INVERSE BORN SERIES FOR SCALAR WAVES * 1. Introduction

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1 Jouna of Computationa Mathematics Vo.3, No.6, 22, doi:.428/jcm.25-m3935 INVERSE BORN SERIES FOR SCALAR WAVES * Kimbey Kigoe Shai Moskow Depatment of Mathematics, Dexe Univesity, Phiadephia, PA 94, USA Emai: [email protected] [email protected] John C. Schotand Depatment of Mathematics, Univesity of Michigan, Ann Abo, MI, 489, USA Emai: [email protected] Abstact We conside the invese scatteing pobem fo scaa waves. We anayze the convegence of the invese Bon seies and study its use in numeica simuations fo the case of a spheicay-symmetic medium in two and thee dimensions. Mathematics subject cassification: 78A46. Key wods: Invese scatteing.. Intoduction The invese scatteing pobem (ISP) fo scaa waves consists of ecoveing the spatiayvaying index of efaction (o scatteing potentia) of a medium fom measuements of the scatteed fied. This pobem is of fundamenta inteest and consideabe appied impotance. Thee is a substantia body of wok on the ISP that has been been compehensivey eviewed in [7 9]. In paticua, much is known about theoetica aspects of the pobem, especiay concening the issues of uniqueness and stabiity. Thee has aso been significant effot devoted to the deveopment of techniques fo image econstuction, incuding optimization, quaitative and diect methods. Thee is aso cosey eated wok in which sma-voume expansions have been used to econstuct the scatteing popeties of sma inhomogeneities. The coesponding econstuction agoithms have been impemented and thei stabiity anayzed as a function of the signa-to-noise-atio of the data [ 6]. In pevious wok, we have poposed a diect method to sove the invese pobem of optica tomogaphy that is based on invesion of the Bon seies [ 3]. In this appoach, the soution to the invese pobem is expessed as an expicity computabe functiona of the scatteing data. In combination with a specta method fo soving the inea invese pobem, the invese Bon seies eads to a fast image econstuction agoithm with anayzabe convegence, stabiity and eo. In this pape we appy the invese Bon seies to the ISP fo scaa waves. We chaacteize the convegence, stabiity and appoximation eo of the method. We aso iustate its use in numeica simuations. We find that the seies conveges apidy fo ow contast objects. As the contast is inceased, the highe ode tems systematicay impove the econstuctions unti, at sufficienty age contast, the seies diveges. The emainde of this pape is oganized as foows. In Section 2, we constuct the Bon seies fo scaa waves. We then deive vaious estimates that ae ate used to study the * Received Novembe, 2 / Revised vesion eceived May, 22 / Accepted May 4, 22 / Pubished onine Novembe 6, 22 /

2 62 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND convegence of the invese Bon seies. The invesion of the Bon seies is taken up in Section 3. In Section 4, the fowad opeatos in the Bon seies ae cacuated fo the case of adiay vaying media. Exact soutions to the pobem of scatteing by sphees and annui ae discussed in Section 5. These esuts ae used as fowad scatteing data fo numeica econstuctions, which ae shown in Section 6. Finay, ou concusions ae pesented in Section Bon Seies We conside the popagation of scaa waves in R n fo n 2. The fied u obeys the equation 2 u(x) + k 2 ( + η(x))u(x) =. (2.) It wi pove usefu to decompose the fied into the sum of an incident fied and a scatteed fied: The incident fied wi be taken to be a pane wave of the fom u = u i + u s. (2.2) u i (x) = e ikx ξ, (2.3) whee k is the wave numbe and ξ S n is the diection in which the incident wave popagates. The scatteed fied u s satisfies and obeys the Sommefed adiation condition ( ) im us iku s 2 u s (x) + k 2 u s (x) = k 2 η(x)u(x) (2.4) =. (2.5) The function η(x) is the petubation of the squaed efactive index, which is assumed to be suppoted in a cosed ba B a of adius a. The soution u can be expessed as the soution to the Lippmann-Schwinge intega equation u(x) = u i (x) + k 2 B a G(x, y)u(y)η(y)dy, (2.6) whee the Geen s function G satisfies the equation 2 xg(x, y) + k 2 G(x, y) = δ(x y). (2.7) Appying a fixed point iteation to (2.6), beginning with the incident wave, gives the we known Bon seies fo the tota fied u u(x) =u i (x) + k 2 B a G(x, y)η(y)u i (y)dy + k 4 B a B a G(x, y)η(y)g(y, y )η(y )u i (y )dydy +. (2.8) Let us define the scatteing data φ = u i u. (2.9)

3 Invese Bon Seies fo Scaa Waves 63 The above seies aows us to expess this data φ as a powe seies in tenso powes of η: The opeatos {K j } ae defined as φ = K η + K 2 η η + K 3 η η η +. (2.) (K j f)(x, ξ) = k 2j B a... B a G(x, y )G(y, y 2 )...G(y j, y j ) u i (y j )f(y,..., y j )dy...dy j, (2.) whee x is the position at which the fied is measued. Note that the dependence of K j f on the incident diection ξ is made expicit. The seies (2.) wi be efeed to as the Bon seies. In ode to anayze the convegence of the Bon seies, we need to find bounds on the nom of the K j opeatos. Assume we measue data on the bounday of a ba of adius R, B R. By poceeding with the same appoach as found in [3], we find that the opeatos K j ae bounded in L : K j : L (B a B a ) L ( B R S n ). Futhemoe, if we define µ = sup x B a k 2 G(x, ) L (B a), (2.2) ν = k 2 B a sup x B R then thei opeato noms satisfy the estimate sup G(x, y)u i (y), (2.3) y B a K j ν µ j. (2.4) We can cacuate µ expicity in thee dimensions: µ = k2 (ka)2 dx =. (2.5) 4π B a x 2 Hee we have used the fact that the Geen s function is given by G(x, y) = eik x y 4π x y. (2.6) We cacuate ν in the next section. As shown in [2] the Bon seies conveges in the L nom when η L < 2 (ka) 2. (2.7) We can simiay bound the seies tems in the L 2 nom, if we view K j as an opeato defined as foows: K j : L 2 (B a B a ) L 2 ( B R S n ). We find, again by the agument in [3], that the opeato noms ae bounded: whee K j 2 ν 2 µ j 2. (2.8) µ 2 = sup x B a k 2 G(x, ) L2 (B a), (2.9) ν 2 = k 2 B a 2 sup x B R sup G(x, y)u i (y) L 2 ( B R ). (2.2) y B a

4 64 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND 3. Invese Bon Seies In the invese scatteing pobem we seek to ecove the coefficient η within the domain B a fom bounday measuements of the scatteing data φ. Foowing [3], we expess η as a foma powe seies in tenso powes of φ of the fom whee K = K +, η = K φ + K 2 φ φ + K 3 φ φ φ +, (3.) K 2 = K K 2 K K, K 3 = (K 2 K K 2 + K 2 K 2 K + K K 3 )K K K, and fo j 2, j K j = K m m= i +...+i m=j K i... K im K... K. (3.2) We wi efe to equation (3.) as the invese Bon seies. We use K + to denote a eguaized pseudoinvese of K. Since K has singua vaues which decay to zeo, it does not have a bounded invese. The foowing theoem on the convegence of the invese seies was poven in [3] and impoved in [] R=2a R=4a R=6a R=8a R=a Radius of convegence ka Fig. 3.. Radius of convegence of the invese seies in the L nom fo diffeent vaues of R. Theoem 3.. (convegence of the invese scatteing seies) The invese Bon seies (3.) conveges in the L nom if K p < /(µ + ν) and K φ Lp (B a) < /(µ + ν), whee µ and ν ae given by (2.2) and (2.3). Simiay, the seies conveges in the L 2 nom if the anaogous inequaities hod with µ and ν instead given by (2.9) and (2.2). In addition, the foowing estimate hods fo p = 2, N η K j φ φ C ((µ p + ν p ) K φ Lp (B a) )N+, (3.3) (µ p + ν p ) K φ L p (B a) j= Lp (B a)

5 Invese Bon Seies fo Scaa Waves 65 whee η is the imit of the invese seies and C = C(µ p, ν p, K p ) does not depend on N o the data φ. Remak 3.. As shown in [3], this theoem can be extended to incude convegence of the invese seies in the L p nom fo 2 p. Using the Geen s function (2.6) and setting the measuement adius R = αa fo some constant α, we have that ν k 2 B a 4πdist(B R, B a ) = (ka)2 3(α ). (3.4) Then the adius of convegence of the invese seies is given by Note that as α, µ + ν (ka) 2 3(α ) + (ka)2 2 = 6(α ) (ka) 2 (2 + 3(α )). (3.5) 2 µ + ν (ka) 2. (3.6) The adius of convegence as a function of ka is shown in Fig. 3. fo vaious vaues of α. In the fa fied the adius is sighty bigge than in the nea fied. 4. Fowad Opeatos fo Radiay-Vaying Media 4.. Two-dimensiona pobem We now cacuate expicity the tems in the fowad seies fo the two-dimensiona case whee Ω = R 2. We take B R to be a disk of adius R centeed at the oigin such that B a B R. We assume that the coefficient η depends ony on the adia coodinate = x. The fundamenta soution is given by which has the Besse seies expansion whee G(x, y) = i 4 H() (k x y ), (4.) G(x, y) = i 4 n= Hee J n ae the Besse functions of the fist kind, H n () > ae defined as We take u i to be a pane wave given by which has the seies expansion e in(θx θy) g n (x, y), (4.2) g n (x, y) = H () n (k > )J n (k < ). (4.3) ae the Hanke functions, and < and < = min( x, y ), > = max( x, y ). (4.4) u i (x) = u i (x) = e ikx ξ, (4.5) n= i n e in(θ θ) J n (k), (4.6)

6 66 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND whee, in poa coodinates, x = (, θ) and θ is the poa ange of the unit vecto ξ on S. The fist tem in the fowad seies is given by If y B R, the functions g n become If we intoduce poa coodinates φ () (θ, θ ) = k 2 B a G(x, x )u i (x )η(x )dx. (4.7) g n (x, y) = g n (x) = H n (kr)j n (k). (4.8) x = (, θ ), (4.9) take x to be on B R and inset the fomuas (4.2), (4.8) and (4.6) into (4.7) and cay out the angua intega, we obtain φ () (θ, θ ) = πik2 2 We can theefoe cacuate the Fouie coefficients φ () m,m 2 = e im(θ θ) H m (kr)j m (k )i m J m (k )η( ) d. (4.) m 2π 2π = 2i m+ k 2 π 3 H m (kr) e imθ im2θ φ () (θ, θ )dθdθ (J m (k )) 2 η( ) d, (4.) which is ony nonzeo when m = m 2, so we put m = m = m 2. We intoduce a escaing of the Fouie coefficients of the fom ( ) ψ m () = H m (kr)2π 3 i m+ φ () m. (4.2) Then, the fist tem in the seies is given by ψ () m = k 2 (J m (k )) 2 η( ) d. (4.3) Repeating this pocess, we can cacuate the second tem in the fowad seies φ (2) (θ, θ ) = k 4 B a B a G(x, x )η(x )G(x, x 2 )η(x 2 )u i (x 2 )dx dx 2. (4.4) As above, we escae the Fouie coefficients ψ (2) m = ik4 π 2 J m (k )η( )H m (k max(, 2 ))J m (k min(, 2 )) η( 2 )J m (k 2 ) 2 d d 2. (4.5) Fo the nth tem in the seies, we have the genea fomua ψ (n) m = in+ k 2n π n 2 n J m (k )η( )H m (k max(, 2 )) J m (k min(, 2 )) η( 2 ) H m (k max( n, n ))J m (k min( n, n )) η( n )J m (k n ) n d d n. (4.6)

7 Invese Bon Seies fo Scaa Waves Thee-dimensiona pobem The setup hee is sticty anaogous to that of the two-dimensiona case. function is given by (2.6) and is expessibe as the seies expansion The Geen s whee G(x, y) = ik = m= Hee j ae the spheica Besse functions and h () kind. If y B R, g takes the fom g (x, y)y m (ˆx)Y m(ŷ), (4.7) g (x, y) = h () (k > )j (k < ). (4.8) ae the spheica Hanke functions of the fist g (x, y) = g (x) = h () (kr)j (k). (4.9) If the incident wave, u i is a pane wave as in the two dimensiona case, then u i has the seies expansion u i (x) = 4π i j (k)y m (ˆx)Y m(ξ). (4.2),m Intoducing spheica coodinates with x B R and making use of (4.7), (4.9) and (4.2), we obtain x = (, ˆx ) and x = (R, ˆx), (4.2) φ () (ˆx, ξ) = k 2 B a G(x, x )u i (x )η(x )dx = 4πik 3 Y m (ˆx) g ( )i j (k )Ym(ξ)η( )d 2. (4.22) Now, taking the Fouie tansfom we have φ (),m, 2,m 2 = Y m (ˆx)Y 2m 2 (ξ)φ () (ˆx, ξ)dˆxdξ S 2 S 2,m = 4πik 3 δ 2 g ( )i j (k )η( )d 2. (4.23) Since the ight hand side no onge depends on m and m 2, we define = = 2 so that φ () = 4πk 3 i + h () (kr) (j (k )) 2 η( ) 2 d. (4.24) As in the two dimensiona case, we can epeat this pocess fo highe ode tems, and find the foowing geneaized fom fo the nth tem in the seies: φ (n) = 4πk 3n i +n h () (kr) j (k )η( )h () (k max(, 2 )) j (k min(, 2 ))η( 2 ) h () (k max( n, n ))j (k min( n, n )) η( n )j m (k n ) 2 2 nd d n. (4.25)

8 68 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND 5.. Spheica scattee 5. Exact Soutions fo Sphees and Annui Fo the pobem of a ba shaped scattee centeed at the oigin, the coefficient η is given by { η < R η() =, (5.) R <. The soution wi then be computed on two subdomains. The fist is the inne disk o sphee B = {x x R }, (5.2) the second is the exteio domain. The system of equations coesponding to (2.) with appopiate inteface matching conditions is given by: 2 u + k 2 u = in B, 2 u 2 + k 2 u 2 = in R n \ B, u = u 2 on B, u ν = u 2 on B, ν (5.3d) (5.3a) (5.3b) (5.3c) whee k 2 = k 2 ( + η ) is the coefficient in the inne egion Disk in two dimensions We use the Besse seies expansion (4.6) fo the incident wave u i whee θ is the poa ange of the incident diection ξ. We expess the soution to (5.3) in a Besse seies expansion, u (x) = a n e in(θ θ) J n (k ), (5.4) n= u 2 (x) = u i (x) + b n e in(θ θ) H n (k). (5.5) n= Appying the inteface conditions aows us to obtain the foowing system to sove fo the coefficients {a n, b n }: [ ] [ ] [ ] Jn (k R ) H n (kr ) an = i n J n (kr ) k J n(k R ) kh n(kr ) kj n (kr. (5.6) ) We thus obtain an expession fo φ fo x B R : φ(θ, θ) = Computing its Fouie coefficients gives φ m,n = 2π 2π n= b n e in(θ θ) b n H n (kr). (5.7) e imθ e inθ2 φ(θ, θ 2 )dθ dθ 2 = (2π) 2 δ m, n b m H m (kr)( ) m. (5.8)

9 Invese Bon Seies fo Scaa Waves 69 Using the fact that the above expession is independent of n we can define φ m = φ m, m = (2π) 2 b m H m (kr)( ) m. (5.9) Using the same escaing as in equation (4.2) gives ( ) ψ m = H m (kr)2π 3 i m+ φ m = 2b m( ) m πi m+. (5.) Sphee in thee dimensions In the thee-dimensiona case, the incident wave u i has the Besse seies expansion (4.2). The soution to the system (5.3) can be expessed as u (x) =,m u 2 (x) = u i (x) +,m a m j (k )Y m (ˆx)Y m(ξ), (5.) b m h () (k)y m (ˆx)Y m(ξ). (5.2) Afte appying the inteface bounday conditions, we obtain a system of equations to sove fo the coefficients {a m, b m }: [ j (k R ) h () (kr ) k j (k R ) k(h () ) (kr ) ] [ am b m ] = 4πi [ j (kr ) kj (kr ) Now, substituting in x B R we get a fomua fo the data function φ(ξ, ˆx) = m, ]. (5.3) b m h () (k)y m (ˆx)Y m(ξ). (5.4) We can compute its Fouie coefficients φ 2m2 m = Y m ( ˆx )Y 2m 2 ( ˆx 2 )φ( ˆx, ˆx 2 )d ˆx d ˆx 2 S 2 S 2 and, as befoe, define 5.2. Annua scattee = δ, 2 δ m,m 2 b m h () (kr) (5.5) φ m = φ mm2 mm = b m h () m (kr). (5.6) The coefficient η is now assumed to be of the fom < R, η() = η R < R 2, R 2 <. (5.7) The domain is then divided into thee subdomains. The fist is the inne disk o sphee B = {x x R }, (5.8)

10 6 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND the second is the midde annuus A = {x R < x R 2 }, (5.9) and the thid is the exteio egion. The system of equations coesponding to (2.) with inteface matching conditions is given by: 2 u + k 2 u = in B, 2 u 2 + k 2 u 2 = in A, 2 u 3 + k 2 u 3 = in R n \ (B A), u = u 2 on B, u ν = u 2 on B, ν (5.2e) u 2 = u 3 on ( A) +, u 2 ν = u 3 ν on ( A)+, (5.2a) (5.2b) (5.2c) (5.2d) (5.2f) (5.2g) whee ( A) + is the oute bounday of A, and k 2 = k 2 ( + η ) is the coefficient in the midde annuus Annuus in two dimensions The incident wave u i is given above in (4.6). The soution to (5.2) can be expessed as u (x) = a n J n (k)e in(θ θ), u 2 (x) = n= n= u 3 (x) = u i (x) + b n J n (k )e in(θ θ) + c n H n (k )e in(θ θ), d n H n (k)e in(θ θ). n= (5.2a) (5.2b) (5.2c) Afte appying the inteface bounday conditions, we obtain the foowing system of equations which can be soved fo the fou coefficients {a n, b n, c n, d n }: = i n J n (kr ) J n (k R ) H n (k R ) J n (k R 2 ) H n (k R 2 ) H n (kr 2 ) kj n(kr ) k J n(k R ) k H n(k R ) k J n(k R 2 ) k H n(k R 2 ) kh n(kr 2 ) J n (kr ) J n (kr 2 ) kj n (kr ) J n (kr 2) The exact soution has the fom: a n b n c n d n. (5.22) ψ m = 2d m( ) m πi m+. (5.23)

11 Invese Bon Seies fo Scaa Waves Annuus in thee dimensions Using the expansion of the incident wave (4.2), the soution to the thee dimensiona annuus pobem can be expessed as u (x) =,m u 2 (x) =,m u 3 (x) = u i (x) +,m a m j (k)y m (ˆx)Y m(ξ), b m h () (k )Y m (ˆx)Y m(ξ) + c m j (k )Y m (ˆx)Y m(ξ), d m h () (k)y m (ˆx)Y m(ξ). (5.24a) (5.24b) (5.24c) In this case, appying the inteface bounday conditions, we have the foowing system of equations which can be soved fo the fou coefficients {a m, b m, c m, d m }: j (kr ) h () (k R ) j (k R ) h () (k R 2 ) j (k R 2 ) h () (kr 2 ) kj (kr ) k (h () ) (k R ) k j (k R ) k (h () ) (k R 2 ) k j (k R 2 ) k(h () ) (kr 2 ) = 4πi j (kr ) j (kr 2 ) kj (kr ) j (kr 2) a m b m c m d m. (5.25) Again, we can define φ m = d m h m (kr). (5.26) 6. Numeica Resuts We now pesent the esuts of numeica econstuctions fo the fou mode systems we have discussed. When computing the tems of the invese seies, we use ecusion to impement the fomua (3.2). The scatteing data is computed fom the fomuas (5.),(5.23),(5.6) and (5.26). The fowad opeatos ae impemented using the fomuas (4.6) and (4.25). We compute the pseudo-invese K = K + by using MATLAB s buit-in singua vaue decomposition. Since the singua vaues of K ae exponentiay sma, we set the ecipicas of a but the agest M = 6 singua vaues to zeo. When computing the data (4.6) and (4.25) we use m = 4 modes and discetize the intega opeatos on a spatia gid of 4 unifomy-spaced nodes in the adia diection. We found that inceasing the numbe of modes and spatia gid points did not significanty change the econstuctions. Fig. 6. shows econstuctions fo ow contast with measuements in the nea-fied. In each case, five tems in the invese seies ae computed. We aso show the of η onto the subspace geneated by the fist M singua vectos, which gives a sense fo what can be econstucted at ow fequencies, fo a paticua eguaization. Note that the seies appeas to convege quite apidy to a econstuction that is cose to the. As the contast is inceased, as shown in Fig. 6.2, the highe ode tems ead to significant impovements compaed to the inea econstuctions. In Fig. 6.3 we pesent econstuctions fo the high contast case, but with measuements caied out in the intemediate fied. In this situation we make use of M = modes. Finay,

12 62 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND inea econstuction inea econstuction inea econstuction inea econstuction Fig. 6.. Numeica esuts fo sma contast measued in the nea fied. Fom top eft: two dimensiona disk, two dimensiona annuus, thee dimensiona sphee, and thee dimensiona annuus. Hee k =.3, k =, R =, R 2 =.5 (fo annuus), and measuements ae at R = 3. We take 6 modes in the eguaized pseudoinvese K inea econstuction inea econstuction inea econstuction.5 inea econstuction Fig Numeica econstuctions fo age contast measued in the nea fied. Fom top eft: two dimensiona disk, two dimensiona annuus, thee dimensiona sphee, and thee dimensiona annuus. Hee k =.3, k =, R =, R 2 =.5 (fo annuus), and measuements ae at R = 3. We take 6 modes in the eguaized pseudoinvese K +. in Fig. 6.4 we show econstuctions of the high contast case with measuements in the fa fied using M = 5 modes. In both cases, the esuts ae compaabe to the nea-fied econstuctions shown in Fig. 6.2.

13 Invese Bon Seies fo Scaa Waves inea econstuction.8.6 inea econstuction inea econstuction.8.6 inea econstuction Fig Numeica econstuctions fo age contast measued in the intemediate fied. Fom top eft: two dimensiona disk, two dimensiona annuus, thee dimensiona sphee, and thee dimensiona annuus. Hee k =.3, k =, R =, R 2 =.5 (fo annuus), and measuements ae at R = 5. We take modes in the eguaized pseudoinvese K inea econstuction inea econstuction inea econstuction inea econstuction Fig Numeica econstuctions fo age contast measued in the fa fied. Fom top eft: two dimensiona disk, two dimensiona annuus, thee dimensiona sphee, and thee dimensiona annuus. Hee k =.3, k =, R =, R 2 =.5 (fo annuus), and measuements ae at R =. We take 5 modes in the eguaized pseudoinvese K Discussion In concusion, we have studied numeicay the convegence of the invese Bon seies fo scaa waves. Exact soutions to the fowad pobem wee used as scatteing data and econ-

14 64 K. KILGORE, S. MOSKOW AND J. C. SCHOTLAND stuctions wee computed to fifth ode in the invese seies. We found that the seies appeas to convege quite apidy fo ow contast objects in both two and thee dimensions. As the contast is inceased, the highe ode tems systematicay impove the econstuctions. We note that the esuts at high contast in both the nea, intemediate and fa fieds ae quaitativey simia. We do not expect that this obsevation wi hod up in the case of eectomagnetic scatteing since soutions to the Maxwe equations in the nea fied decay moe apidy than in the fa zone. Acknowedgments. Shai Moskow and Kimbey Kigoe wee suppoted by the NSF gant DMS John Schotand s wok was suppoted by the NSF gants DMR 2923, DMS 5574 and DMS Refeences [] H. Ammai and H. Kang, Reconstuction of Sma Inhomogeneities fom Bounday Measuements, Lectue Notes in Mathematics, 846. Spinge-Veag, Bein, 24. [2] H. Ammai and H. Kang, Poaization and Moment Tensos: with Appications to Invese Pobems and Effective Medium Theoy, vo. 62, Spinge-Veag, 27. [3] H. Ammai and H. Kang, High-ode tems in the asymptotic expansions of the steady-state votage potentias in the pesence of conductivity inhomogeneities of sma diamete, SIAM J. Math. Ana., 34 (23), [4] H. Ammai and H. Kang, Bounday aye techniques fo soving the Hemhotz equation in the pesence of sma inhomogeneities, J. Math. Ana. App., 296:, (24), [5] H. Ammai, H. Kang, M. Lim, and H. Zibi, The geneaized poaization tensos fo esoved imaging. Pat I: Shape econstuction of a conductivity incusion, Math. Comp., 8 (22), [6] H. Ammai, H. Kang, E. Kim, and J.-Y. Lee, The geneaized poaization tensos fo esoved imaging. Pat II: Shape and eectomagnetic paametes econstuction of an eectomagnetic incusion fom mutistatic measuements, Math. Comp., 8 (22), [7] Chadan, K. and Sabatie, P., Invese Pobems in Quantum Scatteing Theoy, Spinge, 989. [8] Coton, D. and Kess, R., Invese Acoustic and Eectomagnetic Scatteing Theoy, Spinge, 992. [9] Cakoni, F. and Coton, D., Quaitative Methods in Invese Scatteing Theoy: An Intoduction, Spinge, 25. [] Kigoe, K., Moskow, S. and Schotand, J. C., Invese Bon seies fo diffuse waves., Imaging micostuctues, 3-22, Contemp. Math., 494, Ame. Math. Soc., Povidence, RI, (29). [] Make, V., O Suivan, J. and Schotand, J. C., Invese pobem in optica diffusion tomogaphy. IV. Noninea invesion fomuas, J. Opt. Soc. Am. A, 3, (23), [2] Moskow, S. and Schotand, J.C., Numeica studies of the invese bon seies fo diffuse waves, Invese Pobems, 25 (29) [3] Moskow, S. and Schotand, J.C., Convegence and stabiity of the invese scatteing seies fo diffuse waves, Invese Pobems, 24 (28)