INVERSE BOUNDARY VALUE PROBLEM FOR THE ...This inverse problem arises, for example, in reﬂection...

Proceedings of the Project Review, Geo-Mathematical Imaging Group (Purdue University, West Lafayette IN),Vol. 1 (2013) pp. 185-203.

INVERSE BOUNDARY VALUE PROBLEM FOR THE HELMHOLTZ EQUATIONWITH MULTI-FREQUENCY DATA

ELENA BERETTA⇤, MAARTEN V. DE HOOP† , LINGYUN QIU‡ , AND OTMAR SCHERZER§

Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. We develop an explicit iterative reconstruction of the wavespeedusing a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeedscontaining conormal singularities. A conditional Lipschitz estimate for the inverse problem holds for wavespeeds ofthe form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives aframework in which the scheme converges. The stability constant grows exponentially as the number of subdomainsin the domain partitioning increases. To mitigate this growth of the stability constant, we introduce a hierarchy ofcompressive approximations of the solution to the inverse problem with piecewise constant functions. We establishan upper bound of the stability constant, which constrains the compression rate of the solution. Then, tracking thefrequency dependencies through the approximation errors, we arrive at a procedure to select the frequencies suchthat convergence from level to level of our scheme is guaranteed.

1. Introduction. In this paper, we study the inverse boundary value problem for the Helmholtzequation using the Dirichlet-to-Neumann map at selected frequencies as the data. We focus on devel-oping an explicit iterative reconstruction of the wavespeed. This inverse problem arises, for example,in reflection seismology and inverse obstacle scattering problems for electromagnetic waves [6, 24, 7].We consider wavespeeds containing conormal singularities.

Uniqueness of the mentioned inverse boundary value problem was established by Sylvester &Uhlmann [23] assuming that the wavespeed is a strictly positive bounded measurable function. Froman optimization point of view this inverse problem has been extensively studied. We mention, inparticular, the work of [8]. Multi-frequency data and so-called frequency progression have beenintroduced to intuitively stabilize the iterative schemes used in optimization [12, 22, 5, 14].

We give a complete characterization of a multi-level, multi-frequency projected steepest descentmethod guaranteeing convergence. The convergence is derived from stability estimates. ConditionalLipschitz stability is obtained by assuming that the wavespeed is a linear combination of piecewiseconstant functions with an underlying domain partitioning [11]; one can accommodate partial data.Here, we establish Frechet di↵erentiability, obtain the frequency dependencies of the constants,and prove (using complex geometrical optics solutions) an upper bound for the constant in theconditional Lipschitz stability estimate. Here we assume full data. It follows that the stabilityconstant behaves exponentially with respect to the refinement of the domain, ⌦ say, that is, as thenumber of subdomains in the domain partitioning increases.

Conditional Lipschitz stability estimates have been extensively studied in Electrical ImpedanceTomography (EIT). We mention the work of [1, 10]. Usually, the Calderon type problem yieldsonly logarithmic or weaker type stability estimates. The logarithmic stability has been shown to beoptimal assuming su�cient regularity and boundedness of the coe�cient [19]. Selected techniquesdeveloped for EIT carry over in the analysis presented here.

To mitigate the growth of the stability constant, we introduce a hierarchy of compressive ap-proximations of the unique solution to the inverse problem with piecewise constant functions. Wearrive at a multi-level scheme while progressively increasing the number of subdomains subject toa condition which couples the approximation errors and stability constants between neighboringlevels [16]. We use the projected steepest descent iteration proposed in [16]. For transparency, we

⇤Dipartimento di Matematica “Guido Castelnuovo” Universita’ di Roma “La Sapienza”, Roma, Italy([email protected])

†Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette IN 47907, USA., ,[email protected]

‡Center for Computational and Applied Mathemematics, Purdue University, West Lafayette, IN 47907 (mdehoop,[email protected]).

§Computational Science Center, University of Vienna, Nordbergstr. 15, A-1090 Vienna, Austria([email protected]).

185

186 E. BERETTA, M. V. DE HOOP, L. QIU, AND O. SCHERZER

restrict ourselves to the normalized duality mapping. To avoid the use of L1(⌦), which is neithera convex, smooth nor reflexive Banach space, we consider L1(⌦) as a convex subset of Lp(⌦) ora limit of the approximations using lp. We can then choose p to be su�ciently large to obtainestimates relevant to applications. The mentioned upper bound of the stability constant constrainsthe minimal compression rate of the unique solution using piecewise constant functions. Trackingthe frequency dependencies through the approximation errors, we arrive at a procedure to selectthe frequencies such that convergence from level to level of our scheme is guaranteed.

Poschol, Resmerita and Scherzer [20] discussed whether, given a direct problem formulated inL1, the inverse problem should be considered in L1 or whether is it more appropriate to formulatethe problem in some Lp-space with 1 < p < 1 (in fact, p = 2 in [20])? One conclusion there wasthat for numerical realizations Lp approximations are quite advantageous because in the L1 case,one can construct examples of weak star convergent sequences, which are not even convergent inL1. From this perspective a numerical analysis is preferable in Lp-spaces, where the L1 space isembedded and a topology ⌧d is introduced by a pseudometric

d(u, v) =��FuvF

��p+ | kuk1 kvk1 |.

For instance, there exist piecewise constant Ansatz functions, which approximate an L1-functionin the Lp-sense but also the L1-norm (not the function).

Multi-frequency data. The multi-frequency data are obtained from solutions to the corre-sponding boundary value problem for the wave equation by applying a Fourier transform. Let ⌦be a (bounded) Lipschitz domain in R3 and c = c(x) be a strictly positive bounded measurablefunction. We consider the boundary value problem for the wave equation

8><

>:

@2

t u(x, t)� c2(x)�u(x, t) = 0, (x, t) 2 ⌦⇥ (0,1),

u(x, t) = f(x, t), (x, t) 2 @⌦⇥ (0,1),

u(x, 0) = 0, @tu(x, 0) = 0, x 2 ⌦.

The hyperbolic Dirichlet-to-Neumann map, ⇤c�2 , is given by

⇤c�2 : H ! L2(@⌦⇥ R+),

f 7! @⌫uf |@⌦⇥(0,1)

,

where @⌫ denotes the normal derivative at @⌦ and H = {f 2 H1(@⌦ ⇥ R+) | f(x, 0) = 0}. One,indeed, may take the Fourier transform of @⌫uf , since it is a tempered distributions [18], and thusobtain multi-frequency data.

2. Direct problem. We describe the direct problem and some properties of the data, that is,the Dirichlet-to-Neumann map. We consider the boundary value problem,

(2.1)

⇢(�� !2c(x)�2)u = 0, in ⌦,

u = g on @⌦.

If the boundary value g is in H3/2(@⌦) and c(x) is bounded and measurable, then the uniquesolution to (2.1) belongs to H2(⌦). Therefore ru is in H1(⌦) and as a consequence ru |@⌦ belongsto H1/2(@⌦). One can then introduce the Dirichlet-to-Neumann map,

(2.2) ⇤!2c�2 g = ru · ⌫ |@⌦=@u

@⌫|@⌦2 H�1/2(@⌦).

We summarize some results which we will use in the proofs of the properties of the Dirichlet-to-Neumann map.

THE HELMHOLTZ EQUATION WITH MULTI-FREQUENCY DATA 187

Proposition 2.1. Let ⌦ be a bounded Lipschitz domain in R3, c�2(x) be a strictly positive

bounded measurable function and f 2 Lp(⌦), g 2 W 2� 1

p

,p(@⌦) with 1 < p < 1. Assume that 0 isnot a Dirichlet eigenvalue of �� !2c�2 in ⌦. Then, there exists a unique solution u 2 W 2,p(⌦)to the problem

(2.3)

⇢(�� !2c(x)�2)u = f, x 2 ⌦,

u = g, x 2 @⌦.

Moreover, assume that !2

0

kc�2kL1(⌦)

is strictly less than the first Dirichlet eigenvalue of the Lapla-cian on ⌦. Then, for any 0 < ! < !

0

,

(2.4) kukW 2,p

(⌦)

C(kgkW

2� 1

p

,p

(@⌦)

+ kfkLp

(⌦)

)

where C depends on ⌦.The proof makes use of the existence of a W 2,p(⌦) function w such that w = g on @⌦ and

kwkW 2,p

(⌦)

CkgkW

2� 1

p

,p

(@⌦)

, and of the Fredholm alternative; see for example Theorem 3.5.8 in

Feldman and Uhlmann’s notes [17]). For the reader’s convenience, we also mention the followingProposition 2.2 without proof, which we use for the case of L2.

Proposition 2.2. Let ⌦ be a bounded Lipschitz domain in R3, c�2(x) be a strictly positivebounded measurable function and f 2 H�1(⌦), g 2 H1/2(@⌦). Assume that 0 is not a Dirichleteigenvalue of the operator �� !2c�2 in ⌦. Then there exists a unique solution u 2 H1(⌦) of(2.3). Moreover, assume that !2

0

kc�2kL1(⌦)

is strictly less than the first Dirichlet eigenvalue of theLaplacian on ⌦. Then, for any 0 < ! < !

0

,

(2.5) kukH1

(⌦)

C�kgkH1/2

(@⌦)

+ kfkH�1

(⌦)

�,

where C depends on ⌦.Lemma 2.3 (Frechet di↵erentiability). Assume that 0 is not a Dirichlet eigenvalue of ��

!2c�2 in ⌦. The operator, F!, given by

F! : Lp(⌦) \ L1(⌦) ! L(H1/2(@⌦), H�1/2(@⌦)),

c�2(x) 7! ⇤!2c�2 ,

is Frechet di↵erentiable at c�2.Proof. We start from Alessandrini’s identity,

(2.6)

Z

⌦

!2(c�2

1

� c�2

2

)u1

u2

dx = h(⇤!2c�2

1

� ⇤!2c�2

2

)u1

, u2

i

where u1

and u2

are the solutions of the Helmholtz equation with Dirichlet boundary conditionand coe�cient c

1

and c2

, respectively. Let �c�2 2 L1(⌦). We observe, while substituting c�2 andc�2 + �c�2 for c�2

1

and c�2

2

, that

(2.7) h(⇤!2

(c�2

+�c�2

)

� ⇤!2c�2)g , hi = !2

Z

⌦

�c�2 uv dx,

where u and v solve the boundary value problems,⇢

(�� !2(c�2 + �c�2))u = 0, x 2 ⌦,u = g, x 2 @⌦,

and⇢

(�� !2c�2)v = 0, x 2 ⌦,v = h, x 2 @⌦,


respectively. We show that

(2.8) hDF!(c�2)(�c�2)g , hi = !2

Z

⌦

�c�2 uv dx,

where u solves the equation

⇢(�� !2c�2)u = 0, x 2 ⌦,

u = g, x 2 @⌦.

In fact, by (2.7), we have that

(2.9) h(⇤!2

(c�2

+�c�2

)

� ⇤!2c�2)g , hi � !2

Z

⌦

�c�2 uv dx = !2

Z

⌦

�c�2 (u� u)v dx.

By using the Holder inequality twice and the Sobolev embedding theorem, we obtain that

(2.10)

��!2

Z

⌦

�c�2 (u� u)v dx

�� !2k�c�2kLp

(⌦)

ku� ukL2q

(⌦)

kvkL2q

(⌦)

,

where 1

p + 1

q = 1. We note that u� u solves the equations

⇢(�� !2c�2)(u� u) = �!2�c�2 u, x 2 ⌦,

u� u = 0, x 2 @⌦.

Therefore, for 3/2 p 9/4, by the Sobolev embedding theorem and Proposition 2.1, we find that

ku� ukL2q

(⌦)

Cku� ukW

2,

6q

4q+3

(⌦)

(2.11)

Ck!2�c�2 ukL

6q

4q+3

(⌦)

C!2k�c�2kLp

(⌦)

kukL

6q

9�2q

(⌦)

,

with 2 6q9�2q 6. The right-most inequality is obtained by using the Holder inequality. Similarly,

for 9/4 < p < 9/2, we have that

ku� ukL2q

(⌦)

Cku� ukW

1,

6q

2q+3

(⌦)

(2.12)

Ck!2�c�2 ukL

6q

2q+3

(⌦)

C!2k�c�2kLp

(⌦)

kukL

6q

9�4q

(⌦)

,

with 2 6q9�4q 6. For p � 9/2, we get

(2.13) ku� ukL2q

(⌦)

ku� ukW 2,2q

(⌦)

Ck!2�c�2 ukL2q

(⌦)

C!2k�c�2kLp

(⌦)

kukL

2q

3�2q

(⌦)

,

with 2 < 2q3�2q 6. By using the interpolation of Lp spaces,

kukLp

✓

(⌦)

kuk1�✓L2

(⌦)

kuk✓L6

(⌦)

, 8✓ 2 [0, 1],

with p✓ defined by 1/p✓ = (1� ✓)/2 + ✓/6, we conclude that, for any p � 3/2,

(2.14) ku� ukL2q

(⌦)

C!2k�c�2kLp

(⌦)

kuk1�✓L2

(⌦)

kuk✓L6

(⌦)

,

for some ✓ 2 [0, 1].


Then, upon substituting (2.14) into (2.10) and applying the Sobolev embedding theorem andProposition 2.2 to u and v, we conclude that

��h(⇤!2

(c�2

+�c�2

)

� ⇤!2c�2)g , hi � !2

Z

⌦

�c�2 uv dx

��

=

��!2

Z

⌦


��

!2k�c�2kLp

(⌦)

ku� ukL2q

(⌦)

kvkL2q

(⌦)

C!4k�c�2k2Lp

(⌦)

kuk1�✓1

L2

(⌦)

kuk✓1L6

(⌦)

kvk1�✓2

L2

(⌦)

kvk✓2L6

(⌦)

C!4k�c�2k2Lp

(⌦)

kukH1/2

(@⌦)

kvkH1/2

(@⌦)

,

for some constant C and ✓1

, ✓2

2 [0, 1]. This leads to the Frechet di↵erentiability of F! at c�2.We introduce a uniform constant C

0

such that any wavespeed function in the analysis satisfies

kc�2kL1(⌦)

C0

.

We assume that C0

!2

0

is less than the first Dirichlet eigenvalue of the Laplacian on ⌦. The use of auniform constant can be relaxed.

Lemma 2.4. There exists a constant L0

, which depends on ⌦, such that

(2.15) kDF!(c�2)k

L(Lp

(⌦),L(H1

2

(@⌦),H� 1

2

(@⌦)))

L0

!2.

for ! !0

.Proof. We start from Alessandrini’s identity (2.6). By applying the Holder inequality twice, we

find that

|h(⇤!2c�2

1

� ⇤!2c�2

2

)u1

, u2

i| =��Z

⌦

!2(c�2

1

� c�2

2

)u1

u2

dx

��

!2kc�2

1

� c�2

2

kLp

(⌦)

ku1

kL2q

(⌦)

ku2

kL2q

(⌦)

.

By the interpolation of Lp spaces, the Sobolev embedding theorem and Proposition 2.2, we obtain

kuikL2q

(⌦)

CkuikH1

(⌦)

CkuikH1/2

(@⌦)

, i = 1, 2.

Hence,

k⇤!2c�2

1

� ⇤!2c�2

2

kL(H

1

2

(@⌦),H� 1

2

(@⌦))

L0

!2kc�2

1

� c�2

2

kLp

(⌦)

from which (2.15) follows.Lemma 2.5. For any c�2

1

, c�2

2

2 L1(⌦) strictly positive and bounded and 0 ! !0

, thereexists a constant L

0


kDF!(c�2

1

)�DF!(c�2

2

)kL(Lp

(⌦),L(H1

2

(@⌦),H� 1

2

(@⌦)))

L0

!4kc�2

1

� c�2

2

kLp

(⌦)

.

Proof. Let g, h 2 H1/2(⌦) and ui, vi, i = 1, 2, solve the boundary value problems,

⇢(�� !2c�2

i )ui = 0, x 2 ⌦,ui = h, x 2 @⌦,

⇢(�� !2c�2

i )vi = 0, x 2 ⌦,vi = g, x 2 @⌦,


resepcectively. By usingidentity (2.8) and applying the Holder inequality twice, we have

|h(DF!(c�2

1

)(�c�2)�DF!(c�2

2

)(�c�2))g , hi|

=

��!2

Z

⌦

�c�2 (u1

v1

� u2

v2

) dx

��

!2k�c�2kLp

(⌦)

(ku1

� u2

kL2q

(⌦)

kv1

kL2q

(⌦)

+ ku2

kL2q

(⌦)

kv1

� v2

kL2q

(⌦)

).

We note that u1

� u2

solves the equations⇢

(�� !2c�2

1

)(u1

� u2

) = !2(c�2

1

� c�2

2

)u2

, x 2 ⌦,u1

� u2

= 0, x 2 @⌦.

Using an argument similar to the one in the proof of Lemma 2.3, we derive that

ku1

� u2

kL2q

(⌦)

C!2kc�2

1

� c�2

2

kLp

(⌦)

ku2

kL2q

(⌦)

C!2kc�2

1

� c�2

2

kLp

(⌦)

kgkH

1

2

(@⌦)

and, analogously,

kv1

� v2

kL2q

(⌦)

C!2kc�2

1

� c�2

2

kLp

(⌦)

kv2

kL2q

(⌦)

C!2kc�2

1

� c�2

2

kLp

(⌦)

khkH

1

2

(@⌦)

.

Hence

|h(DF!(c�2

1

)(�c�2)�DF!(c�2

2

)(�c�2))g , hi| C!4k�c�2kLp

(⌦)

kc�2

1

� c�2

2

kLp

(⌦)

kgkH

1

2

(@⌦)

khkH

1

2

(@⌦)

,

which gives that

kDF!(c�2

1

)�DF!(c�2

2

)kL(Lp

(⌦),L(H1

2

(@⌦),H� 1

2

(@⌦)))

C!4kc�2

1

� c�2

2

kLp

(⌦)

.

3. Stability of the inverse problem. We invoke the followingAssumption 3.1. ⌦ ⇢ Rn is a bounded domain satisfying

|⌦| A

Here and in the sequel |⌦| denotes the Lebesgue measure of ⌦. We assume that @⌦ is of Lipschitzclass and we fix an open portion ⌃ of @⌦ which is of Lipschitz class with constants r

0

and L.A domain partitioning of ⌦ is given by

(3.1) DN , {{D1

, D2

, . . . , DN} |N[

j=1

Dj = ⌦ , (Dj \Dj0)� = ;}.

Assumption 3.2. The wavespeed function c(x) satisfies

kckL1(⌦)

B1

, kc�1kL1(⌦)

B2

,

where B1

and B2

are positive constants, and is of the form

c(x) =NX

j=1

cj�Dj

(x),

where cj , j = 1, . . . N are unknown numbers and Dj are known open sets in Rn which satisfy thefollowing assumption.


Assumption 3.3. The Dj , j = 1, . . . , N are connected and @Dj are of Lipschitz class. Thereexists one set, say D

1

, such that @D1

\ @⌦ contains an open portion ⌃1

of Lipschitz class withconstants r

0

and L. For every j 2 {2, . . . , N} there exist j1

, . . . , jM 2 {1, . . . , N} such that

Dj1

= D1

, DjM

= Dj

and, for every k = 1, . . . ,M ,

@Djk�1

\ @Djk

contains a non-empty open portion ⌃k of Lipschitz class with constants r0

and L such that

⌃1

⇢ ⌃,⌃k ⇢ ⌦, 8k = 2, . . . ,M.

Furthermore there exists Pk 2 ⌃k, at which Dk�1

satisfies the interior ball condition with radius3r

0

16

, and a rigid transformation of coordinates such that Pk = 0 and

⌃k \Qr0

/3 = {x 2 Qr0

/3 | xn = �k(x0)},Dj

k

\Qr0

/3 = {x 2 Qr0

/3 | xn > �k(x0)},Dj

k�1

\Qr0

/3 = {x 2 Qr0

/3 | xn < �k(x0)},

where �k is a C0,1 function on B0r0

/3 satisfying

�k(0) = 0

and

k�kkC0,1

(B0r

0

/3

)

L.

For simplicity, we call Dj1

, . . . , DjM

a chain of domains connecting D1

to Dj.Definition 3.4. Let ⌦ be a bounded open subset of R3 and of Lipschitz class and ⌃ be a open

portion of @⌦. We define H1/2co (⌃) as

H1/2co (⌃) = {g 2 H1/2(@⌦) | supp g ⇢ ⌃}

and H�1/2co (⌃) as the topological dual of H1/2

co (⌃); we denote by h·, ·i the dual pairing between H1/2co (⌃)

and H�1/2co (⌃).

We define the local Dirichlet-to-Neumann map ⇤(⌃)

q as

⇤(⌃)

q : H1/2co (⌃) ! H

�1/2co (⌃)

g 7! @u

@⌫

��⌃

,

where u solves (2.1) and ⌫ is the exterior unit normal vector to @⌦.Note that Lemma 2.3, 2.4, 2.5, 4.1, 4.2 and 4.3 also hold true for the local Dirichlet-to-Neumann

map.We define the nonlinear operator-valued map F!,D

N

as

(3.2)F!,D

N

: span(�D1

, . . . ,�DN

) ! L(H 1

2 (⌃), H� 1

2 (⌃))

c 7! ⇤(⌃)

!2c�2

.

The convergence of our multi-level (frequency, domain partitioning) iterative scheme relies onthe Frechet di↵erentiability of this map, its boundedness, and the stability of its inverse.


Lemma 3.5 ([9]). Let ⌦ satisfy Assumption 3.1 and ck, k = 1, 2 be two piecewise constantfunctions of the form

ck(x) =NX

j=1

ck,j�Dj

(x), k = 1, 2

which satisfy Assumption 3.2 and Dj , j = 1, . . . , N satisfy Assumption 3.3. Then, there exists aconstant C

0

= C0

(r0

, L,A,B,N,!0

), such that

(3.3) kc�2

1

� c�2

2

kLp

(⌦)

C0

k⇤(⌃)

1

� ⇤(⌃)

2

kL(H1/2

co

(⌃),H�1/2

co

(⌃))

!2

,

where ⇤(⌃)

k = ⇤(⌃)

!2c�2

k

for k = 1, 2.

We note that !�2k⇤(⌃)

1

� ⇤(⌃)

2

kL(H1/2

co

(⌃),H�1/2

co

(⌃))

does not blow up as ! tends to zero.

Remark 3.6. The lower bound of the Frechet derivative, DF!, has been used to study thestability properties [4, 2, 3, 11]. A standard proposition states the following. Let M

1

and M2

bepositive numbers and N be a positive integer. Let A and K be an open subset and a compact subsetof RN , respectively. Assume that K ⇢ A,

dist�K,RN \ A

�� M

1

and K ⇢ BM2

(0).

Let B be a Banach space and let F : A ! B be such that:(i) F is Frechet di↵erentiable;(ii) the Frechet derivative DF : A ! L(RN ,B) is uniformly continuous with a modulus of conti-

nuity �1

(·);(iii) F|K is injective;(iv) (F|K)

�1 : F (K) ! K is uniformly continuous with a modulus of continuity �2

(·);(v) DF is injective in K, namely, there is a positive number C

0

such that

minx2K,|h|=1

kDF (x)(h)kB � C0

;

then

kx1

� x2

kRN CkF (x1

)� F (x2

)kB for every x1

, x2

2 K,

where C = max{ 2M1

��1

2

(�1

)

, 2

C0

}, for �1

= 1

2

min{�0

,M2

} with �0

= ��1

1

(C0

2

).

Thus a positive estimate of the lower bound of the Frechet derivative implies a Lipschitz typestability estimate for the inverse. It holds true also in infinite dimensional spaces. For a proof infinite dimensional spaces, we refer to [3].

Proposition 3.7. Let ck, k = 1, 2 satisfy the assumptions in Lemma 3.5. Then there existconstants C

0

and K1

, such that

(3.4) kc�2

1

� c�2

2

kLp

(⌦)

C0

eK1

N7/2

k⇤1

� ⇤2

kL(H1/2

(@⌦),H�1/2

(@⌦))

!2

,

where ⇤k = ⇤(@⌦)

!2c�2

k

for k = 1, 2 and C0

and K1

depend only on A,L, r0

, B1

and B2

.

Proof. We start the proof with two facts. The first fact is that, if the bounded measurablefunction c�2(x) is of the form

c(x) =NX

j=1

cj�Dj

(x),


then its Hs0 norm can be bounded by its L1 norm for every 0 < s0 < 1/2. That is, kckHs

0 C,where the constant C depends only on A,B

1

, B2

, L, r0

and N . The second fact is on the existenceof so-called complex geometrical optics (CGO) solutions of the Helmholtz equation. Assume thatc�2 2 L1(⌦), kc�2kL1

(⌦)

B2

, then the equation

��u� !2c�2u = 0

has a solution u, which is of the form

u(x) = expix·⇣(1 + r(x)),

where r 2 H1(⌦) satisfies

krkL2

(⌦)

C

|⇣|!2kc�2kL2

(⌦)

,

krrkL2

(⌦)

C!2kc�2kL2

(⌦)

,

for a constant C which only depends on ⌦. For a proof, we refer to [23].Now, we claim that

(3.5) !2kc�2

1

� c�2

2

kL2

(⌦)

CEN1/2�

✓k⇤

1

� ⇤2

kE

◆

where

(3.6) �(t) =

⇢| log t|� 1

7 for 0 < t < 1

e ,et for t � 1

e ,

and where E = !2kc�2

1

� c�2

2

kL1(⌦)

. To proof (3.5), we start from Alessandrini’s identityZ

⌦

!2(c�2

1

� c�2

2

)u1

u2

= h(⇤1

� ⇤2

)u1

|@⌦, u2

|@⌦i,

for any uj 2 H1(⌦) solution of ��uj � !2c�2

j uj = 0 for j = 1, 2. Let ⇠ 2 R3 and let ⇣1

and ⇣2

beunit vectors of R3 such that {⇣

1

, ⇣2

, ⇠} is an orthogonal set. We take

⇣1

=sp2

r1� |⇠|2

2s2⇣1

+1p2s

+ i⇣2

!,

⇣2

= � sp2

r1� |⇠|2

2s2⇣1

� 1p2s

+ i⇣2

'

!.

Then, ⇣j · ⇣j = 0 for j = 1, 2 and |⇣1

| = |⇣2

| = s and ⇣1

+ ⇣2

= ⇠. Assume that uj are solutions to��uj � !2c�2

j uj = 0 for j = 1, 2 of the form

u1

(x) = eix·⇣1(1 + r1

(x)), u2

(x) = eix·⇣2(1 + r2

(x))

provided that |⇣| = s � max(C0

B2

, 1) with

krjkL2

(⌦)

C0

sB

2

for j = 1, 2 and C0

= C0

(⌦). Inserting the solutions u1

and u2

in Alessandrini’s identity we derive��

Z

⌦

!2(c�2

1

� c�2

2

)ei⇠·xdx

��(3.7)

k⇤1

� ⇤2

kku1

kH1/2

(@⌦)

ku2

kH1/2

(@⌦)

+

��Z

⌦

!2(c�2

1

� c�2

2

)ei⇠·x(r1

+ r2

+ r1

r2

)dx

��

k⇤1

� ⇤2

kku1

kH1

(⌦)

ku2

kH1

(⌦)

+ CE(kr1

kL2

(⌦)

+ kr2

kL2

(⌦)

+ kr1

kL2

(⌦)

kr2

kL2

(⌦)

).


If ⌦ ⇢ BR(0) then

kujkH1

(⌦)

CseRs, j = 1, 2.

Let s be large enough so that s eRs. Then, for s � C 0, we have

(3.8) |!2(c�2

1

� c�2

2

) (⇠)| C

✓e4Rsk⇤

1

� ⇤2

k+ E

s

◆

where the !2c�2

j ’s have been extended to all R3 by zero. Hence we get

(3.9) k!2(c�2

1

� c�2

2

) k2L2

(R3

)

C⇢3(e8Rsk⇤1

� ⇤2

k2 + E2

s2) +

Z

|⇠|�⇢

|!2(c�2

1

� c�2

2

) (⇠)|2 d⇠

The we have that

k!2(c�2

1

� c�2

2

)k2Hs

0(⌦)

CNE2,

where C depends on A,L, r0

and on s0 and hence

⇢2s0Z

|⇠|�⇢

|!2(c�2

1

� c�2

2

) (⇠)|2 d⇠ Z

|⇠|�⇢

|⇠|2s0|!2(c�2

1

� c�2

2

) (⇠)|2 d⇠

Z

R3

(1 + |⇠|2)s0|!2(c�2

1

� c�2

2

) (⇠)|2 d⇠ CNE2.

HenceZ

|⇠|�⇢

|!2(c�2

1

� c�2

2

) (⇠)|2 d⇠ CNE2

⇢2s0

for any s0 2 (0, 1/2). Inserting last bound in (3.9) we derive

k!2(c�2

1

� c�2

2

) k2L2

(R3

)

C⇢3(e8Rsk⇤1

� ⇤2

k2 + E2

s2) +

CNE2

⇢2s0.

Picking up ⇢ = s2

3+2s

0 and inserting it into last relation we obtain

k!2(c�2

1

� c�2

2

)k2L2

(⌦)

CNE2

e16Rs

✓k⇤

1

� ⇤2

kE

◆2

+1

s4s

03+2s

0

!

for s � C 0. Let us choose

s =1

16R

��lnk⇤

1

� ⇤2

kE

��

where we have assumed that

k⇤1

� ⇤2

kE

< c

so that s � C 0. Under this assumption,

k!2(c�2

1

� c�2

2

)kL2

(⌦)

CN1/2E

0

@✓k⇤

1

� ⇤2

kE

◆+

��lnk⇤

1

� ⇤2

kE

�� 2s

03+2s

0

1

A


On the other hand if

k⇤1

� ⇤2

kE

� c,

then

k!2(c�2

1

� c�2

2

)kL2

(⌦)

N1/2E CN1/2Ek⇤

1

� ⇤2

kE

Finally, choosing s0 = 1/4, the claim (3.5) follows.Now observing that

kc�2

1

� c�2

2

kL1(⌦)

C(r0

)kc�2

1

� c�2

2

kL2

(⌦)

,

we derive

k!2(c�2

1

� c�2

2

)kL1(⌦)

CN1/2E�

✓k⇤

1

� ⇤2

kE

◆,

so that

E CN1/2E�

✓k⇤

1

� ⇤2

kE

◆,

which gives

E 1

��1(CN�1/2)k⇤

1

� ⇤2

k,

and finally recalling the expression of � we get

E C0

eK1

N7/2

k⇤1

� ⇤2

k,

where C0

and K1

depends only on A,B1

, B2

, L, r0

.

Approximation error estimates. For a given domain partitioning, we introduce the function

' : N 7! C�1

0

e�K1

N7/2

,

which will play the role of compression rate.Definition 3.8. For a family of given domain partitionings {DN}, the corresponding family

of admissible sets {AN} is given by

AN , {c 2 L1(⌦) | distLp

(⌦)

(c�2, span(�D1

,�D2

, . . . ,�DN

)) '(N)}.

Definition 3.9. For a given admissible set AN , the approximation error ⌘!,DN

is defined by

⌘!,DN

: Y ! R+

y 7! dist(y, F!,DN

(AN )).

For any given data, y, from Lemma 2.4, it immediately follows that

(3.10) ⌘!,DN

(y) L0

!2'(N).


4. Formulation in lp. Here, we describe the direct problem and some properties of theDirichlet-to-Neumann map in terms of a coe�cient space identified as a subspace embedded inlp. We revisit the forward operator-valued map and define

(4.1)F!,D

N

: lp ! L(H1

2

co(⌃), H� 1

2

co (⌃))

(c�2

1

, . . . , c�2

N , 0, . . . ) 7! ⇤(⌃)

!2

PN

j=1

c�2

j

�D

j

.

The following lemma is an analogue of Lemma 2.3.Lemma 4.1 (Frechet di↵erentiability). Let ⌦ satisfy Assumption 3.1 and c, �c�2 be functions

of the form

c(x) =NX

j=1

cj�Dj

(x), �c�2(x) =NX

j=1

�c�2

j �Dj

(x),

which satisfy Assumption 3.2 and Dj , j = 1, . . . , N satisfy Assumption 3.3. Assume that 0 isnot a Dirichlet eigenvalue of �� !2c�2 in ⌦. Then, the map F!,D

N

given in (4.1) is Frechetdi↵erentiable at c�2.

Proof. We follow and adapt the proof of Lemma 2.3. We continue from (2.9). By using theHolder inequality twice and the Sobolev embedding theorem, we find that

(4.2)

��!2

Z

⌦


��

!2

��c�2

NX

j=1

(|Dj |�Dj

)�1/p

��Lp

(⌦)

��

NX

j=1

(|Dj |�Dj

)1/p(u� u)v

��Lq

(⌦)

!2 maxj

|Dj |1/p k�c�2klpku� ukL2q

(⌦)

kvkL2q

(⌦)

.

We note that u� u solves⇢

(�� !2c�2)(u� u) = �!2�c�2 u, x 2 ⌦,u� u = 0, x 2 @⌦.

Again, following the proof of Lemma 2.3, we obtain the estimate

(4.3) ku� ukL2q

(⌦)

C!2 maxj

|Dj |1/pk�c�2klpkuk1�✓L2

(⌦)

kuk✓L6

(⌦)

,

for some ✓ 2 [0, 1]. Then, substituting (4.3) into (4.2) and applying the Sobolev embedding theoremand Proposition 2.2 to u and v, we conclude that

��h(⇤!2

(c�2

+�c�2

)

� ⇤!2c�2)g , hi � !2

Z

⌦

�c�2 uv dx

��

=

��!2

Z

⌦


��

!2 maxj

|Dj |1/p k�c�2klpku� ukL2q

(⌦)

kvkL2q

(⌦)

C!4 maxj

|Dj |2/p k�c�2k2lpkukH1/2

(@⌦)

kvkH1/2

(@⌦)

,

for some constant C. This implies the Frechet di↵erentiability of F!,DN

at c�2.The following lemma is an analogue of Lemma 2.4.


Lemma 4.2. Let ⌦ satisfy Assumption 3.1 and c1

and c2

be functions of the form

ci(x) =NX

j=1

ci,j�Dj

(x), i = 1, 2

which satisfy Assumption 3.2 and Dj , j = 1, . . . , N satisfy Assumption 3.3. Let 0 ! !0

. Then

there exists a constant L0

, which depends on ⌦ , such that

(4.4) kDF!,DN

(c�2

1

)kL(H

1

2

(@⌦),H� 1

2

(@⌦))

L0

!2 maxj

|Dj |1/p.

Proof. We start from the Alessandrini’s identity

(4.5)

Z

⌦

!2(c�2

1

� c�2

2

)u1

u2

dx = h(⇤!2c�2

1

� ⇤!2c�2

2

)u1

, u2

i

where u1

and u2

are the solutions of the Helmholtz equation with wavespeeds c1

and c2

, respectively,subject to the Dirichlet boundary condition. Then, by applying the Holder inequality twice, we havethat

|h(⇤!2c�2

1

� ⇤!2c�2

2

)u1

, u2

i| =��Z

⌦

!2(c�2

1

� c�2

2

)u1

u2

dx

��

!2

��(c�2

1

� c�2

2

)NX

j=1

(|Dj |�Dj

)�1/p

��Lp

(⌦)

��

NX

j=1

(|Dj |�Dj

)1/pu1

u2

��Lq

(⌦)

!2 maxj

|Dj |1/p kc�2

1

� c�2

2

klpku1

kL2q

(⌦)

ku2

kL2q

(⌦)

.

By the Sobolev embedding theorem and Proposition 2.2, we obtain that

kuikL2q

(⌦)

CkuikH1

(⌦)

CkuikH1/2

(@⌦)

, i = 1, 2.

We conclude that

k⇤!2c�2

1

� ⇤!2c�2

2

kL(H

1

2

(@⌦),H� 1

2

(@⌦))

L0

!2 maxj

|Dj |1/pkc�2

1

� c�2

2

kl3

2

and (4.4) follows.In the following lemma, an analogue of Lemma 2.5, we discuss the Lipschitz continuity of

DF!,DN

.Lemma 4.3. Let ⌦ satisfy Assumption 3.1 and c

1

and c2

be two functions of the form

ci(x) =NX

j=1

ci,j�Dj

(x), i = 1, 2

which satisfy Assumption 3.2 and Dj , j = 1, . . . , N satisfy Assumption 3.3. Let 0 ! !0

. Thenthere exists a constant L

0


kDF!,DN

(c�2

1

)�DF!,DN

(c�2

2

)kL(lp,L(H

1

2

(@⌦),H� 1

2

(@⌦)))

L0

!4 maxj

|Dj |2/pkc�2

1

� c�2

2

klp .

Proof. We let �c�2 have the same form as c1

and c2

. Then

|h(DF!,DN

(�c�2)�DF!,DN

(c�2

2

)(�c�2))u1

, u2

i|

=

��!2

Z

⌦

�c�2 (u1

v1

� u2

v2

) dx

��

!2 maxj

|Dj |1/pk�c�2klp(ku1

� u2

kL2q

(⌦)

kv1

kL2q

(⌦)

+ ku2

kL2q

(⌦)

kv1

� v2

kL2q

(⌦)

).


Using an argument similar to the one in the proof of Lemma 4.1, we find the estimate

ku1

� u2

kL2q

(⌦)

C!2 maxj

|Dj |1/pkc�2

1

� c�2

2

klpku2

kL2q

(⌦)

(4.6)

Cmaxj

|Dj |1/pkc�2

1

� c�2

2

klp ku2

kH

1

2

(@⌦)

and analogously

kv1

� v2

kL2q

(⌦)

C!2 maxj

|Dj |1/pkc�2

1

� c�2

2

klpkv2kL2q

(⌦)

(4.7)

Cmaxj

|Dj |1/pkc�2

1

� c�2

2

klpkv2kH

1

2

(@⌦)

.

Hence

|h(DF!,DN

(c�2

1

)(�c�2)�DF!,DN

(c�2

2

)(�c�2))g , hi| C!4 max

j|Dj |2/pk�c�2klpkc�2

1

� c�2

2

klpku1

kH

1

2

(@⌦)

ku2

kH

1

2

(@⌦)

,

which gives that

kDF!,DN

(c�2

1

)�DF!,DN

(c�2

2

)kL(lp,L(H

1

2

(@⌦),H� 1

2

(@⌦)))

C!4 maxj

|Dj |2/pkc�2

1

� c�2

2

klp .

The key reason to follow a formulation in lp is that the dependencies of the constants on thedomains are ‘weaker’ in the sense that they depend on a fractional power of the measures of thesets in the domain partitioning only.

5. Projected steepest descent iteration for Lp(⌦). In this section, we described the pro-jected steepest descent iteration proposed in [16] for Lp and the corresponding constants. Wesummarize some basic notions associated with iterative methods in Banach spaces.

We let 1 < p, q < 1 be conjugate exponents,

1

p+

1

q= 1.

For p > 1, the subdi↵erential mapping Jp = @fp : X ! 2X⇤of the convex functional fp : x 7! 1

pkxkp

defined by

(5.1) Jp(x) = {x⇤ 2 X⇤ | hx, x⇤i = kxk kx⇤k and kx⇤k = kxkp�1}

is called the duality mapping of X with gauge function t 7! tp�1. Generally, the duality mapping isset-valued. For a detailed introduction to the geometry of Banach spaces and the duality mapping,we refer to [15, 21].

Definition 5.1. Let X be a uniformly smooth Banach space. The Bregman distance �2

(x, ·)of the convex functional x 7! 1

2

kxk2 at x 2 X is defined as

(5.2) �2

(x, x) =1

2kxk2 � 1

2kxk2 � hJ

2

(x), x� xi, x 2 X,

where J2

denotes the normalized duality mapping of X.In the next lemma, we summarize some facts we need here concerning the duality mapping and

Bregman distance for Lp(⌦).Lemma 5.2. For the Banach spaces Lp(⌦), the following holds true:

(a) The normalized duality mapping J2

for Lp(⌦) is single-valued and defined by

J2

: Lp(⌦) ! Lq(⌦)

f(x) 7! kfk2�pLp

(⌦)

|f(x)|p�2 f(x),


(b) For all f, f 2 Lp(⌦), we have that

(5.3) �2

(f, f) � 1

2kf � fkpLp

(⌦)

.

The Bregman distance �2

is similar to a metric, but, in general, does not satisfy the triangleinequality nor symmetry. For the Hilbert space L2(⌦),

�2

(f, f) =1

2kf � fk2L2

(⌦)

.

To constrain the iterates to a subset where Lipschitz type stability holds, we use the non-expansiveBregman projection for Lp spaces [16]. A comprehensive introduction to this topic can be found in[13].

Definition 5.3. Let X be a uniformly smooth Banach space. Given a closed convex set Z ⇢ Xand Bregman distance �

2

, which is defined in Definition 5.2, the Bregman projection of a pointx 2 X onto Z is the point

(5.4) PZ(x) = argminy2Z

�2

(y, x).

In the case of Lp(⌦), the Bregman projection is given by

PZ(f) = argming2Z

⇢1

2kfk2Lp

(⌦)

+1

2kgk2Lp

(⌦)

� kgk2�pLp

(⌦)

Z

⌦

|g(x)|p�2g(x)f(x) dx

�.

Remark 5.4. Assume that {Dj}Nj=1

is a domain partitioning of ⌦ as in Section 3. Let Z bedefined by Z = span{�D

1

, . . . ,�DN

}. For p = 2, we have that

PZ(f) =NX

j=1

gj �Dj

with gj =1

|Dj |

Z

Dj

f(x) dx.

The following projected steepest descent iteration is taken from [16]. We identify, if we restrictour analysis to lp,

(5.5)

L = L0

!2 maxj

|Dj |1/p,

L = L0

!4 maxj

|Dj |2/p,

C = C0

!�2eK1

N7/2

,

and F!,DN

with (4.1).Algorithm 5.5. We fix some abbreviations first: For c�2

k , k = 0, 1, 2, . . ., fixed denote

(5.6) Rk = F (c�2

k )� y� , Tk = DF (c�2

k )⇤j2

(F (c�2

k )� y�) , rk = kRkk , tk = kTkk .

Moreover, we define

(5.7)

C := LC2 ,

⇢ :=1

2(2CL)�2

✓1 +

q1� 8C⌘ � 4⌘C

◆2

.

and for k = 0, 1, . . .

(5.8)

uk := �Cr2k + (1� 2C⌘)rk � ⌘ � C⌘2 ,

vk := t�2

k ukr2

k(rk � ⌘)� 1

2t�2

k u2

kr2

k ,

wk := Lt�2

k ukr2

k ,

µk := t�2

k ukrk .


The algorithm is given by(S0) Choose a starting point x

0

2 Z such that

(5.9) �2

(x0

, z†) < ⇢,

(S1) Compute the new iterate via

(5.10)c�2

k+1

= J⇤2

(J2

(c�2

k )� µkTk)

c�2

k+1

= PZ(c�2

k+1

).

Set k k + 1 and repeat step (S1).Due to the projection applied, all iterates belong to the ‘stable subset’ (corresponding with a

domain partitioning), Z, which in general can only o↵er an approximation to the unique solution.While the dimension of Z should be low to ensure a large radius of convergence, the approximationshould be compressive. In [16] we introduced a multi-level approach to enable a gradual refinementof the domain partitioning defining subsets Zn where n stands for the level; Zn is given by

(5.11) Zn = {c�2 2 span(�D1

, . . . ,�DN

n

)}.

As the level index n increases, the number of subdomains, Nn, grows, and hence, the n-level domainpartitioning DN

n

refines. In the following algorithm, c�2

n,k denotes the kth iterate at level n. Weidentify F!

n

,DN

n

with Fn and ⌘!n

,DN

n

with ⌘n. For each frequency !n, the noisy data are written

as y�. We identify Ln, Ln and Cn with (5.5) replacing ! by !n. See Lemma 2.4, Lemma 2.5 andProposition 3.7. For simplicity, we omit the subscript in the operator norm. It is natural to let thefrequency !n increase with increasing n.

Algorithm 5.6.(S0) Use c

0,0 as the starting point. Set n = 0.

(S1) Iteration. Use Fn and Zn as the modelling operator and convex subset to run Algorithm 5.5with the discrepancy criterion given by

(5.12) Kn = min{k 2 N | kFn(c�2

n,k)� y�k (3 + ")⌘n},

where " > 0 is a given uniform tolerance constant.STOP, if n = N , a given number.

(S2) Set cn+1,0 = cn,Kn

; Refine the domain partitioning to Dn+1

; Choose frequency !n+1

such that

the corresponding parameters Ln+1

, Ln+1

, Cn+1

and the approximation error ⌘n+1

satisfy thefollowing inequality

(5.13) (3 + ")⌘n < 2�1/2(Ln+1

Cn+1

)�1

0

@1 +

q1� 8Cn+1

⌘n+1

2Cn+1

� 2⌘n+1

1

A� ⌘n+1

;

Set n = n+ 1 and go to step (S1).

Main result. Our multi-level scheme starts at a low frequency. We cannot only choose fre-quency as the index for the levels because only one parameter appears not to be su�cient to satisfy(5.13) and arrive at a convergent scheme.

To achieve high accuracy reconstruction with multi-frequency data, we design an algorithm suchthat the starting point c�2

0,n+1

at level n+1, which equals the result after the iterations at level n, iswithin the (n + 1)�level convergence radius ⇢n+1

. Therefore, the iterations can continue until thedesired accuracy is obtained. In the above, (5.13) yields a su�cient multi-level condition balancing


the competition between the approximation error and the convergence radius of neighboring levels[16]. Here, we establish that subject to a minimal compression rate of the unique solution usingpiecewise constant functions we can find frequencies such that this multi-level condition is satisfied.

Theorem 5.7. Let DNn

, n = 1, . . . , nmax

be a sequence of domain partitionings, and ANn

thesequence of corresponding admissible sets, where n

max

is a positive integer and depends on r0

, Land |⌦|. Let the unique solution satisfy the compression rate,

(5.14) dist(c†,[mn=1

ANn

) C�1

0

e�K1

m7/2

, m = 1, . . . , nmax

.

There exists a set of selected frequencies, {!n}nmax

n=1

, such that the multi-level condition (5.13) issatisfied. Then, Algorithm 5.6 has the property that the n-level iteration result, c�2

n,Kn

, satisfies

kc�2

n,Kn

� c�2

† kLp

(⌦)

3C�1

0

e�K1

N7/2

n , 8n = 1, . . . , nmax,3

2 p < 1.

Proof. Following Proposition 3.7, we set '(N) = C�1

0

e�K1

N7/2

. Given the frequency, !n, forlevel n, we choose the frequency !n+1

such that the following inequalities are satisfied:

(5.15) !6

n+1

('(Nn+1

))3((3 + ")!2

n'(Nn) + !2

n+1

'(Nn+1

)) (23/2L2

0

L0

)�1,

and

(5.16) !6

n+1

(8L0

L0

('(Nn+1

))3)�1.

Then, on the one hand, from Lemma 2.4 and 2.5, (3.10) and (5.15), we conclude that

(5.17) 23/2Cn+1

(Ln+1

Cn+1

)((3 + ")⌘n + ⌘n+1

)

(23/2L2

0

L0

)!6

n+1

('(Nn+1

))3((3 + ")!2

n'(Nn) + !2

n+1

'(Nn+1

)) 1.

On the other hand, (5.16) with Lemma 2.5 and (3.10), gives that

1� 8Cn+1

⌘n+1

> 0.

Hence,

1 +q1� 8Cn+1

⌘n+1

� 4Cn+1

⌘n+1

> 1,

which together with (5.17) gives (5.13).For any true wavespeed c† and starting model c

0,0, if the first frequency is su�ciently low, thenour multi-level projected steepest descent scheme converges. That is, by letting ! tend to 0, theconvergence radius ⇢ tends to infinity. The following theorem gives a precise statement. Note thatthis statement does not give a lower bound of the frequency which is uniform for all c† and c

0

.Theorem 5.8. Let the admissible set AN be as defined in Definition 3.8 for some given N .

Assume that c0

2 AN . Then, for any c† satisfying

c�2 2 L1(⌦) \ Lp(⌦), kckL1(⌦)

B1

, kc�1kL1(⌦)

B2

,

there exists a frequency su�ciently small such that the convergence radius ⇢ is larger than kc�2

0

�c�2

† kLp

(⌦)

.Proof. Note that the convergence radius, derived in [16], for Lp is

(5.18) �2

(x0

, z†) < ⇢ :=1

2(2CL)�2

✓1 +

q1� 8C⌘ � 4⌘C

◆2

.


The proof is divided into two steps. In the first step we provide a uniform lower bound of

1 +q

1� 8C⌘ � 4⌘C.

Note that Lemma 2.5 and Proposition 3.7 implies that C is uniformly bounded. Then the desiredlower bound follows by noticing that

lim!!0

⌘ = 0.

In the second step we show that

lim!!0

LC2L = 0.

The above indentity is proved by combining Lemma 2.5,2.4 and Proposition 3.7 to arrive at

LC2L C!2,

for some constant C. This completes the proof.

6. Discussion. We apply a projected steepest descent iterative method to the inverse bound-ary value problem of the Helmholtz equation using the Dirichlet-to-Neumann map as the data. Wegive explicit conditions for the convergence, derived from a conditional Lipschitz stability estimatefor the inverse problem. The asymptotic behavior of the stability constant plays an important rolein the convergence radius and required compression rate for the unique solution.

With su�cient regularity of the wave speed function, a Lipschitz stability estimate can beobtained as a variation of the arguments for the general logarithmic type stability estimates. Forexample, see [1] for the analogue of the EIT problem. To be more precise, assuming that c�2

i , i = 1, 2are the linear combinations of finitely many known C1 functions

1

, . . . , N , we have that

kc�2

1

� c�2

2

kL1(⌦)

CNk⇤1

� ⇤2

kL(H1/2

(@⌦),H�1/2

(@⌦))

.

However, here we have been focused on the presence of discontinuities. With the same behavior ofthe stability constant, we can repeat the analogous treatment for C1 true wave speed reconstructionwith AN constructed from {

1

, . . . , N}.

Acknowledgements. The authors thank the members, BGP, ExxonMobil, PGS, Statoil, To-tal, of the Geo-Mathematical Imaging for partial support. The work of E. Beretta was partiallysupported by MIUR grant PRIN 20089PWTPS003. The research of M. de Hoop and L. Qiu was sup-ported in part by National Science Foundation grant CMG DMS-102531. The work of O. Scherzerhas been supported by the Austrian Science Fund (FWF) within the national research networksPhotoacoustic Imaging in Biology and Medicine, project S10505 and Geometry and SimulationS11704.

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INVERSE BOUNDARY VALUE PROBLEM FOR THE ...This inverse problem arises, for example, in reﬂection...

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