Scattering by Media

+(2R') -3/2 ~

Ik(~')-n(~')l.

(52)

T.y>p 2

t := -

3 In IISn - S~ Iloo and p " t 2/7. 7(4R2 + 1)

Then the inequalities IISn --S~ Iloo < 1 and t ___M1 + 2k are satisfied by the definition of e, and we also have p>2. Finally, we obtain from p = t 2/7 < t, the definition of t and inequality (54)

The Cauchy-Schwarz inequality implies that Y__, I~(~)- n(~)l y.T>p 2

< ( E (I+T'T)21n(T)-n(y)I2) 1/2

IIn-~lloo

y.y>p2

x(

~{.y>p2

<

C

-v~

.

<_ c{e(8R2+2)tl]Sn

1/2

(,+,,/2) 1

:

<_C{ (IISn-S~ (53)

2 S~lloo+ ~ }

]oo)1/7+

N c(-lnllSn-S~

(-In ]]Sn-[S~llco)-l/7 }

]0o)-1/7

Scattering by Media

because x _< (-ln(x)) -1 for 0 < x < I, and we have proved the theorem. 9 Before we proceed with the continuous dependence of n on Uoo,n we want to mention a relation between Sn and the Dirichlet-to-Neumann map An, which maps the Dirichlet data ula/32 of a solution of Au + k2nu = 0 in B2 = B(0, R2) to its Neumann data 3u/av on 3B2. Provided the Dirichlet problems in B2 for the Helmholtz equation with refractive indices n and ~ have unique solutions, Eq. (48) yields the equation An - A~ = 2(S~~ - S~ ~).

(55)

This allows us to conclude that an analogous result to Theorem 6.1 is valid with Sn replaced by An. In order to replace Sn by uoo,n in the preceding theorem we must introduce a norm on the far field patterns. To this end we define the Fourier coefficients

.lmpq "= I f~I

(56)

l,p = O, 1, ...,

-l_
87

This last formula allows us to express the Fourier coefficients

[ ~2w~,,oo(.~,y)Yf'(~)ds(~) with the help of latmpq, and Theorem 4.3 of Chapter 1.2.1 yields the desired expansion k2 4re ~-~ il-p ldlmpq

w~'(x' Y) -

l,m,p,q

(1)

(kR )

y/, x

y

(58)

Formula (58) provides the tool to prove continuous dependence of w~, on n, that is, s

s

IIw, - w~ Iloo _
-p
Due to the rapid decay of these Fourier coefficients (see Stefanov (1990)) the norm Iluo~, II2 := n

Z

l,m,p,q

(21+1 2t+3 2p+1 2p+3 12 ekR ) (ekR) Ilal,npq (57)

is well defined. We intend to reconstruct the kernel of the integral operator Sn with the help of a series expansion involving the la/mpq. This allows us to conclude the continuity of the mapping uoo,n ~ Sn that together with Theorem 6.1 implies the desired continuity of n on uoo,n. According to (14) and (15), w~( - , y) - wS( 9, y) is the scattered wave for the incident wave (I)(-,y). From the Funk-Hecke formula ((71) in f7 of Chapter 1.2.1), by superposition, we infer that

I uoo(~c,d)Y~,(d)ds(d) is the far field pattern for the incident wave

l ~ eik<7x Y~,(d)ds(d) = ~4ref i jp(klxl) Y~(~). Hence, in view of the addition theorem ((40) in ~4 of Chapter 1.2.1), using superposition again we see that s the far field pattern wn, oo(~, y) for the incident wave d)(., y) is given by

w~.,oo(~, y) - ik Z

such that for all n, [z ~ U the estimate

I I - - ~II~.B _
h(v~)(klYl)YJ(5,)

such that for all n, ~ e U the estimate

x4nT ~

holds.

P,q

88

w

Scattering by Media

The Reconstruction of the Refractive Index

This section is devoted to Nachman's method (Nachman, 1988) to reconstruct the Fourier coefficients ( n - 1 ) ^ (7), 7 e F, from a knowledge of Uoo,n. His idea is to recover a certain boundary integral operator from a knowledge of Uo~,n in a first step. We will recover the Robin-to-Dirichlet map, that is, the mapping An that assigns to the Robin data a u / a v - iu of a solution of Au + k2nu = 0 in B(0, R2) its Dirichlet data u on 8B(0,R2). Next, we find a uniquely solvable Fredholm integral equation of the second kind for the Robin data of the special solutions u(., ~) from Lemma 4.1. Since the integral operators occurring in this integral equation only contain the modified fundamental solution ~r and An, we are able to compute a u ( . , ~ ) / a v - iu(.,~) from uo~,n. Finally, knowing the Robin and the Dirichlet data of u(-, ~) allows us to find ( n - 1)^(7) in a way similar to the reasoning of Theorem 4.2. Throughout this section we assume supp(1 - n) c B - B(0, R), and B2 = B(0, R2) denotes a ball with radius R2 > R. THEOREM 7.1. Suppose the far field pattern Uoo,n related to a refractive index n is known. Then the Robin-to-Dirichlet operator

An" C~

cl'~X(aB2)

can be computed from uo~,n. Proof. First, we note that An is well defined. To this end we observe that, as a consequence of Green's first integral theorem, the Robin boundary value p r o b l e m t o f i n d u e C 2 ( B 2 ) n C l ( B 2 ) w i t h A u + k 2n u - O in B2 and O u / 8 v - iu - f on 8B2 has at most one solution. Then we define the operator K'~'C~ C~ u(~)B2)by

(K'nq))(x) "= 2

I

aw(x,y) ~q)(y)ds(y),

aB2 C)V(X)

x ~ aB2,

(59)

where w(x, y) denotes the Green's function (14) for the refractive index n. Note that if u is defined as in (45), that is, u(x) = 2[

w(x,y)(p(y)ds(y),

J OB2

X E ~ 3,

(60)

then we can infer from the properties of the singlelayer potential with kernel 9 (see Theorem 5.2 of Chapter 1.2.1) that the first derivatives of u can be extended uniformly H61der continuous from the interior and exterior of B2 to ~B2 and that Ou_/av = Kn~+q), au+/g)v = Kn~P- ~ on c)B2. Therefore, if q) e C~ is a solution to

(I+Kn-iSn)q)=f,

(61)

where Sn is the operator from (46), then U]B2 with u defined by (60) is a solution to the Robin boundary value problem. The mapping properties of the operators S and K' together with the smoothness of w s from Eq. (15) imply the compactness of Sn and K~ in C~ Moreover, the uniqueness of the Robin problem and of the exterior Dirichlet problem together with the jump relations imply the injectivity of the operator (I + K n -iSn). Hence, Eq. (61) has a unique solution and so has the Robin boundary value problem. Furthermore, we have An - S n ( I +K'~iSn) -1. We already know how to obtain the kernel of the integral operator Sn from uoo,n (see (58)). The kernel of the integral operator Kn is

aO(x, y) awS(x, y) Ov(x~ + Ov(x-----------~' x, y E OB2, x Cy. Hence it can also be computed from the Fourier coefficients of Uo%n. This ends the proof of the theorem. 9 Before we are able to derive an integral equation for the Robin data of the special solutions u(x, ~)= ei~x(1 + v(x, ~)), which were constructed in Lemma 4.1, we must introduce the analogues of the operators S, K, K' and T with kernel ~ instead of ~. We assume that the vector ~ e C 3 always satisfies ~. ~ = k 2 and IIm(~)l > M1 > 0 with the positive constant M1 being large enough to ensure the existence of the solutions u(., ~). By W; we mean the modified fundamental solution defined in (5). Now, the boundary operators S~,/C~,/C~ and T~ are defined by

-y) (T~f)(x)-: 2~)v(x) I aB2 ~ ( x~)v(y) f(y)ds(y), (S~f)(x) .- 21 (lCff)(x)

8B2

2[

~(x-

y)f(y)ds(y),

8eg~(x - y)

aB2 ~)v(y) f(y) as(y), a~I'~(x- Y) (lC'~f)(x) :- 21 aB2 8V(x) fly) ds(y),

x ~ aB2.

Since ~ and eikil/(4rti 9 i) only differ by a smooth function, the boundary operators above inherit the properties of the analogous operators with kernel (see ~5 of Chapter 1.2.1). Similarly, the jump relations and mapping properties of single- and doublelayer potentials and their derivatives defined with the kernel ~ are the same as those with the kernel ~. Finally, we introduce the operator

An,~ 9 C~

C~

1

An, ~f := ~ { (T~ - ilC'~)Anf - (IC'~- I)f -i(1C~ - iS~)Anf + iS~f}, with the Robin-to'Dirichlet map An.

Scattering by Media

LEMMA 7.1. Suppose u(-, ~) is the special solution to the Helmholtz equation with refractive index n that was constructed in Lemma 4.1. Then its Robin data f:= (3u(., ~)/3v)- iu(., ~) are a solution to

fix)- (An, {f)(x) - aei~x 3v

ie i~'x,

(62)

x ~ aB2.

Proof. For R < R2 < R1 < R' we consider the modified Lippmann-Schwinger equation (33) in B(O, R1) for u(-,~). For R2 < Ix] < R1 it can be rewritten as

The operator An,( is compact in C~ and I-An,~ has a trivial nullspace. Especially, Eq. (62) has a unique solution, namely the Robin data of u(., ~). THEOREM 7.2.

Proof. It remains to prove the injectivity of I An,~. To this end assume f e C~ is a solution to f = An, ~f. We define v in B2 to be the solution of Av + k2nv = 0 in B2 that has the Robin data 3v/3v - iv = f. In the exterior of B2, for R2 < ]xl _
8eg~ (x - Y) - iW~ (x - y) ) (Anf)(y) v(x)"-IaB{ (av(y)

u(x, ~) = ei~'x + Ia~2 (a~I'~(x-y) u(y, ~) - ~p~(x- y) av(y)

au(y, ~) 3v ) ds(y),

because the boundary integral and the volume integral in Eq. (33) coincide due to Green's second integral theorem. Now we replace

If, for a given f e C0'~X(SB2), v is the solution to Av + k 2 n v - 0 in B2 with 3 v / 3 v - i v - f on 3B2, then the mapping P assigning v to f is a linear compact operator from C~ to C(B2). Moreover, Green's representation formula (11) in ~2 in Chapter 1.2.1 applied in B2 with the fundamental solution We to the function v yields

3v

~k 2 [

J B2

3~(x-y) 3v(y)

~(x-y)(1-n)(y)v(y)dy,

v(y) } as(y) xeB2.

(65)

Reordering terms as in (64) and using the jump relations we obtain

3v 2f= 2 ~-~ - 2iv = 2 f - 2An, ~f -2k2I

B2

3v+ 8/32(W~(x-y) - ~ ( Y ) -

in (63) and use the jump relations to compute (3u+(., r iu+(., ~). This ends the proof. 9

{q*~(x- y)-o-~(y)-

(3q~('-Y) -i~r 0V(')

3v(y)

v+(y)) ds(y)=0

W((x-y)(1-n)(y)v(y)dy, J B2

x ~ B2,

and must vanish in B2 according to the proof of Lemma 4.1. This finally implies that f = 3v_/3viv_=O. 9 After knowing how to compute

f_ 3u(., ~_____~) iu(. ~) and u(. ~)= Anf ~v ' ' from the far field pattern, i.e., the Cauchy data of u(., ~), we can use Green's second integral theorem and the idea of Theorem 4.2 for the computation of (n-1)^(7), 7~ F. As in the Uniqueness Theorem 4.2 for a fixed vector 7 6 F we choose the unit vectors dl, d2 E ~3 such that d l . d2 = dl "7 = d2 "7 = 0 and define the vectors ~t and ~t as in (35), (36) for sufficiently large t.

THEOREM 7.3. Fix 7 ~ F, define ~t and ~t as above and suppose the Cauchy data (3u(., ~t)/3v) and u(., ~t) of the special solutions are known. Then lim

a~'~(. - y) - itt'~ (. - y))(1 - n)(y)v(y)dy, 3v([)

3~?~(x - y)

~k 2 [

v(y)

I

t-+oo OB2

An,~f

whence An, ~ is a compact operator.

v+(y)) ds(y)

for x E B2 we infer that v is a solution to the homogeneous modified Lippmann-Schwinger equation

Hence, An, (f can be represented as

- -k2,[B2 (

av(y)

I (64)

aB2

v(x) = -k 2 [ ~ ( x - y ) ( 1 - n ) ( y ) v ( y ) d y J B2 I 3v+ 3?~(x-y) for x e B2. From the relation

- - ( 3 ~ {((Ax n- Yf ))(-yi )e-Pe{P( ~x -( Yx -) y) ) f ( y ) 3 v ( y )

I

Then we show that v+ = v_ and 3v_/3v = 3v+/3v, and as in (65), we obtain the representation

+

a~'r (x - y) au(y, ~) av(y) u(y, ~)- ~e~(x- y) - -

v(x) =

(X - y)f(y) } ds(y).

~

(63)

89

f~ei~t'x - ~u(x'~t)) as(x) /. 8V U(X, ~t) -- d ~''x 3v = k2(2R')3/2(n - 1)^(7).

Proof. The proof is an immediate consequence of Green's second integral theorem together

90 Scattering by Media

with u(x, ~t) = 8i~t'x( 1 + V(X, ~t)) and IIv(', ~t)]lL2 ~ 0 as t ~ oo. 9 Although we have given a constructive method to recover n, let us point out that it is nontrivial to numerically implement it and in fact nobody to date has done so. For example the kernel ~ is highly oscillating in certain directions while exponentially growing or decaying in other directions. This poses severe difficulties in solving Eq. (62) numerically.

w

Sampling and Optimisation Methods

In view of the remark at the end of the previous section we might ask whether it is possible to obtain less information on n with simpler techniques. For example we might be content to reconstruct the support o f m : = l - n from U~,n. For the sake of simplicity we assume that n is real-valued, n(x) < 1 for all x ~ D := {x ~ ~3:n(x) ~ 1}, the exterior of D is connected and D has a smooth boundary. Moreover, throughout the whole section we suppose that the homogeneous interior transmission problem

z ~ ~)D, the norms Ilvlloo, D must become unbounded as z approaches ~)D (otherwise v and w = (I + k Z T ) - l v remain bounded, whence f = w - v would remain bounded). This in turn means that Ilg(',z)llL2(~) ~ oo when z approaches ~)D from the interior. Our considerations suggest solving Eq. (66) for points z from a grid, and recovering ~)D at points where the norms Ilg(-, z)IIL2(~) sharply increase. We call numerical procedures using this idea sampling methods. Unfortunately, in general, Eq. (66) has no solution for z e D, and it is only possible to prove the analogous assertion to Theorem 14.1 of the previous chapter. THEOREM 8.1. Under the assumptions for n made at the beginning o f the section, for every 8 > 0 and z ~ D there exists a function g(.,z) ~ L2(~) such that IlFg(.,z)- r

z)llL2(~) _
and ]]g(.,Z)]IL2(~) ~ oo,

Z ~ aD,

and the Herglotz wave function v with kernel g(., z) becomes unbounded as z ~ 8D.

Kirsch (1999) replaced Eq. (66) with the equation Av § k2v = O, Aw +k 2nw - 0 in D, 8w 8v w - v = f , 8v 8 v - h onbD,

(F*F)l/4g=~oo(.,z)

with f - h = 0 has only the trivial solution. As in ~ 14 of the previous chapter we try to solve the equation (66)

Fg=r

with 1 e_ikSc.z,

Sc E ~,

being the far field pattern of a point source located at z E E 3. Since Fg is the far field pattern corresponding to the Herglotz wave function with kernel g as incident wave, by Rellich's Lemma 3.1 of Chapter 1.2.1, the solvability of Eq. (66) is equivalent to the existence of a Herglotz wave function whose scattered field coincides with r in the exterior of D. Obviously, for z ~ D Eq. (66) cannot possess a solution because a scattered field is always smooth in the exterior of D, whereas r z) has a singularity at z. Let us assume for the moment that for z e D Eq. (66) has a solution g ( . , z ) ~ L2(~). Then, setting v to be the Herglotz wave function with kernel g(-, z) and w to be the total field for the incident field v, we conclude that v, w are a solution to the interior transmission problem with f= r and h = ~)r on ~)D. Applying Green's formula to w - v in D reveals that v is related to w via the Lippmann-Schwinger equation v = (I § k 2T)w. Here T denotes the operator defined in (26). Hence, since f becomes unbounded as

(67)

and succeeded in proving the equivalence of z lying in D and the solvability of (67). According to Theorem 3.6, our assumption that n is real-valued implies F to be a compact, normal operator in L2(~). Furthermore, since the homogeneous interior transmission problem has only the trivial solution, F is injective by Theorem 3.3. Then the spectral theorem for compact, normal operators ensures that F has an orthonormal basis of eigenfunctions ~j ~ L2(~) with eigenvalues kj # 0, j e N. From ~j we derive the function q)j e L 2 (D, m) via q)/ .- - ~1A I l ~ j ,

j ~ ~,

where A ~ is the adjoint operator of the operator A introduced in (27) and V/k//means the square root with positive imaginary part. Note that, due to the factorisation from Theorem 3.9, q)j satisfies k2

A (I +kZ T~ )q)i = - -~ ~f-~j.

(68)

In order to state another result about the functions q)i we define two closed subspaces in LZ(D,m), namely //1 := {rE C2(D)nL2(D,m):Av+k2v=O inD}, Hn := {we C2(D)nL2(D,m):Aw+k2nw=O in D}.

The fact that they are closed is a consequence of regularity theorems for elliptic partial differential equations. Since n is real-valued, the adjoint for the

Scattering by Media

operator T : L 2 ( D , m ) ~ L 2 ( D , m ) from (26)is given by T~u = T-~. Furthermore, the bijectivity of (I+k2T) implies the bijectivity of (I+ k 2 T ~) in L2(D,m). For a function w ~ l-In, using the properties of the volume potential we conclude that v " - ( I + k2T~)w satisfies Av + k 2 v - O. On the other hand defining w "(I + k2T~)-lv for v 9 Hi implies that Aw + k2nw - O. Hence, the restriction

91

where ( . , - ) denotes the inner product in L2(~). Conversely, starting from ~1/9 L2 (f~) satisfying o0 I(~, ~j)l 2 I)~jl

we define 09 := -(4x/k2)(~l/, ~ j ) I v / ~ and 00

(I +k2T~)lttn. Hn ~ H1

v := (I+k2T ~) ~(zi~p j e t-11 j=l

of (I + k 2 T 11)to Hn is bounded and has a bounded inverse, too. Since a Herglotz wave function with kernel 9 i belongs to/-/1, we can infer from relation (29) that

and compute Av = ~ as above. Observing that the eigenvalues and eigenfunctions of F*F are given by ]kjj2 and ~j, respectively, we can conclude that the range of (F'F) 1/4 and the set

k2

k 2 T~) -1H~j ~ Hn,

e g2(n): ~ I(~, ~l//)l2 i=1

and therefore g0i 9 H~. The following technical theorem whose proof is given in Kirsch (1998, 1999) states a useful fact about the functions g0i. THEOREM 8.2. Suppose n satisfies the assumptions from the beginning of this section. Then the functions ~Pi 9 ~-ln, j 9 ~ form a Riesz basis in t-In; that is, each function Ip 9 Hn can be written as a series o0

j=~

We are now in a position to characterise a point z 9 D with the help of the operator (F'F)1~4. THEOREM 8.4. Define ~o0(Sc, z) := (4/1:)-1 e -ikSc'z for s 9 ~, z 9 ~3. Then (F*F)I/4g = ~o0(.,z)

~[~10cjl2 < oo, j=~

and each square summable sequence (czi)i c C defines an element (p = E ~jipj 9 Hn. THEOREM 8.3. Suppose n satisfies the assumptions from the beginning o f this section. Then the ranges o f Al~l:/-h ~ L2(~) defined in (2 7) and of (F*F)l/4:L2(~2) ~

L2(~)

coincide. Proof. Assuming ~1/= Av, v 9 i.e., ~ lies in the range of Alul, we set w := (I + k2T~)-lv 9 l-In and use relation (68) together with the series expansion o0

w =

00

E i jl 2 < oo

j=l

has a solution if and only if z 9 D. Proof. According to the previous theorem it suffices to show that ~o0(.,z) lies in the range of A]~tl if and only if z 9 D. For z ~ D the only radiating solution to the Helmholtz equation with far field pattern 9 o0(., z) is ~(., z), which has a singularity in the exterior of D. Since the range of A only contains far field patterns of continuous functions in the exterior of D we conclude that ~o0 (., z) cannot be in the range of A. On the other hand for z 9 D one can show (see Kirsch (1999)) that there are functions v 9 H1, w 9 that satisfy (I + k2 T )w = v in D and k2 ( Tw) law = -(~)(" , Z)13w on 3D. Then u(x)

- k 2 [ m(y)~(x, y)w(y)dy,

x ~ •3 \ D,

JD

must coincide with ~(., z) in the exterior of D because the exterior Dirichlet problem has a unique solution. This yields uo0 = ~o0(-, z), whence Av = ~o0 (., z ) . .

j=l

of the previous theorem to arrive at ~ll = A(I + k2 T~)w = - - ~

o9 j=l

whence

~ I(~,~J)12

< oo

coincide. Hence, we have proved the assertion. ,,

o0

~p= ~, (~jipj with

I'~il

(k2) 2

tlli,

Let us conclude this section by a brief review of other methods for numerically recovering the refractive index n. A straightforward approach is to reformulate the problem as an optimisation problem. We recall that the total field of the direct scattering problem with incident wave u i is the solution to the Lippmann-Schwinger equation u + k 2 Tnu = u i

92

Scattering by Media

in B with the Lippmann-Schwinger operator from (16). The corresponding far field pattern is then given by Pnu with

k2 I B e-ik~c'y(1 - n(y) ) u(y) dy, (Pnu)(~c)=--4-~

~c~ .

Given the far field patterns uoo(-, d i) corresponding to the incident waves ui(x, dj) e ikx'di with different directions dj, j - 1 , . . . , J , we might try to minimise the functional

T':m ~ uoo, which maps m := 1 - n e C~ to the far field pattern uoo associated with n. Then our inverse medium problem is to find m ~ C~ compactly supported in B such that ~'(m)= uoo holds. Since we proved that m is uniquely determined by a knowledge of uoo(~c,d) for all ~, d ~, we assume that uoo is a function on ~ x ~. For the Newton method we pick an initial approximation m0 and define recursively m i : - mj_l + h, j ~ ~, where h is the solution to the linear equation

J

j=l

+llVnuj-uoo(',dj)l122(~,} over suitable sets for the unknown functions uj and n. We refer the reader to the references given in the monograph of Cohon and Kress (1998). However, increasing the number J of directions will increase even more the number of unknowns in the large optimisation problem. One possibility to remedy this drawback was suggested by Colton and Monk (1988). In a first step a solution g of the equation Fg = r is approximated. From our considerations for the sampling method we know that this means we try to superimpose incident plane waves in such a way that the resulting scattered field coincides with (I)(.,0) in the exterior of B. Note that, in general, this problem has no solution but that it is possible to find g ~ L2(f2) such that IIFg9 oo(',0)llL2{~} becomes arbitrarily small (similar to Theorem 8.1). Then defining the Herglotz wave function vg with kernel g we try to find n and u such that

{ IIu + k 2 mnu - vgl122(B) +llk2(mnu)lsB +~(", 0)I~3BII2L2(~3B)}

is minimised. Here the second term of the sum ensures that the scattered part of the solution coincides with ~(., 0) in the exterior of B. Now, the number of unknowns in this optimisation problem is independent of J. Again details and further references can be found in the monographs of Colton and Kress (1998) and Kirsch (1996).

w 9.

The Newton Method

Similarly to the inverse obstacle problem we can try to approximate the solution to the inverse medium problem with the help of an iterative method. We confine ourselves to the Newton method. To this end we introduce the refractive-index-to-far-field operator

(69)

T" (mj-1)h = Uoo - l:'(mj_l ).

E { Ilu + k 2 Tnuj - u s

Here T"(m) denotes the Fr~chet derivative of T' at m. In the sequel we suppose that B c C := (-~,~)3. In view of the Lippmann-Schwinger equation (8) and the far field representation (10) we can write k2 r [F(m)](~, d) - -4-~ ]_c e-ik~c'Ym(y)u(y, d) dy,

~c, d ~ ~2, (70)

where u is the solution to

u(x, d) + k 2 [ ~(x, y)m(y)u(y, d) dy Jc = ui(x, d),

x ~ C, d ~ ~2.

(71)

From (71) we find that the Fr~chet derivative v of u with respect to m (in direction h) satisfies v(x, d) § k 2 [ ~(x, y)m(y)v(y, d) dy =-k2[

Jc

Jc ~(x,y)h(y)u(y,d)dy,

x 6 C , d6~2.

(72)

Together with (70) for the Fr6chet derivative of T' at m in direction h we compute

k2 I c e-ikSc'yh(y)u(y'd) dy {[F'(m)]h} (~,d)= -~--~ 4g c

e-ik~'Ym(y)v(y,d) dy,

~c,d e ~.

(73)

It is easily seen that [F'(m)]h is the far field pattern of the function v from (72). Therefore, [~-"(m)]h can be characterised as the far field pattern of the radiating solution v to the equation Av + k 2nv = k 2hu

in [~3. Equation (70) reveals that ]P is completely continuous. Hence, T"(m) is a linear and compact operator. THEOREM 9.1. r'(m): C 0o, ~ (B)

0, For m ~ C O ~(B) the operator L 2 (~2 x ~2) is injective.

Proof. Suppose [F'(m)]h = 0. Then for each d ~ ~ the far field pattern of the function v(., d) from (72) vanishes and Rellich's lemma yields v ( . , d ) = O, 8 v ( . , d ) / S v = 0 on ~)B. Therefore, for an arbitrary

Scattering by Media

solution to Aw + k2nw = 0 in B by Green's second integral theorem we have 0 =I

aB

( w d3v(. ),

av

- v(- d) 3w) ds '

-a-g

93

r~ = 0 the trigonometric polynomial ~ is not compactly supported. However, the refractive index h(x) :- 1-Fn(x), x e C, h ( x ) : - 1, x r C, has a far field pattern that coincides with uoo for all ()~r,dr), ~eZ.

= k2 IB hu(.,d)w dx. Recalling from Theorem 2.1 that span {u(., d): d e f~ } is dense in the set of all solutions to Au + k2nu = 0 in B we can replace u(.,d) by an arbitrary solution ffJ to Aff~+ k2ngv = 0 in B. Then we can conclude h = 0 analogously to the proof of Theorem 4.2. 9 Counting the degrees of freedom in the domain and the range of the Fr&het derivative we cannot expect that F'(m) has a dense range in L2(f~ x f2) because P'(m) maps functions depending on three real parameters into functions depending on four real parameters. This can also be seen by the fact that the L 2 adjoint

[r'(O)] *" L2(.Q x D.) --+ L2(B) is not injective. The function y0 (~) + y0 (d), ~, d e f~, is a nonzero element in the nullspace of IT"(0)]*. Equation (69) is ill posed. Moreover, the computation of F'(m) must be performed at each step. Therefore, Gutman and Klibanov (1994) suggest using a simplified and regularised Newton method. We modify the presentation given by Kirsch (1996) to match our operator ~'. If we replace T"(m/_l) by F'(0) in (69) we obtain T" (O)h = uoo - T'(mj_l ),

(74)

and from (73) the Fr6chet derivative T"(O) is given by

{[r'(o)]h} (~:,a)= - ~

c e-ik;cYh(y)e iky'd dy. (75)

Defining Z'= {y= (Y1,Y2,Y3)~ Z/3"maxlYjl < 2k/x/3}, given 7 e Z we choose d r and xr e f2 such that k(~ r dr) =7. Then Eq. (74)reads I c h(y)e -iv'y dy = ~4rt [/V(mj-1)- uoo] (x~,,d~,) ; i.e., we know the first Fourier coefficients of the solution h to (74). Now, for x e ( - x , x ) 3 we define a regularised solution as the trigonometric polynomial 1 ~ [r(mi_l)- uo~] (~v, dr) eir'x" h(x) .- 2rtZk2 ~,ez Note that the use of F'(0) leads to equations that we already know from the Born approximation. A similar analysis to Kirsch (1996) shows that for a sufficiently small m this regularised and simplified Newton method converges linearly to a trigonometric polynomial r~. With the exception of

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