CRC Research Notes in Mathematics, No. 427

Roland Potthast

Point sources and multipoles in inverse scattering theory

CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C. © 2001 by Chapman & Hall/CRC

CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board H. Amann, University of Zürich R. Aris, University of Minnesota G.I. Barenblatt, University of Cambridge H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware D. Jerison, Massachusetts Institute of Technology B. Lawson, State University of New York at Stony Brook

B. Moodie, University of Alberta S. Mori, Kyoto University L.E. Payne, Cornell University D.B. Pearson, University of Hull I. Raeburn, University of Newcastle, Australia G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

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Library of Congress Cataloging-in-Publication Data Potthast, Roland. Point sources and multipoles in inverse scattering theory / Roland Potthast. p. cm. -- (Chapman & Hall/CRC research notes in mathematics series ; 427) Includes bibliographical references and index. ISBN 1-58488-252-2 (alk. paper) 1. Scattering (Mathematics) 2. Inverse problems (Differential equations) I. Title. II. Series. QA377 .P64 2001 515′.353--dc21

2001025323

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Visit the CRC Press Web site at www.crcpress.com © 2001 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-252-2 Library of Congress Card Number 2001025323 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

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7

Linear sampling methods

The idea to use point-sources for the reconstruction of the unknown scatterer has been exploited by Colton and Kirsch and many others since 1995 to develop and investigate a class of inversion methods, known as linear sampling methods. We will introduce the original linear sampling method as proposed by Colton and Kirsch [6] and Colton, Monk, Piana and Potthast [10], [13] in our rst Section 7.1. Section 7.2 will present the main ideas of a modi ed approach by Kirsch [41], [42], who used the spectral theory of the far eld operator to study the linear sampling method and obtained characterizations of the unknown support of a scatterer by the expansion of the far eld pattern of point-sources with respect to the eigenvalues and eigenvectors of the far eld operator. We will conclude with two sections about anisotropic media: in Section 7.3 we use the linear sampling method to detect the support of an orthotropic medium from the knowledge of the far eld operator; Section 7.4 treats the full threedimensional anisotropic inverse electromagnetic scattering problem as proposed by Colton and Potthast [14]. 7.1

The original linear sampling method

To describe the original linear sampling method consider the direct acoustic scattering problem as introduced in De nition 2.1.1 or the direct acoustic inhomogeneous medium scattering problem as given by De nition 2.2.1. In the last case we consider a piecewise smooth index of refraction n. We are interested in determining D from the knowledge of the far eld patterns for all incident plane waves u1 (^x; d) for x^; d 2 . In order to do this, we investigate the far eld equation (F g)(^x) = 1 x; z ); x^ 2 ; true (^

(7.1.1)

where the operator F : L2 ( ) ! L2 ( ), de ned by (F g)(^x) :=

Z



u1 (^x; d)g(d) ds(d); x^ 2 ;

(7.1.2)

is known as the far eld operator and 1 true denotes the far eld pattern of the fundamental solution (1.2.40). The far eld operator F has smooth kernel, is thus compact in L2 ( ) and in general equation (7.1.1) does not have a solution. We temporarily assume that we are able to solve (7.1.1) and in a second step show how to modify the approach to treat the general case. Let g(
; z ) be a solution of (7.1.1) with z in the interior of the scattering © 2001 by Chapman & Hall/CRC

208

7. Linear sampling methods

obstacle D. Then the incident eld ui = vg with the Herglotz wave function vg (x) :=

has the scattered eld vgs (x) :=

and far eld pattern

Z



Z

ui (x; d)g(d) ds(d); x 2 IRm

(7.1.3)

us (x; d)g(d) ds(d); x 2 IRm

(7.1.4)

vg1 (^x) =

Z

u1(^x; d)g(d) ds(d);

= (F g)(^x): (7.1.5) Since the far eld patterns of vgs and (
; z ) coincide, by Rellich's lemma we know that the scattered elds in the exterior of the scatterering domain D must be identical. We rst consider the case of an impenetrable sound-soft scatterer, i.e., we consider the Dirichlet boundary condition (2.1.4). On the boundary @D of the scatterer D we have vgs = vg . Then, from the boundedness of the Herglotz wave operator H : L2( ) ! C (@D); g 7! vg ; (7.1.6) we obtain jjg(
; z )jjL2 ( )  cjjvg jjC (@D) = cjjvgs jjC (@D) = cjj(
; z )jjC (@D) (7.1.7) with some constant c. Thus for z ! @D the norm jjg(
; z )jj of the solution g(
; z ) of (7.1.1) will blow up in the same way as the singularity of the fundamental solution . Since for the calculation of g only the far eld pattern u1(^x; d) for x^; d 2 is needed, we can formulate a method to nd the unknown scatterer D: Definition 7.1.1 (Linear Sampling Method.) For z on a grid which covers the region where the unknown scatterer D is supposed solve equation (7.1.1) with an appropriate regularization and calculate the norm jjg (
; z )jj of the solution. Then, the boundary of the unknown scatterer is found as a level curve where jjg (
; z )jj is large.

Since for the determination of the unknown scatterer the main eort is the calculation of g(
; z ) as a solution to a linear equation on a grid of sampling points, the method is called linear sampling method. © 2001 by Chapman & Hall/CRC

7.1 The original linear sampling method

209

In general the equation (7.1.1) does not have a solution. But under appropriate assumptions on the scatterer we can show as in Lemma 3.1.2 that H has dense range in C (@D). In this case given  > 0 for each z 2 D there is a density g~(
; z ) with  jjH g~(
; z ) ( (
; z ))jjC (@D)  jjFjj ; (7.1.8) and

jjF g~(
; z ) 1 (7.1.9) true (
; z )jjL2 ( )   2 where F : C (@D) ! L ( ) denotes the mapping of a function f onto

the far eld pattern of the solution to the exterior Dirichlet problem with boundary values f . From (7.1.8) we obtain for g~ or vg~(
;z), respectively, the behavior (7.1.7). Using a regularization method to solve the equation (7.1.1) may be considered as a way to numerically solve the equation F g(
; z ) = vg~1(
;z)(
; z ) (7.1.10) 1 with the right-hand side given by  (
; z ) which has a data error of size . We now turn our attention to the inhomogeneous medium case. The index of refraction is assumed to be piecewise continuously dierentiable with a jump discontinuity across the smooth boundary @D; compare De nition 2.2.5. Our aim is to show that the far eld equation (7.1.1) has an approximate solution g such that kgkL2( ) and kvg kL2 (D) become unbounded as z tends to the boundary @D of the support  = 1 n. The proof of this result is more diÆcult than for impenetrable obstacles and involves examining a special interior transmission problem (cf. [8, 43]) instead of an interior Dirichlet or Neumann problem for the Helmholtz equation as in the case of obstacle scattering. Our analysis will be based on a projection theorem for Hilbert spaces where the inner product is replaced by a bounded sesquilinear form. Let X be a Hilbert space with the scalar product (
;
) and norm k
k induced by (
;
). Let h
;
i be a bounded sesquilinear form on X such that jh; ij  kk2 (7.1.11) for all  2 X where is a positive constant. For a subspace Z  X we de ne Z ? to be the orthogonal complement of Z with respect to (
;
) and Z ?s to be the orthogonal complement of Z with respect to h
;
i. By the Lax-Milgram Theorem there exists a unique bounded linear operator M : X ! X such that h; i = (M; ) (7.1.12) 1 for all ; 2 X , M is bijective and the norm of M is bounded by 1 . The case of a penetrable inhomogeneous medium.

© 2001 by Chapman & Hall/CRC

210 Lemma 7.1.2

7. Linear sampling methods

For every closed subspace Z

where Z ? \ MZ = f0g.

 X we have the decomposition

X = Z ? + MZ

Proof. De ne G := Z ? + MZ and let  2 G? . Then  2 Z [ (MZ )? , i.e., hh; i = (Mh; ) = 0 for all h 2 Z . Setting h =  by (7.1.11) we obtain  = 0 and hence X = Z ? + MZ . Now assume that for g 2 X we have g = 1 + Mh1 = 2 + Mh2. Then for := 1 2 and h := h1 h2 we have 0 = + Mh with 2 Z ? and h 2 Z . Therefore 0 = ( + Mh; h) = (Mh; h) = hh; hi and hence h = 0. This implies = 0 and the proof is complete. 2 Now let P0 be the orthogonal projection operator in X onto the space Z with respect to the scalar product (
;
) and let PM be the projection operator onto MZ as de ned by Lemma 7.1.2. By the closed graph theorem, PM is a bounded operator. Lemma 7.1.3

For every closed subspace Z  X we have M 1 Z ? = (M  Z )? = Z ?s :

. The rst equality follows from the fact that  2 (M  Z )? if and only if (; M  h) = (M; h) = 0 for every h 2 Z and hence M 2 Z ? , i.e.,  2 M 1 Z ?. The second equality follows from the fact that (; M  h) = (M; h) = h; hi. 2 Proof

We are now in a position to show that every  2 X can be uniquely written as a sum  = v + w with v 2 Z ?s and w 2 Z , i.e., X = Z ?s s Z where s is the orthogonal decomposition with respect to the sesquilinear form h
;
i. Theorem 7.1.4

decomposition

For every closed subspace Z

 X we have the orthogonal

X = Z ?s s Z: The projection operator P : X ! Z ?s de ned by this decomposition is bounded in X .

© 2001 by Chapman & Hall/CRC

7.1 The original linear sampling method

Proof

that i.e.,

Hence where

211

. For  2 X we de ne ^ := M. Then from Lemma 7.1.2 we have ^ = (1 PM )^ + PM ^; M = (1 PM )M + PM M:  = M 1 (1 PM )M + M 1 PM M = v + w

v := M 1 (1 PM )M 2 M 1 Z ? = Z ?s w := M 1 PM M 2 Z: We have thus shown that X = Z ?s + Z . To show the uniqueness of this decomposition, suppose v + w = 0 with v 2 Z ?s and w 2 Z . Then

0 = jhv; wij = jhw; wij  kwk2 which implies that w = v = 0. Finally, since from the above analysis we obtain P = M 1 (1 PM )M and PM is bounded, we have that P is bounded. 2 We will now turn our attention to the problem of showing the existence of a unique weak solution v; w of the interior transmission problem 4w + 2 n(x)w = 0; 4v + 2 v = 0 in D (7.1.13) @v = @ (x; z ) on D @ @x

(7.1.14)

(x; y)(y)w(y)dy = v(x); x 2 D

(7.1.16)

w v = (x; z );

@w @

where n = 1  and D are as de ned above (see also De nition 2.2.5), and (
; z ) is given by (1.2.40) with z 2 D. We will further assume that there exists a positive constant c such that Im (x)  c (7.1.15) for x 2 D. To motivate the following de nition of a weak solution of (7.1.13), (7.1.14) we note that if a solution v; w 2 C 2 (D)\C 1 (D) to (7.1.13), (7.1.14) exists, then from Green's formula and (7.1.13) we have that w(x) + 2

Z

D

and (7.1.14) will be satis ed provided 2

Z

D

(x; y)(y)w(y)dy = (x; y); x 2 @B

© 2001 by Chapman & Hall/CRC

(7.1.17)

212

7. Linear sampling methods

where B is a ball centered at the origin with D  B . This last statement follows from Rellich's lemma and the unique continuation principle. Definition 7.1.5

the linear space

(Weak interior transmission problem.)

Let Z be

Z := fu 2 C 2 (IRm ) : 4u + 2 u = 0 in IRm g and Z the closure of Z in L2 (D). With the volume potential V ' de ned by (2.2.2) a pair v; w with v 2 Z and w 2 L2 (D) is said to be a weak solution of the interior transmission problem (7.1.13), (7.1.14) with the point source z 2 D if v and w satisfy the integral equation

(I + 2 V )w(x) = v(x); x 2 D and the boundary condition

2 (V w)(x) = (x; z ); x 2 @B:

Before proceeding to establish the existence of a unique weak solution to the interior transmission problem (7.1.13), (7.1.14), we make a few preliminary observations. We rst note that condition (7.1.15) implies that the multiplication operator (M)(x) := (x)(x) for  2 L2 (D) is continuously invertible in L2 (D) and that the sesquilinear form Z h; i := (y)(y) (y)dy (7.1.18) D

satis es the assumption (7.1.11). We also note that from the Jacobi-Anger expansion we have that the space of Herglotz wave functions with kernel g 2 L2 ( ) is a dense subset of Z . Finally, by the uniqueness of the solution to the exterior Dirichlet problem for the Helmholtz equation and the unique continuation principle, we see that if v; w is a weak solution of the interior transmission problem with point source z 2 D then 2 (V w)(x) = (x; z ) for all x 2 IRm n D. For every source point z 2 D there exists at most one weak solution of the interior transmission problem.

Theorem 7.1.6

Proof. Let w and v be the dierence between two weak solutions of the interior transmission problem. Then from the boundary condition © 2001 by Chapman & Hall/CRC


          
    




     
     







              


  


      


 

          
 
  
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