−d, d]

3

_.

Wedenoteby

C

P

theChebyshevpolynomialofdegree

P > 0

whi hisgivenbythe formula

(60)

C

P

(Z) = cos(P arccos (Z)),

Z ∈ [−1, 1].

For

1 ≤ p ≤ P

,its zerosare denoted by

Z

P

p

= cos



(p −

1

2

)

π

P



.Tothe zeros

Z

P

p

,

thatallbelongto

[−1, 1]

,weasso iatetheLagrangianinterpolationpolynomial

(61)

I

P

p

(z) =

P

Y

k=1

k6=p

z − Z

P

k

Z

P

p

− Z

k

P

for

p ∈ [1, P ]

and

z ∈ [−1, 1].

Ontheinterval

[−1, 1]

,anyfun tion

u

an beapproximatedbytheformula

(62)

u(z) =

P

X

p=1

u(Z

p

P

) I

P

p

(z) + ε

P

(z).

These interpolation polynomialsare optimal in the sense that theyminimize the

L

∞

normerror (63)

kε

P

k

L

∞

([−1,1])

=

 2

π

log(P + 1) + 1

 (π/2)

P

P !

ku

(P )

k

L

∞

([−1,1])

.

Thanks to this family of unidemensional interpolation fun tions, we dene the tensorialinterpolationfun tions

p

m

: B −→ C

onthebox

B

(64)

p

m

(x

0

+ z d) = I

P

p1

(z

1

) I

p2

P

(z

2

) I

p3

P

(z

3

)

with

z

= (z

1

, z

2

, z

3

) ∈ [−1, 1]

3

_.

andtheinterpolationpoints

X

m

∈ B

(65)

X

m

= x

0

+ d (Z

P

p1

, Z

P

p2

, Z

P

p3

)

wherewehavedenotedby

m ∈ [1, M]

,with

M = P

3

,theintegerdenedby

(66)

m = p

3

P

2

_{+ p}

2

P + p

1

with

p

1

, p

2

and

p

3

∈ [1, P ]

Itfollowsthat afun tion

f : B −→ C

an beapproximatedinthefollowingway

(67)

f (x) ≃

M

X

m=1

p

m

(x) f (X

m

)

Referen es

[1℄ N.S.Bakhvalov,Numeri alMethods,Nauka,1973.inRussian.MR0362811

[2℄ S. Chaillat and M. Bonnet, Re ent advan es on the fast multipole a elerated boundary elementmethodfor3Dtime-harmoni elastodynami s,WaveMotion,50:7(2013),10901104. MR3144050

[3℄ J. Demmel and W. Kahan, A urate singular values of bidiagonal matri es, So iety for Industrialand Applied Mathemati s,Journal onS ienti and Statisti al Computing,11:5 (1990),873912.MR1057146

[4℄ Z.Drma and K.Veseli ,Newfast and a urateJa obi SVD algorithmI,SIAMJ. Matrix Anal.Appl.,35:2(2008),13221342.Zbl1221.65100

[5℄ A.GumerovandR.Duraiswami,FastmultipolemethodsfortheHelmholtzequationinthree dimensions,Elsevier,Amsterdam,2004.

[6℄ N.Halko,P.Martinsson, andJ.A.Tropp,Finding stru turewith randomness:probabilisti algorithmsfor onstru tingapproximate matrixde ompositions, SIAMReview,53:2(2011), 217288.MR2806637

[7℄ I.Silvestrov, D.Neklyudov, C.Kostov, and V.T heverda. Full-waveform inversion for ma ro velo ity model re onstru tion in look-ahead oset VSP: numeri al SVD-based analysis, Geophysi alProspe ting,61:6(2013),10991113.

[8℄ G.MingandS.C. Eisenstat,E ient algorithmsfor omputinga strongrank-revealing QR fa torization,SIAMJournalonS ienti Computing,17:4(1996),848869.Zbl0858.65044 [9℄ J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several

variables,A ad.Press,NewYork,1970.MR0273810

[10℄ P. P. M. De Rijk, A one-sided Ja obi algorithm for omputing the singular value de ompositiononave tor omputer,SIAMJ.S i.Stat.Comp.,10(1998),359371. [11℄ S.Rjasanow,Adaptive rossapproximationofdensematri es,InIABEM2002,International

Asso iationforBoundaryElementMethods,UTAustin,TX,USA,May28-30,2002. [12℄ O. S henk and K. Gartner, On fast fa torization pivoting methods for sparse symmetri

indenite systems,ETNA. Ele troni Transa tionson Numeri alAnalysis[ele troni only℄, 23(2006),158179.MR2268558

[13℄ I.Silvestrov,D.Neklyudov,M.Pu kett,andV.T heverda,Resolutionandstabilityanalysis of oset VSP a quisition s enarios with appli ations to full-waveform inversion, In SEG Te hni alProgramExpandedAbstra ts2012,2012.

[14℄ W.Menke,Geophysi al DataAnalysis:Dis reteInverse Theory,A ademi Press,In .,New York,1984.

SergeyAlexandrovi hSolovyev

Instituteof PetroleumGeologyandGeophysi sSBRAS, pr.Koptyuga,3,

630090,Novosibirsk,Russia

E-mailaddress:solovevsaipgg.sbras.ru

S

ebastienTordeux

InriaBordeauxSud-Ouest,Equipe-ProjetMagique-3DIPRA-LMA Universit

ede PauetdesPaysdel'Adour BP1155,64013PauCedex,

Universit

ede Pau,Fran e E-mailaddress:sebastien.tordeuxgmail. o m