A Comparative Assessment among Iterative Linear Solvers Dealing with Electromagnetic Integral Equations in 3D Inhomogeneous Anisotropic Media

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

A COMPARATIVE ASSESSMENT AMONG ITERATIVE LINEAR

SOLVERS DEALING WITH ELECTROMAGNETIC INTEGRAL

EQUATIONS IN 3D INHOMOGENEOUS ANISOTROPIC MEDIA

G. Franceschini, A. Abubakar, T. M. Habashy, and A. Massa

January 2007

dealing with Ele tromagneti Integral Equations in 3D

Inhomogeneous Anisotropi Media

G. Fran es hini

(1)

, A. Abubakar

(2)

, T. M. Habashy

(2)

,and A. Massa

(1)

(1)

Department of Informationand Communi ation Te hnologies University of Trento,Via Sommarive 14,38050 Trento - Italy Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it, {gabriele.fran es hini}dit.unitn.it Web: http://www.eledia.ing.unitn.it

(2)

S hlumberger-Doll Resear h

36 Old Quarry Road, Ridgeeld, CT-06877 -USA Tel. +1203 431 5000, Fax +1 203 438 3819

dealing with Ele tromagneti Integral Equations in 3D

Inhomogeneous Anisotropi Media

Gabriele Fran es hini, AriaAbubakar,Tarek M. Habashy, and Andrea Massa

Abstra t

This paper deals with full-ve torial, three-dimensional, ele tromagneti s attering problemsformulatedintermsofintegrals atteringequations. Theweakformulation is applied in order to ee tively deal with inhomogeneous anisotropi media and the arising set of algebrai linear equations is solved through some of the most re ent andee tiveiterativelinearsolversforallowingadetailedassessmentoftheir performan es whenfa ing withthree-dimensional omplex s enarios.

Ge-Modeling ele tromagneti elds in realisti three-dimensional (3D) s enarios is an ap-pealing resear h topi . However, the solution of the arising ele tromagneti s attering problems is a riti al issue sin e it requires the use of numeri al pro edures with a non-negligible omputationalload, espe iallywhen large and omplex realisti ongurations are taken intoa ount.

As far as two-dimensional (2D) geometries and dissipative obje ts are on erned, Ri h-mond proposed in [1℄ the use of the Method of Moments (MoM). A stiness matrix, whose dimensionsdepend onthe size ofthe investigationdomain,the working frequen y, and the ontrast of the diele tri obje ts, is generated and su essively inverted. The need of inverting su h a matrix strongly limits the appli ation of the MoM espe ially whenthe problemsize grows. Inordertoredu ethe omputationalloadandthe required large amount of omputer memory, the so- alled k-Spa e Method has been introdu ed [2℄. Su h an approa h ombines an iterative approa h and the Fast Fourier Transform (FFT) algorithm for e iently omputing the spatial onvolution operator that o urs in the integral s attering equations [3℄. A further improvement, on erned with 2D-TM ongurations, has been su essfully obtained by applying the Conjugate Gradient Fast Fourier Transform (CG-FFT) [4℄[5℄[6℄[7℄[8℄[9℄.

Con erning 2D-TEs enarios,the problemhas been addressedbyZwamborn and van den Berg [10℄developinga weak-formof the integrals attering. Asu essive extensionto3D isotropi ases has been presented in [11℄ and further improved for allowing a more easy and ee tive numeri al implementation[12℄[13℄.

In thispaper, the approa hpresented in[13℄ isreformulatedfordealingwith 3D inhomo-geneousanisotropi media. Thesolutionofthearisingalgebrai linearsystemisaddressed by means of a set of iterative solvers and the performan es of the Conjugate Gradient (CG) approa h, the BiConjugate Gradient (BiCG) method, the Stabilized BiConjugate Gradient (BiCGStab) te hnique, the Quasi Minimal Residual (QMR) method, and the Generalized Minimal Residual (GMRES) algorithm are then ompared. To the best of authors' knowledge,althoughdetailedanalyzeshavebeen arriedoutonweak-form-based

out.

The paper is organized as follows. In Se tion 2, the integral formulation of the three-dimensional anisotropi problemis presented. The resultsof a omparativestudy among ee tive iterative linear solvers are shown in Se tion 3. Some on lusions are eventually presented inSe tion 4.

2 Mathemati al Formulation

Letus onsideradatadomain

R

and a omputational(orinvestigation)domain

D

gener-ally omposedbyaninhomogeneousanisotropi mediumofnitedimensionandembedded in a homogeneousisotropi ba kground of onstant permittivity

ε

b

, ele tri ondu tivity

σ

′

b

,andpermeability

µ

b

. Su has enarioisilluminatedbyaknownele tromagneti sour e dened onasupport

S

and des ribed through theimpressed ele tri ,

J



r

S



,

r

S

_{∈ S}

,and magneti ,

K



r

S



, urrent densities.

By onsidering a Cartesian oordinate system, a time-harmoni temporal dependen e, and non-magneti materials,the ele tri eld satises the followingequation

▽ × ▽ × E (r) − k

2

b

E

(r) =

k

_b

2

Q

(r) · E (r) + iωµ

b

J



r

S



_{− ▽ × K}



_r

S



_,

(1) where

k

2

b

= iωµ

b

σ

b

,

σ

b

= σ

′

b

− iωε

b

, and

Q

(r)

is the ontrast fun tion des ribing the

anisotropi investigationdomain

Q

(r) =

1

σ

b

















σ

xx

(r) − σ

b

σ

xy

(r)

σ

xz

(r)

σ

xy

(r)

σ

yy

(r) − σ

b

σ

yz

(r)

σ

xz

(r)

σ

yz

(r)

σ

zz

(r) − σ

b

















,

r

∈ D

(2)

By applying the Green's theorem and the radiation ondition at innity, the problem mathemati allydes ribed through Eq. (1)isreformulatedinintegral formbywritingthe

E

b

(r) = Ω

h

E

(r)

i

= E (r) −

h

k

2

_b

I

+ ▽ ▽

i

· A (r) ,

(3)

where

E

b

(r)

istheele tri eldinahomogeneousandunbounded ba kgroundof omplex ondu tivity

σ

b

and permeability

µ

b

,

I

is the unit dyadi and

A

(r) =

Z

D

g

(r, r

′

_{) Q (r}

′

_{) · E (r}

′

_{) dr}

′

(4)

is the ele tri ve tor potential,the s alarGreen fun tion

g

(r, r

′

₎

being

g

(r, r

′

_{) =}

exp (ik

b

|r − r

′

|)

4π |r − r

′

_|

.

(5)

In order tonumeri ally ompute

E

(r)

,the domain

D

is partitionedusing a uniformgrid ofre tangular ellsofside

∆x × ∆y × ∆z

wherethe ontrast

Q

isassumedtobe onstant. A ording to the pro edure des ribed in [13℄, the integral operators are dis retized by applying the weakening pro edure in order to ope with the singularity and the spatial dierentiation operators are al ulated by using the nite dieren e rule. In order to properly deal with anisotropi media, the ele tri ve tor potential

A

(r)

is numeri ally-evaluated asfollows

A

(h)

(r

m,n,p

) = ∆x∆y∆z

P

M

m

′

₌₁

P

N

n

′

₌₁

P

P

p

′

₌₁

g

(r

m,n,p

, r

m

′

_,n

′

_,p

′

)

P

k=x,y,z

Q

(h,k)

(r

m

′

,n

′

,p

′

) E

(k)

(r

m

′

,n

′

,p

′

)

h, k

= x, y, z; m = 1, ..., M; n = 1, ..., N; p = 1, ..., P ;

(6)

where

r

m,n,p

identies the generi enter-point of a volumetri sub-domain belonging to the investigation area;

M

,

N

and

P

are the numbers of dis retizations along

x

b

,

y

b

and

b

z

, respe tively;

E

(k)

denotes the

k

-th omponent of

E

, and

g

(r

m,n,p

, r

m

′

,n

′

,p

′

)

is omputed as in[13℄. Byapplying the onvolutiontheorem of Dis rete FourierTransform(DFT),it turns out that

A

(h)

_(r

m,n,p

) = ∆x∆y∆zDF T

−1

n

DF T

[g (r

m,n,p

)] B

(h)

(r

m,n,p

)

o

h

= x, y, z; m = 1, ..., M; n = 1, ..., N; p = 1, ..., P ;

(7) where

B

(h)

_(r

m,n,p

) = DF T

hP

k=x,y,z

Q

(h,k)

(r

m,n,p

) E

(k)

(r

m,n,p

)

i

h

= x, y, z; m = 1, ..., M; n = 1, ..., N; p = 1, ..., P.

,

(8)

thus allowing a omputationally-e ient omputation through FFT routines.

After dis retization, the predi tion problem is then re asts as the solution of the arising linear system of

U

= 3 × M × N × P

equations where the ele tri eld of the ba k-ground,

E

(k)

b,syn

(r

m,n,p

)

,

k

= x, y, z

,

m

= 1, ..., M

,

n

= 1, ..., N

,

p

= 1, ..., P

, is a known quantity be ause of the knowledge of the ele tromagneti sour e. Towards this end, be ause of the well-posed nature of the forward problem at hand [17℄, ee tive linear iterative solvers [namely, the well-known Conjugate Gradient (

CG

) approa h, the Bi-Conjugate Gradient(

BiCG

) method [14℄[18℄ and its stabilizedversion (

BiCGStab

)[15℄, the Quasi-MinimalResidual(

QM R

)approa h[19℄,andthe GeneralizedMinimum

Resid-ual (

GM RES

) method or its restarted implementation [20℄ (

R

− GMRES

)℄ aimed at

minimizing the distan e

ρ

i

(

i

being the iteration index) between the estimated solution and the a tual one, an be protably used thus avoiding time- onsuming inversion pro- edures. More in detail, let us dene the residual ve tor

u

i

, an array of dimension

U

whose omponents are given by

u

i(k)

_m,n,p

= E

_b

i(k)

(r

m,n,p

) − E

(k)

b,syn

(r

m,n,p

)

k

= x, y, z; m = 1, ..., M;

n

= 1, ..., N; p = 1, ..., P,

(9)

E

_b

i(k)

(r

m,n,p

)

being omputed through (3) on the basis of the trial solution estimated at

the

i

-th iteration,

E

(k)

i

(r

m,n,p

)

, of the iterative pro ess. Then, the distan e is omputed as

ρ

i

=

r

P

k=x,y,z

P

M

m=1

P

N

n=1

P

P

p=1







u

i(k)

m,n,p







2

r

P

k=x,y,z

P

M

m=1

P

N

n=1

P

P

p=1







E

b,syn

(k)

(r

m,n,p

)







2

.

(10)

andminimizedbygeneratinga onvergentsequen eoftrialsolutions

n

E

i

(k)

(r

m,n,p

) ; i = 1, ..., I

o

a ording to asuitable iterativeapproa h. Finally, on e the distribution of

E

(k)

_(r

m,n,p

)

,

k

= x, y, z; m = 1, ..., M; n = 1, ..., N; p =

1, ..., P ; ∀k = x, y, z, m = 1, ..., M, n = 1, ..., N, p = 1, ..., P

is determined,also the

s at-tered magneti eld at

r

R

_{∈ R}

anbeeasily omputedthrough the followingrelationship

H

scatt



r

R



= σ

b

▽

R

× A



r

R



.

(11)

3 Numeri al Validation

In this se tion, the performan es of the set of representative linear iterative solvers are ompared by onsidering, as a referen e ben hmark, the ele tromagneti problem mod-eling the system for the ele tromagneti indu tion well logging largely used in the oil exploration.

As far as the ele tromagneti sour e is on erned, it is a point magneti dipole dire ted along the

ν

-dire tion and represented through a null ele tri density (

J

= 0

) and an impulsive magneti urrent (

K



r

S



= δ



r

S



ν

. Consequently, the ele tri eld in the

ba kground is given by

E

b,syn

(r

m,n,p

) = −▽ × g



r

m,n,p

, r

S



ν

= −h



r

m,n,p

, r

S



× ν

m

= 1, ..., M; n = 1, ..., N; p = 1, ..., P ;

(12) where

h

(k)



_r

m,n,p

, r

S



= −

g

(

r

m,n,p

, r

S

₎

|

r

m,n,p

−r

S

|

r

(k)

−r

S

(k)

|

r

m,n,p

−r

S

|

h

1 − ik

b







r

m,n,p

− r

S







i

k

= x, y, z; m = 1, ..., M; n = 1, ..., N; p = 1, ..., P .

(13)

Inthersttest ase,theprobingsystem onsistsof: (a)anele tromagneti sour eworking at frequen y

f

0

= 1 KHz

and lo ated at

r

S

_{= (−50, 0, 0) m}

[

r

S

_{∈ D}

℄, (b)

N

R

_{= 41}

re eivers lo ated at

r

R

j

=

h

50, 0.0, 5.0 ×



j

−

N

R

−1

2

i

m

,

j

= 0, ..., (N

R

_{− 1)}

. Con erning the ubi al omputationaldomain

D

, itis inhomogeneous,

l

D

= 50 m

in side, and it has beenpartitionedintoagridof

32×32×32

ells. Twohomogeneousisotropi ubi obje ts

l

obj

= 12.5 m

-sided and hara terized by a ondu tivity

σ

(1)

xx

= σ

(1)

yy

= σ

(1)

zz

= 10

m

S

and

σ

(2)

xx

= σ

yy

(2)

= σ

zz

(2)

= 10

−2 S

m

lie in

D

at the lo ations

C

1

= (−12.5, −12.5, −12.5) m

and

C

2

= (12.5, 12.5, 12.5) m

, respe tively. The isotropi ba kground is homogeneous with a

onstant ondu tivity equalto

σ

b

= 0.1

S

m

.

As faras the iterativepro ess is on erned,the minimizationhas been stopped whenthe ondition

(ρ

i

<

10

−7

₎

wassatisedandthepredi tionresultsare showninFig. 1interms of thevalues ofthe magneti eld omponentsatthe measurementpointsin

R

. Asit an beobserved, the plots related toea h solver are almost indistinguishable, but signi ant dieren es turns out interms of the omputational load and onvergen e.

Figure2showsthebehavioroftheerrorfun tion

ρ

i

versustheiterationnumber

j

pointing out its monotoni de reasing when the

CG

,

GM RES

and

R

− GMRES

te hniques are used. On the other hand, the remaining approa hes (and in parti ular the BiCG te hnique) seem to be unstable with fast variations in the values of

ρ

i

. However, its should noti ed that the onvergen e ratio of the BiCG method signi antly improves ompared to the standard CG redu ing ten times the number of iterations for rea hing the onvergen e threshold [see Table I where the average

CP U

-time per iteration (

t

i

) together with the onvergen e index (

I

conv

), the initialization time (

t

0

), and the total

CP U

-time (

T

) are given℄. Moreover, with referen e to Figure 3 and Table I, it turns

out that the

CG

, the

BiCG

, and the

QM R

need of the same amount of

CP U

-time per iteration

t

i

sin etheir omputational ostsareduetotheevaluations(i.e.,twoevaluations at ea h iteration)of the operator

Ω

and the remainingve tor/s alar produ t operations

require a negligibleamount of

CP U

-time (if ompared tothe

Ω

evaluation).

A signi ant improvement of the omputational performan es is obtained by using the

BiCGStab

te hnique sin e, in addition to a slight redu tion of the time per iteration

t

i

due to the smaller number of ve torial produ ts at ea h iteration, the total number of iterations is almost halved when ompared to those of the

BiCG

approa h. A further improvementisa hievedbythe

GM RES

te hniquesin eitrequiresonlyone omputation oftheoperator

Ω

periterationeventhoughthe

CP U

-timegrowslinearlywiththenumber of iterations be ause of the in reasing of the dimension of the Hessenberg matrix. Su h a behavior results in the quadrati dependen e of the omputation time

T

i

as shown in Fig. 4.

In order to avoid the drawba k related to the matrix storage of the

GM RES

, the

R

−

GM RES

methodhasbeenevaluated,aswell,bysetting

I

res

= 20

. Although,onaverage,

t

i

de reases, su h an approa h presents a slower onvergen e ratio (Fig. 2) than the

GM RES

and furthermore, anextra timeis needed at ea hrestart asshown in Figure3.

Su h an event further onrms the reliability and the omputationalee tiveness of the

GM RES

.

In the se ond test ase, the water-oil onta t model shown in Figure 5 is onsidered. In su h a ase, the omputational domain of size

6.4 × 6.4 × 12.8 m

3

and dis retized

into

32 × 32 × 64

ells onsists of an isotropi deviated water layer with ondu tivity

σ

water

xx

= σ

yy

water

= σ

water

zz

= 5

m

S

(white olor in Fig. 5) and an anisotropi water-oil onta t region (bla k olor in Fig. 5) [

σ

oil

xx

= σ

yy

oil

= 0.333

m

S

,

σ

oil

zz

= 0.05

m

S

℄ in a ro k ba kground (brown olor in Fig. 5) of ondu tivity

σ

r

= 1.0

S

m

. Both transmitter and re eivers havethe same lo ationsof theprevious example,but the operatingfrequen y is equal to

f

0

= 26.3 KHz

.

Figure 6 shows the behavior of the predi ted magneti eld at the lo ations of the re- eivers. As expe ted, whatever the approa h, the eld behavior is faithfully estimated. Consequently,the omputationalee tiveness turnsouttobethe index ofsu ess among the dierent solutionte hniques.

From the omputational point-of-view and with referen e to Figures 7-9 and Table II, similar on lusionstothose on erned withthe rsttest ase holdtrue. However, despite

gen e ratio is on average faster than that of the previous test ase (see Fig 7), but the

CP U

-time per iteration in reases (Tab. II). Thanks to the non-negligible redu tion of

the required iterations(

I

conv

)in omparisonwith those ofthe other te hniques, the total amountof

CP U

-time required by the

BiCGStab

is omparablewith that needed for the

GM RES

method.

As far as the

R

− GMRES

solver is on erned,

I

res

has been xed to

I

res

= 5

. This

is a smaller value than that in the rst example, sin e the amount of memory used per iteration signi antly grows. Consequently, the initializationtime at ea h restart has a signi ant inuen e on the total

CP U

-time and therefore, the

R

− GMRES

method is slowerthan itsstandard implementation.

Finally,itshouldbeobservedthatinsu hanexampletheratiobetweenthe omputational osts of the fastest and the slowest algorithmsigni antly enlarges (

T

CG

T

GM RES

≃ 19

- Test Case 2 vs.

T

CG

T

GM RES

≃ 8.3

- Test Case 1) be ause of the slower onvergen e of the

CG

method(

I

CG

conv

= 1556

- Test Case 1 vs.

I

CG

conv

= 110

- Test Case 2).

4 Con lusions

In this paper, a omparativeassessment amongiterativelinear solvers whendealingwith three-dimensional inhomogeneous and anisotropi media has been arried out. As ex-pe ted, the numeri al results onrmed the ee tiveness of the onsidered approa hes on erning the a ura yin the ele tromagneti predi tion. On the other hand, the nu-meri al study showed that the

GM RES

method is the fastest solver even though the orresponding

CP U

-timeperiterationlinearly in reases. Moreover, therequired amount of memory depends on the number of iterations for rea hing onvergen e in a propor-tional way. Consequently, the

GM RES

turns out to be a suitable te hnique only when a large amount of memory is available or when small-s aleproblems are dealt with. On the ontrary, the obtained results pointed out that it is protable to use the

BiCGStab

algorithmwhenthe omputationaldomainbe omeslargerandlarger. Asamatteroffa t, su h an approa h presented omparable or better performan es than the

R

− GMRES

,

[1℄ J.H.Ri hmond,"S atteringbyadiele tri ylinderofarbitrary rossse tion,"IEEE

Trans. Antennas Propagat., vol. 13,pp. 334-341, Mar.1965.

[2℄ N. N.Bojarski,"K-spa eformulationof thes atteringprobleminthetimedomain," J. A oust. So .Am.,vol.72, pp. 570-584,1982.

[3℄ D. Lesselier and B. Du hene, "Buried, 2-D penetrable obje ts illuminated by line sour es: FFT-based iterative omputationsoftheanomalouseld,"Progressin Ele -tromagneti s Resear h,PIER, vol.5, pp. 351-389,1991.

[4℄ T. K. Sarkar, E. Arvas, and S. M. Rao, "Appli ation of FFT and the onjugate methodforthesolutionofele tromagneti radiationfromele tri allylargeandsmall ondu ting bodies," IEEE Trans. Antennas Propagat., vol. 34, pp. 635-640, May 1986.

[5℄ R. Kastner, "A onjugate gradient pro edure for analysis of planar ondu tors with alternating pat hand apertureformulation,"IEEE Trans.Antennas Propagat., vol. 36, pp. 1616-1620, Nov. 1988.

[6℄ T. J.Peters and J.L. Volakis,"Appli ationof a onjugategradient FFTmethodto s attering fromthin planar materialplates," IEEE Trans. Antennas Propagat., vol. 36, pp. 518-526,Apr. 1988.

[7℄ M. F. Catedra, J. G.Cuevas,and L. Nuno, "A s heme toanalyze ondu ting plates of resonant size using the onjugate gradient method and fast Fourier transform," IEEE Trans.Antennas Propagat., vol.36, pp. 1744-1752, De . 1988.

[8℄ A. F. Peterson, S. L. Ray, C. H. Chen, and R. Mittra, "Numeri alimplementations of the onjugate gradient method and the CG-FFT for ele tromagneti s attering," Progress in Ele tromagneti sResear h, PIER, vol. 5,pp. 241-300,1991.

[9℄ J.L. VolakisandK.Barkeshli, "Appli ationsofthe onjugategradientFFTmethod to radiation and s attering," Progress in Ele tromagneti s Resear h, PIER, vol. 5,

FFTmethodfortwo-dimensionalTE s attering problems,"IEEE Trans.Mi rowave Theory Te h., vol. 39,pp. 953-960, Jun. 1991.

[11℄ P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the onjugate gradient FFT method for s attering problems," IEEE Trans. Mi rowave Theory Te h., vol. 40,pp. 1757-1766, Sep. 1992.

[12℄ A.AbubakarandP.M.vandenBerg,Three-dimensionalnonlinearinversionin ross-well ele trode logging,Radio S ien e, vol. 33,no. 4,pp. 989-1004,1998.

[13℄ A.AbubakarandP.M.vandenBerg,"Iterativeforwardandinversealgorithmsbased on domain integral equations for three-dimensional ele tri and magneti obje ts," Journal of ComputationalPhysi s, vol.195, pp. 236-262,2004.

[14℄ Z. Q. Zhang and Q. H. Liu, "Three-dimensional weak-form onjugate- and bi onjugate-gradient FFT methods for volume integral equations," Mi rowave Opt. Te hnol. Lett., vol. 29,pp. 350-356, Jun. 2001.

[15℄ Z.Q.ZhangandQ.H.Liu,"Appli ationsoftheBCGS-FFTmethodto3-Dindu tion well logging problems," IEEE Geos i. Remote Sensing, vol. 41, pp. 998-1004, May 2003.

[16℄ J. A. Kong, Ele tromagneti Wave Theory,New York,USA: John Wiley,1990.

[17℄ P. M. van den Berg, "Iterative s hemes based on the minimization of the error in

eld problems," Ele tromagneti s, vol. 5,no. 2-3, pp. 237-262,1985

[18℄ H. Gan and W. C. Chew, "A dis rete BCG-FFT algorithm for solving 3D

inhomo-geneous s atterer problems ," JEMWA, vol. 9,pp. 1339-1357, 1995.

[19℄ R. Freund and N. Na htigal, "QMR: A quasi-minimal residual method for

non-hermitianlinear systems," Numer. Math.,vol. 60, pp. 315-339,1991.

[20℄ Y. Saad and M. H. S hultz, "GMRES: a generelized minimal residual algorithm

•

Figure 1. Test Case I - Real (a)( )(e) and imaginary (b)(d)(f) part of the

x

(a)(b),

y

( )(d),and

z

(e)(f) omponentsof the magneti eld inthe data domain

S

.

•

Figure 2. Test Case I - Normalizederror

ρ

i

versus iterationnumber

i

.

•

Figure 3. Test Case I -

CP U

-timeperiteration(

t

i

).

•

Figure 4. Test Case I - Total

CP U

-time

T

i

versus iterationnumber

i

.

•

Figure 5. Test Case II -Condu tivity distributionof the water-oil onta t model.

On the left hand side two orthogonal volume sli es of the are shown. On the right handside,2D ondu tivitydistributionsat

y

= 0

and

x

= −0.1 m

. Multi- omponent dataare olle tedalongtheverti alaxis(i.e.,

z

-axis)that oin ideswiththewellbore axis.

•

Figure 6. Test Case II - Real (a)( )(e) and imaginary (b)(d)(f) part of the

x

(a)(b),

y

( )(d),and

z

(e)(f) omponentsof the magneti eld inthe data domain

S

.

•

Figure 7. Test Case II - Normalizederror

ρ

i

versus iterationnumber

i

.

•

Figure 8. Test Case II -

CP U

-time per iteration(

t

i

).

•

Figure 9. Test Case II - Total

CP U

-time

T

i

•

Table I. Test Case I - Computationalindexes.

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

0

5

10

15

20

25

30

35

40

Re [H

x

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-1.0

-0.5

0

0.5

1.0

0

5

10

15

20

25

30

35

40

Im [H

x

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

(a) (b)

-1.5

-1.0

-0.5

0

0.5

1.0

0

5

10

15

20

25

30

35

40

Re [H

y

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-1.0

-0.5

0

0.5

1.0

0

5

10

15

20

25

30

35

40

Im [H

y

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

( ) (d)

-5.0

-4.0

-3.0

0

-2.0

-1.0

0

5

10

15

20

25

30

35

40

Re [H

z

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-1.0

-0.8

-0.6

0

-0.4

-0.2

0

5

10

15

20

25

30

35

40

Im [H

z

(r

j

)] [x 10

8

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

(e) (f)

0

200

400

600

800

1000

1200

1400

1600

10

−8

10

−6

10

−4

10

−2

10

0

10

2

i

r

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

CG

0

200

400

600

800

1000

1200

1400

1600

0

1

2

3

4

5

6

7

8

i

t

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

CG

0

50

100

150

200

250

0

200

400

600

800

1000

1200

i

T

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

1

2

3

4

5

y = 0 m

−2

0

2

−6

−4

−2

0

2

4

6

1

2

3

4

5

x = −0.1 m

−2

0

2

−6

−4

−2

0

2

4

6

1

2

3

4

5

σ

(S/m)

x

−→

y

−→

z

−

→

z

−

→

-0.5

0

0.5

1.0

1.5

0

4

8

12

16

20

24

Re [H

x

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-1.5

-1.0

-0.5

0

0.5

0

4

8

12

16

20

24

Im [H

x

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

(a) (b)

-0.5

0

0.5

1.0

1.5

0

4

8

12

16

20

24

Re [H

y

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-1.5

-1.0

-0.5

0

0.5

0

4

8

12

16

20

24

Im [H

y

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

( ) (d)

-0.5

0

0.5

1.0

1.5

0

4

8

12

16

20

24

Re [H

z

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

-2.0

-1.0

0

1.0

2.0

3.0

0

4

8

12

16

20

24

Im [H

z

(r

j

)] [x 10

3

]

j

Restarted GMRES

GMRES

QMR

BiCGStab

BiCG

CG

(e) (f)

0

20

40

60

80

100

120

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

i

r

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

CG

0

20

40

60

80

100

120

0

2

4

6

8

10

12

14

16

i

t

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

CG

0

5

10

15

20

25

30

35

0

50

100

150

200

250

300

350

400

i

T

i

Restarted−GMRES

GMRES

QMR

BiCGStab

BiCG

M ethod

I

conv

t

0

[sec]

t

i

[sec]

T

[sec]

R

− GM RES

233

4.5

3.1

736

GM RES

136

4.8

4.0

545

QM R

172

6.9

6.7

1171

BiCGStab

106

4.8

6.2

659

BiCG

166

5.2

6.8

1133

CG

1556

4.9

6.7

10420

M ethod

I

conv

t

0

[sec]

t

i

[sec]

T

[sec]

R

− GM RES

31

15.4

7.4

245

GM RES

25

15.2

6.7

183

QM R

27

15.3

13.6

382

BiCGStab

15

15.3

13.0

210

BiCG

27

15.5

13.9

389

CG

110

15.5

13.7

1524