Inverse problems and related topics 9781584881919, 1584881917, 9781138404083

Table of contents :
Content: A Finite Difference Model for Calderon's Boundary Inverse ProblemInverse Problems for Equations with MemoryParameter Estimation of Elastic MediaThe Probe Method and its ApplicationsRecent Progress in the Inverse Conductivity Problem with Single MeasurementA Moment Method on Inverse Problems for the Heat EquationSome Remarks on Free Boundaries of Recirculation Euler Flows with Constant VorticityAlgorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible BeamsNumerical Solutions of the Cauchy Problem in Potential and ElastostaticsInverse Source Problems in the Helmholtz EquationsA Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse ProblemsImpedance Computed Tomo-ElectrocardiographyAn Inverse Problem for Free Channel ScatteringSurface Impedance Tensor and Boundary Value ProblemAysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville ProblemExact Controllability Method and Multidimensional Linear Inverse Problems

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Inverse problems and related topics

CHAPMAN & HALUCRC Research Notes in Mathematics Series Main Editors H. Brezis, Universite de Paris R.G. Douglas, Texas AcceM University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board H. Amann, University of Zurich R. Aris, University of Minnesota G.I. Barenblatt, University of Cambridge H. Begehr, Freie Universitet 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. Raebum, 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

Submission of proposals for consideration Suggestions for publication, in the form of outlines and representative samples, are invited by the Editorial Board for assessment. Intending authors should approach one of the main editors or another member of the Editorial Board, citing the relevant AMS subject classifications. Alternatively, outlines may be sent directly to the publisher's offices. Refereeing is by members of the board and other mathematical authorities in the topic concerned, throughout the world. Preparation of accepted manuscripts On acceptance of a proposal, the publisher will supply full instructions for the preparation of manuscripts in a form suitable for direct photo-lithographic reproduction. Specially printed grid sheets can be provided. Word processor output, subject to the publisher's approval, is also acceptable. Illustrations should be prepared by the authors, ready for direct reproduction without further improvement. The use of hand-drawn symbols should be avoided wherever possible, in order to obtain maximum clarity of the text. The publisher will be pleased to give guidance necessary during the preparation of a typescript and will be happy to answer any queries. Important note In order to avoid later retyping, intending authors are strongly urged not to begin final preparation of a typescript before receiving the publisher's guidelines. In this way we hope to preserve the uniform appearance of the series.

G Nakamura, S Saitoh, J K Seo and M Yamamoto (Editors)

Inverse problems and related topics

CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

A CHAPMAN & HALL BOOK

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in hardback 2019 © 2000 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13:978-1-58488-191-9 (pbk) ISBN-13: 978-1-138-40408-3 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// wwvv.copyright.comi) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://wvvw.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Library of Congress Cataloging-in-Publication Data Inverse problems and related topics / G. Nakamura ... [et al.], (editors). p. cm. — (Chapman & HalUCRC research notes in mathematics ; 419) Proceedings of a seminar held at the Kobe Institute, Japan, Feb. 6-10, 1998. Includes bibliographical references. ISBN 1-58488-191-7 (alk. paper) 1. Inverse problems (Differential equations)—Congresses. I. Nakamura, Gisaku, 1928— II. Chapman & Hall/CRC research notes in mathematics series ; 419. QR374.159 2000 99-462330 515'.35—dc21 CIP Library of Congress Card Number 99-462330

TABLE OF CONTENTS

Preface 1. A finite difference model for CalderOn's boundary inverse problem K. Amano

1

2. Inverse problems for equations with memory A. L. Bukhgeim and G. V. Dyatlov

19

3. Parameter estimation of elastic media D. P. Chi and J. Kim

37

4. The probe method and its applications M. Ikehata

57

5. Recent progress in the inverse conductivity problem with single

measurement H. Kang and J. K. Seo

69

6. A moment method on inverse problems for the heat equation M. Kawashita, Y. Kurylev and H. Soga

81

7. Some remarks on free boundaries of recirculating Euler flows with

constant vorticity S. C. Kim

89

8. Algorithms for the identification of spatially varying/invariant stiff-

ness and dampings in flexible beams N. Nakano, A. Ohsumi and A. Shintani

97

9. Numerical solutions of the Cauchy problem in potential and elas-

tostatics K. Onishi and Q. Wang

115

10. Inverse source problems in the Helmholtz equation S. Saitoh

133

11. A numerical method for a magnetostatic inverse problem using the edge element T. Shigeta and K. Onishi

145

12. Impedance computed tomo-electrocardiography K. Shirota, G. Nakamura and K. Onishi

155

13. An inverse problem for free channel scattering 165

T. Takiguchi 14. Surface impedance tensor and boundary value problem

181

K. Tanuma

15. Asymptotics for the spectral and Weyl functions of the operatorvalued Sturm-Liouville problem 189

I. Trooshin

16. Exact controllability method and multidimensional linear inverse problems M. Yamamoto

209

VI

PREFACE Inverse problems arise in many fields and are important in practical applications to science and engineering. New and efficient mathematical methods have been devised and developed over the last 20 years. The number of researchers in these fields is growing enormously. In order to establish a good network for collaboration and to increase the number of Japanese and Korean researchers in these fields, the JapanKorea joint scientific seminar on " Inverse Problems and Related Topics" was held at the Kobe Institute, Japan, during February 6-10, 1998. It was just after the economic crisis in Korea. The joint seminar was a timely opportunity to present many interesting and important new results. Some presentations were quite unique and original. The number of participants was 54 and there were many valuable and stimulating discussions among the participants. This volume constitutes the Proceedings of the joint seminar. The readers will notice that there are several Japanese and Korean groups actively working and doing interesting work on inverse problems. We would be very happy if this book would represent and introduce the existence of these groups and their work. The joint seminar was proposed as a Joint Project under the Japan-Korea Basic Scientific Promotion Program supported by JSPS (Japan Society for the Promotion of Science) and KOSEF (Korean Science and Engineering Foundation). We wish to thank the JSPS and KOSEF for their support. We would like to thank Professor Robert P. Gilbert for his recommendation of our Proceedings to this reputed series. We express our sincere thanks to our colleagues Kazuo Amano, Masaru Ikehata, Jin Cheng of Gunma University for their help for the Proceedings and Mr. Yasushi Ota, of Osaka University for his assistance for the joint seminar. We also would like to thank the staff at Chapman & Hall / CRC Press and Mrs. Noriko Kimura of Gunma University for their help in preparation and publishing the Proceedings. . October, 1999 G. Nakamura S. Saitoh J.K. Seo and M. Yamamoto

vii

A FINITE DIFFERENCE MODEL FOR CALDERON'S BOUNDARY INVERSE PROBLEM * KAZUO AMANOt Abstract. The author gives a finite difference model of the inverse problem for a certain class of elliptic partial differential equations which appear in the study of electrical impedance tomography, and shows an approximate representation of Dirichlet-to-Neumann map in terms of the coefficients of differential equation and Dirichlet data (Theorem 1.1). Using this representation, he proves that if a certain system of nonlinear algebraic equations is numerically solvable, then it is possible to reconstruct the coefficient of Calderon's inverse problem (Theorem 1.2). His result enables to circumvent the severe instability of inverse problem by using a hybrid symbolic-numeric method (Sections 3 and 4). Some numerical examples (Examples 4.3-4.10) show that his scheme would be able to recover the deeper region of given object without the notorious resolution problem (cf. Kotre's survey [9)). Key words. boundary inverse problem, electrical impedance tomography, finite difference method, hybrid symbolic-numeric method AMS subject classifications. 35J25, 35R30, 65N06

1. Introduction. Calderon [3] discussed an inverse problem of the Dirichlet problem V(-y Vu) = 0 in D u=

on 8D,

where D denotes a bounded domain of multi-dimensional Euclidean space and 7 is a real bounded measurable function defined in D with positive lower bound. Roughly speaking, the problem is uniqueness and reconstruction of the coefficient 7 when Dirichlet condition is given and Neumann condition is observable. Though there are vast references on Calderon's inverse problem (cf , for example, [6], [9] and their references), it seems that only few researchers have been interested in finite difference method for this problem so far (cf. [4], [7, Chapter 10] and their references). In this paper, we establish a new type of finite difference scheme for Calderon's problem and show that his problem is approximately equivalent with a certain system of nonlinear algebraic equations. It is to be remembered that the forward or regular problem is translated into a system of linear algebraic equations by virtue of finite difference method. We discuss an inverse problem of the Dirichlet problem (1.2)

— a (a(x, y) — au ) — a (a(x ,y) — au = 0 ax ax ay ay u=4) on

in

D

8D.

The coefficient a(x, y) E Coo (R2) , a(x, y) > 0 , is defined by the convolution (1.3)

ff

r1)

li

*Received by the editors October 1, 1998; accepted by the editors. The author was supported by Grant-in-Aid for Scientific Research (C)(2) No.10640102, Ministry of Education, Science, Sports and Culture, Japan. tDepartment of Mathematica, Faculty of Engineering, Gunma University, Kiryu, Gunma, 3768515, Japan ([email protected]. ac. jp).

2

KAZUO AMANO

for a fixed small b > 0 , where X(X

=

,

and

e—(r2 +V2)

X6(x,Y)=

x

\ )•

The fundamental property of Friedrichs' mollifier shows a(x, y) is approximately equal to the original coefficient 7(x, y) when 6 > 0 is sufficiently small. d•(x, y) is a Coo function defined in R2 . For the sake of simplicity, we assume that D is a unit open interval of R2 , i.e., D = (0,1) x (0, 1) . Our result remains valid for a broader class of bounded domains with piecewise smooth boundary and it is not so difficult to generalize the following theorems in multi-dimensional space. Throughout this paper, we assume that u(x,y) E C°°(D) is a solution of (1.2). We assume the existence of smooth solution and concentrate on the search of its symbolic representation. Let us take a sufficiently large natural number N and put i

U(i,j) = u(xi,w),

(xi, yj) = (7, 7), A(i, j) = loga(xi, for 0 < i ,j < N and

a. 8v(x' •yji )

= 0(xi, Yi), 41(i, :7)

(0,0), (0, N), (N , 0), (N, N) , where v denotes the for (xi, E 8D satisfying (i, j) &A-t(0 , 0), ,i), (1, o), . Then there exists a family inner normal vector on of polynomials PN (i, j, k, t), 0 < i , j, k, P < N in variables A(m, n), 0 < m, n < N for which the following theorems are valid. THEOREM 1.1. Assume that u E Coo (D) is a solution of the Dirichlet problem (1.2). Then, for any 0 < k, t < N , we have

r=

U (k,t) — (1.4)

E

j)

j, k , t)1

(x.,y,)Er

< 0(N-2 1og

(sup 101) OD

i PN(i, j, k,

0 1 . Here 0(N -2 log N) depends only on N , lower and upper bounds of y and C8 norms of Dirichlet and Neumann conditions, and the last term E o 3, with smooth boundary Oft. Consider the boundary value problem (2.1) (2.2)

O, , x E St, t E IR,

k(x, //ion xR = g(x,t), x E

an, t E lit

Here k(x ,t) is a smooth function given for x E Cl, t > 0. We suppose k(x, t) to be extended by zero to t < 0. To state the solvability theorem for the problem (2.1), (2.2), we introduce the weighted spaces 1/4(S2 x IR) and 1-17(0S2 x IR) that are constituted by the functions u such that e-7tu E 1-18 (Ct x IR) and e-7tu E H8 (011 x IR) respectively. Here li8 (S2 x IR) and H8 (ac2 x IR) are the standard Sobolev spaces. We denote the norms in the weighted spaces by II 117,8 and ( • )7,8. We now state the solvability theorem for the direct problem: THEOREM 2.1 [BDI]. Let K > 0. There are positive constants 'yo and C depending only on Cl and K and such that for arbitrary y > yo, k E C(S/ x [0, oo)), 11klic < K, and g E H4(011 x R) there is a unique solution u E 11- 4 (ft x IR) to the problem (2.1), (2.2); moreover, the following estimate holds: + (0„u)7,0 < (here a,, stands for the normal derivative). One can derive Theorem 2.1 from the well-known results on solvability of general boundary value problems for hyperbolic equations (see, for instance, [S, VG]). By Theorem 2.1, we can define the Dirichlet-to-Neumann map H : x R) H7°(ao x IR) by the rule Hg ovulas-axR, where u is the solution to the problem (2.1), (2.2).

The inverse problem is to find k(x, t) from the given Dirichlet-to-Neumann map. We prove a conditional stability estimate for this problem: THEOREM 2.2. Suppose that ki E C3 (S) x [0, oo)), j = 1, 2, are two functions such that Pi k's < K, with some positive constant K (which determines the correctness class). There is -yo > 0 depending only on St and K such that, for every 7 >'y0, the corresponding Dirichlet-to-Neumann maps Hi are defined as operators from H7 (ail x IR) to H,°(asi x R) and the following estimate holds: Ilk1 — k2I17,0 5_

where w(e)

— H2II),

C(loge-1)-1/(2n-1-3),

with C depending only on Cl, K, and -y.

-4 0,

26

ALEXANDRE L. BUKHGETM AND GLEB V. DYATLOV COROLLARY 2.3.

The assertion of Theorem 2.2 remains valid if we restrict the

domain of H3 to .1172 (Osl x R) that is the completion of the set of functions in Co (oft x R) supported in Oft x (0, co) in the norm ( • )-y,z•

Observe that the solution to (2.1), (2.2) with g E H2.7 (aft x IR) vanishes for t < 0. So Corollary 2.3 means that we have the same result for the initial-boundary value problem for (2.1) with the zero initial condition and the boundary condition (2.2). Proof. We can work with functions in Cr (OD x R), since this space is dense in 1/72 (afi x IR). Take g E x R). Since g has compact support, there is T E R such that g(x, t) = 0 for t < —T. Put g(x,t) := g(x, t — T) and observe that g(x,t) = 0 for t < 0. Let u be the solution to (2.1), (2.2). It is easily verified that u(x, t) = u(x, t — T) is the solution to (2.1), (2.2) with g substituted for g. We have (11 - 07,o = 0o-47,o = e-7T (0,,u).0:, = e--YT (Hg)7,o and (0.7,2 = e-77 (g)7 ,2. Hence the norm of the operator H calculated over the space of functions §, E x R) such that g(x,t) = 0 for t < 0 is not less than that calculated over H7 (Oft x IR). 2.2. Auxiliary propositions. First we suppose that lico is the one given by Theorem 2.1. In the process of the proof we possibly increase this value; moreover, we always assume that 7 > "Yo. For functions u(x, t) such that ue-71 E L2(f/ x R) (or ue-14 E L2(as2 x R)) we introduce the Fourier—Laplace transform in time as follows: (Fu)(x,0) = 11(x,0) :=

f e —ite u(x, t) dt,

where 9 = a — i-y, a E R. Observe that the Parseval identity holds: (2.3)

(u(x, t))-y,o = 1121(x, a — i7)IIL2(osixR),

where ft is considered as a function of x and a. Applying the Fourier—Laplace transform to the problem (2.1), (2.2), we obtain (2.4) (2.5)

Ati(x,0)+ (kx,e) + 92)fi(x,e) = 0,

ii(x,O)lorixR = §(x, 8).

Thus, il(x, 9) is a solution to the family of the stationary problems (2.4), (2.5) that depends on the parameter 9 = a — i-y, a E R. Let q.,(x) E C3 (f2), j = 1,2. Consider the following equation in the domain f2: (2.6)

Av.,(x) + q.,(x)v.,(x) = 0.

Suppose that the Dirichlet problem for (2.4) in ft has at most one solution. For each qj (x) we define the so-called (stationary) Dirichlet-to-Neumann map Ai as follows: A7 g := e„v)Ian,

where vj is the solution to (2.6) with the condition vlon = g. The operator Ai is a bounded operator from H 1(asi) into H° (an) which ensues from the general results on the solvability of elliptic equations (see, for example, [LM]).

INVERSE PROBLEMS FOR EQUATIONS WITH MEMORY

27

LEMMA 2.4. The following identity is valid for sufficiently smooth solutions vi

to (2.6), j = 1, 2:

(2.7)

fo (42 — )vi v2 dx f

(Ai — A2)v2 dS .

Oft

Here and in the sequel the notation like Ai vi means that we act by the operator Al on the trace vIlasz. In the case of real-valued potentials qi (x) this identity was proven in [A] and the case of general elliptic operators was considered in [Il] (see also [BDI]). Consider the equation (2.8)

(A + 02)v(x) + q(x)v(x) = 0.

To prove Theorem 2.2, we need some generalization of one theorem of J. Sylvester and G. Uhlmann [SU] (see also [I2]). THEOREM 2.5. Let q(x) E C8(1) with a natural s. There are constants C1 and C2 depending only on S2 and s such that, for every ( E en satisfying the conditions and ICI > liql1c.(c ) , the equation (2.8) has a solution of • +0z = 0, the form

(2.9)

v(x, () = e(*x(1 + w(x,()),

where w(x, () satisfies the estimate C2

(2.10)

iiw( • ,C)iiii.(n) < iiqiic.(n)• — ICI The constants C1 and C2 depend only on the domain Si and s. The dot in the statement of the theorem stands for the "inner product without complex conjugation," i.e., x • y = xlyl + • • • + xnyn. Theorem 2.4 was proven in [I2] for s = 0. The case of an arbitrary s is considered by analogy (see [BDI]). Below, the letter C with subscripts denotes constants that depend only on the quantities indicated in the subscript. Constants with the same subscript may differ. COROLLARY 2.6. Under the conditions of the last theorem, we have iiv(• ,()I1H.(n) < Crt,secni(1.

2.3. Proof of Theorem 2.2. Suppose that ki(x,t), j = 1, 2, are functions satisfying the conditions of Theorem 2.2. We turn to the equation (2.4) with k = We denote the solution to (2.4) by 6.,(x, t). Denote the corresponding Dirichlet-toNeumann maps by A. By Lemma 2.4, we have the identity dx = f

(x , 0)(AT — ADii2(x , 0) dS .

al/

By Theorem 2.5, the equation (2.4) with k = k, possesses a solution vi (x, (3 , 0) = e(1'x(1 + wi (x, (2 ,6)),

where E en , (j • (3 + 02 = 0, K./ I > the estimate (2.11)

( • , 0) ic3(n) 1Ni1 > V2, and wi satisfies

ilwi( • ,C.i, 9)111/3(o) 0 depending only 11 and K such that for arbitrary -y > yo and k E C(12 x [0, oo)) with iikiic < K the problem (3.1), (3.2) has a unique solution u E H.72,1(1/ x R) for an arbitrary g E 1/4'1(50 x R). H2,1 (f-2

Moreover, the estimate

IIuII ,2,1 + (avu)2-y ,o < C(9)7,2,1 is valid for some constant C depending only on Il and K. Thereby, by analogy with § 2, we can define the Dirichlet-to-Neumann map H : H°(852 x IR). The inverse problem is to find k(x , t) from the given Dirichlet-to-Neumann map H. As in the hyperbolic case we prove conditional stability: THEOREM 3.2. Suppose that ki E C3(0 X [0, CO)), j = 1, 2, are two functions such

H-y2 '1 (an x R)

that Ilkik3 < K, with some positive constant K (which determines the correctness class). There is -yo > 0 depending only on SI and K and such that, for every -y > 2,0, the corresponding Dirichlet-to-Neumann maps Hi are defined as operators from 1-1,2,i (an x IR) to H° (an x R) and the following estimate holds:

Ilk1 where co(e)

- Ha),

C(log 6 -1)-1 / (2n-1-3)

6 -4 0,

with C depending only on 11, K, and -y. Moreover, the result remains valid if we restrict the domain of Hi to the space 1-12,1(8n x IR) defined by analogy with the space H71(852 x IR) (see 2.1). Proof. The proof repeats almost verbatim that of Theorem 2.2. The only difference is the choice of (j.

INVERSE PROBLEMS FOR EQUATIONS WITH MEMORY

31

Applying the Fourier-Laplace transform to (3.1), we arrive at the equation ilk/ - ou - k(x,0)/1(x,O) = 0. So now we have to choose (3 from the conditions CI

ICA

+ C2

= -ie, (i • (i - i0 = 0, PCJI 5 Afi•

Cillki( • ,19)11c3(n),

As in §2, searching (i in the form (i = p • 4" = 0, we obtain the system A2 _ p2

+ (-1)i(ip + A), with A, p E Rn, A • C = .

2A • p = a.

Take A to be an arbitrary vector such that A•C = 0 and A2 = -y + +1-2, r > •V2-. Afterwards, choose µ from the conditions p • C = 0, p2 = L2l + r 2 , and A • p = We can fulfill the last condition by varying the angle between A and pi, since iAliPi > In this case we have + +ial 2r2, K3 12 = so we can proceed as in the proof of Theorem 2.2. § 4. SOLUTION OF MEMORY RECONSTRUCTION PROBLEMS BY THE NEWTON METHOD Here we consider the following two problems that appeared earlier (see [L]): Problem 1. Find the pair of functions (u, h) in the problem (4.1)

ut - Au - f h(t - r)Bu(r) dr = f (t), t E [0, T]; u(0) = uo

from the additional information (4.2)

4.[u(t)] = g(t), t E [0, T]. Problem 2. Find the pair of functions (u, h) in the problem

(4.3)

utt - Au - I h(t - r)Bu(r) dr = f (t), t E [0, 2 ]; u(0) = uo, ut(0) = ui

from the additional information (4.4)

4)[u(t)] = g(t), t E [0,4

Here A and B are closed densely-defined operators in a Banach space X; u and f are functions from [0, T] into X; h and g are scalar functions on [0, T]; uo and ui are given elements on X; and 4) is a continuous linear functional on X. Let Y be the Banach space with the norm ligily = liglix + We consider Problems 1 and 2 in the following two cases: (i) B(dom A) C dom A and B E r(Y);

32

ALEXANDRE L. BUKHGEIM AND GLEB V. DYATLOV

(ii) domB 3 dom A (for instance, B = A). Suppose that A has an inverse operator and generates the strongly continuous semi-group e", t > 0, (see [F]). A typical example here is A = A - a, a > 0, and X = L2. In the case (i) the following theorem is valid: THEOREM 4.1. If the data of the problem (4.1), (4.2) satisfy the smoothness conditions f E C3([0, T], X), g E C2([0,T]), uo E dom A3, f (0) E dom A2,

f'(t) E dom A, t E [0,T];

the agreement conditions g(0) = fluo], g'(0) = 4'[Auo + f(0)]; and the condition 44[Buo] 0 0, then Problem 1 has a unique (global) solution u E C2 ([0, 7],Y), h E C([0,TD. Moreover, the solution can be found by the Newton method and the convergence rate is described by the estimates Ilan — 11C2 ([0,MY) < C1C3ecTT

(C2)2' —1 2n 22: —1

lihn hlIc([o,T]) < CieuT (C2)

where C3 = (IIA-111 + 1)(T3 - 1)(T - 1)-1, C, > 0 and C2 E [0,1] are arbitrary constants, and a depends on C1 and C2 .

Now, suppose that A has an inverse and generates the strongly continuous family ch(tA) (see [IMF]). Then, for example, in the case (i) the following theorem is valid for Problem 2: THEOREM 4.2. If the data of the problem (4.3), (4.4) satisfy the smoothness conditions f E C5 ([0, T], X), g E C3 ([0,7]), f (0) E dom A2,

uo E dom A4, u1 E dom A3,

(0) E dom A2 , f"(0) E dom A, t"(0) E dom A;

the agreement conditions

(1)[ul], g"(0) = 4:1)[Auo + f(0)], gm(0) = (I)[Aui + h(0)Buo + PM];

g(0) = cl,[u0],

i(0)

and the condition (D[Buo] 0 0, then Problem 2 has a unique (global) solution u E C4 ([0, T],Y), h E C1([0,T]). Moreover, the solution can be found by the Newton method and the convergence rate is described by the estimates

(rI \r-1

Ilan - UIIC4([0,71,Y) < CiC3eTk‘-'21

2n

/Ty 1"-1

Ilhn hlIC00,T0 < C1C4euTk

`-'2 ) 2n

where C3 = (IIA-111+ 1)(T5 - 1)(T = T +1, C1 > 0 and C2 E [0,1] are arbitrary constants, and cr depends on Cl and C2.

33

INVERSE PROBLEMS FOR EQUATIONS WITH MEMORY

We can prove similar theorems (with slightly different conditions on the data and the same estimates) for Problems 1 and 2 in the case (ii). The proofs of these results are rather technical and bulky, so we refer the reader to [BK]. We only observe that we reduce Problems 1 and 2 to a nonlinear system of integral equations as in [B1, 1] and then apply the Newton method. As an example, consider the Cauchy problem (4.5)

utt — Au — f h(t — 7-)u(x,r) dr = 0, x u(x, 0) = 0, ut(x, 0) = (5(x),

E R3 , 0 < t
o+

40

DONG PYO CHI and JINSOO KIM

and when F(tco) = 0, sup F(ko) = o(/3). If G(K) = IKIz for p > 0, then for every 0+ there exist local solutions no„ E M of F(K) + OnG(k) converging to Ko as re goes to infinity.

On

We can easily verify that all the assumptions in Lemma 2.1 are satisfied due to the continuity of the parameter-to-solution map which is the result of Theorem 4.1 in §4. Thus we have the following theorem which states that the minimizers of the regularized minimization problems exist and converge to the minimizer of the unregularized minimization problem as 3 —+ 0+. THEOREM 2.2. For each /3 > 0 there exists a solution ko of regularized minimization problem (2.2). Also we have lim 0-1 (sup J(Ka) — J(k0 )) = 0

/3—,0+

provided that there exists a minimizer ko of the minimization problem (2.1). Moreover, if J(Ko) = 0 then sup ,./(ko) = o(/3). For every On -+ 0+ there exist local solutions is 43 of the regularized minimization problem (2.2) with /3 = On converging to ko as n goes to infinity. Proof. Since the observation operator A is continuous, the parameter-to-output map 4) = A o u is also continuous by Theorem 4.1. We can easily show that all the assumptions in Lemma 2.1 are satisfied with F = J and G(•) = 111' 11.11"-, (S1)13 0 Remark 1. When N = 1 the identification of K by the point observation operator instead of the mean-value observation operator is possible. In other words, we may take Ai(v) = v(ri). In this case, the governing equation becomes an acoustic wave equation and the component of the parameter-to-output map is given by 4)i(k)(t) (ri , t). Since the imbedding H1 (1/) y C°(f1) is continuous by Sobolev imbedding theorem, A : L2(I; ) is continuous. Thus the conclusions in Theorem 2.2 still remain valid for this choice of A. Remark 2. The observation operator A corresponds to destructive measurements. By locating the receivers on the boundary and using a suitable interpolation, one can establish non-destructive measurements without any change of the analyses in this chapter. Let us consider the linearization of 4). Let ko be a reference parameter in P. In §4 we ensure that 4)(k) = 4.(ko) + (D4'),o (ok) + o (115kIl[H— ( -2)13) as ok —> 0 where ok = K — ko . Let 4.0 be the observed value. Then using the linearization of 4i the output least-squares functional is defined by f(dk) -= 2 II~ e

— (D1.),0 (on)11L2 (/ilvviv,.)

where 4), = 4)0 — 4)(ko). The output least-squares formulation is to minimize the distance between the discrepancy value 4), and the Gateaux derivative of 4). The condition (DJ)6,, = 0 leads to

((I) 4))*,,o (d k), (DI), (a)) L2 (1,1R ,N,) for all a tion

E [Hm (1l)]3 .

(D4 )Ko (0 )) 1,2 (1

=

Thus Newton-Kantorovich technique leads to the normal equa(D4)):.° (D4)),,0 (ok) = (D4))% ( 4), )

41

Parameter Estimation of Elastic Media

and the corresponding iteration is defined by tC(i+1) = K(t)

{(1)4 ):(+)(D4))K(') -1 (D4 ):(, )(4)0

CIC(2) ))

with an initial guess K(') E P for i E NU {0}. In the case that the inverse of (D40:„ (D4)),,(,) does not exist, an obvious difficulty arises. In order to avoid this difficulty, one can solve an alternative least-squares problem based on LevenbergMarquardt method [6]. If we minimize J subject to the constraint that 11(5K11[H-A]3 = (50, we choose bk to minimize the cost function 1

1

J((5K,13) -= - li(De - (D4)),,O(bK)112L2(/;RNN- + 20 (IISKII?Hm(o)13 2 )

6O)

where i3 > 0 is a Lagrange multiplier. Then we have the modified normal equation {(D4?):0 (D4)),,, +/3} (bK) = (D.1):.(4 e ). Since all the eigenvalues of the operator (D4i):0 (D.:1,)„0 +13 are greater than or equal to (3, we have more stable iterative scheme:

Ko+1) = K(1) + {(D4)):( , (D4)),(,) + 01-1 (D(D);oo (4'0 - 4)(K(i) )) with an initial guess K(0) E P for i E NU {0}. These methods reconstruct the cost functional according to the linearization of 4) and are popular in parameter estimation. By the quadratic approximation of Jo, one can establish an iterative algorithm. Since the cost to evaluate D2 41 is very expensive, we approximate the second-order term using the linearization of 4). If we here use the linearization of 4. in the quadratic approximation of Jo then the resulting algorithm is the same as the NewtonKantorovich method applied to the regularized output least-squares functional reconstructed by the linearization of 4). Instead of linearizing 4) and then formulating J, we approximate J up to second order and then ignore higher order terms of 4). By Theorem 4.5 we have the following quadratic approximation of Jo:

Jo (OK) = (KO

(DJO)Ko (bK) + 2 (D 2 J0)„ (bK, 6K) + 0(1161qHm (c))13

Thus we obtain, up to the first-order terms of 4), { (DC% (D4))„0 + )31 (SK = (DC:o (430 - a. > 0 a.e. on

r

for all nonzero y E RN . One can easily verify that these conditions are satisfied by the matrix (1.2) given as an example of the absorbing boundary condition. Before going into the proof of the Frechet differentiability, we demonstrate that the map u : P C [Hm(1)]3 -4 U defined by u(K) = u, is continuous, which plays an essential role in the proofs of Theorem 2.2 and the fact that the map u : P C [Hm (11)]3 U is twice Frechet differentiable. Let us define d(O,Ki,n2) = Oki) - lbk2) for a map ip : P U and Ki , E P. THEOREM 4.1. Let K i = (Pi, Ai 'Pi) and K2 = (P2, A2, ii2) be two elements in P. nj and g E [147 k+2,2 ; L2 (r))1 N , then for every q > 0 there If f E [wk+2,2(i; Lzor\IN exist 6 > 0 such that Ilkl - K211[H-(c)13 < 6 implies ak

E k+2

d(u k2) Otk "

5 C17

1=k

at

f

at' EL2 (1J,2 0-miN

g at' [L2 (.1;L2 (r))1K)

where C is a positive constant independent of q, f and g. Proof. We note that d(u, , K2 ) satisfies the equation (1.1a)-(1.1c) with - div aA 2 -AI 412 f = (P2 (uk2), g = (p2 A K2 - p1A.1 )uK 2,t + y • aA2-A1,P2-ill (uK2)• The corresponding variational formulation is (4.1)

G(d(u, tcl, K2), v) = ((P2 - p1)u,,u(K2), v) + (aA2-A1,112-1,11 (U"), grad v) + ((P2A,, - P1AK1 ) nfrt2,t,

for all v E [H' (C2)] N . Adding the inequality (3.4) with d(u, K1 , K2 ) to the equation (4.1) with v = dt(u, Ki, K2)(• ,t) and integrating the resulting inequality from 0 to t

45

Parameter Estimation of Elastic Media

with respect to the time variable, we obtain 2

1 1 „ ,_.., , to-L V,ki,k2)(•,00L2 ( miN + 7z: .-. 2 II Pia

al divd(u,Ki,k2)(*,t) I tL 2 (s-n iN A

LIE(d(u, ki , KM., Wil .2[L2(c)r + iid(u, Ki , k2)(., t)iqc,20-opt + ilVF

5_ f0

((P2 — P1)uK2,tt(• s), dt(u, K 1 , k 2)(., s)) ds (crA2-A,422-12,(nK2(•, s)), grad dt (u,

s), dr (u, ki, K2)(., s)) r ds

+ f “P2A,, — AKI +

, tc2 )(•, s)) ds

K2) (• s)[L2(c)iN ds + f t

t

, K2) (• qr,2 (Q)]N ds.

Applying integration by parts in the time variable to the second and the third terms of the right-hand side in the above inequality and using Korn's inequality, we get 1161( 17

Kl) tC2)( • / tVL2(MIN

Il dt

E2

E3

-1-c I Ildt (U1 1

k2)(, S,)11 L,2 (

1J

+C

0

Kl, K2)(•,t,)11 2(c.2)yv

± B2

Ild(u,

-2)i N ds

K2)(- s)q1,2 0-or ds

where El = C f ((P2 — Pi) utr2,t1(• s), d t (u, ki, k2)(', s)) ds, 0

\,,,2 _,I (U"(•,t)), grad d(u,

E2 = C E3 = Bl =

k2)(• ,t))

f (Cr A2- Al ,P2 -P1 (UK2,t (.1 s)), grad d(u,

K2)(•, s)) ds,

0

(•, t), d(u, kl, k2)(•, t))r,

C ((P2AK2 — Pl Ak,

B2 = —c f ((P2AK2 — 0

AK, )u.2,tt (-, s), d(u,

K2)(• s)) r ds.

We shall bound each term of the right-hand side in the above inequality. For E1 we have k2 qHm(1-2)1311UK2.tt qL2 (/;L2 (0))1N

1E11 ft

0

11 dt(U1 K1

N2)('I SA? .LL2(11)iiv ds.

Since E2 = C ((A2

div u„, (•, t), div d(u, k l , k 2)(• ,t))

+2C (( 12 — P1 )E(12,2 (*, t)), E(d(n, K1 k2)( • / t)))

46

DONG PYO CHI and JINSOO KIM

we obtain 1 E21 < C11K1 —

ii?

± 3Ila(u,ki,K2)(•,t_)11r,l (s., )1 ,„• Similarly we get 1E31 5 CIIKI — K211H-(0)J311u.2,0L L2(/;H1(c))1N +

f

Ild(u, ki,K2)(.,$)11rw (miN ds.

Since A is continuous, by trace inequality there exists 81 > 0 such that 1 1311

012 II UK2,t1IFLL...(/;111 (0))1N

1 ,

UI K1 , t)11rH1(0)IN 3 Il dK2)(•

whenever I K1 — K211[H-0-NN < 51 . Similarly there exists 82 > 0 such that 1B21 5_ 0/2 11uN2,ttqL2(/ ; H1(-2))1N +

s...r„, (0)1N ds )11

11deu,K1,

whenever Ilk! — K211[H—( -2)1N < 82. We set 8 = minfoi, 82}. Then by Gronwall's inequality and Theorem 3.1 we get k2)(.1 t)11rL2 (MIN -Flid(u,„1,K2)(., )11

Ildt(U7 2

< cri2

E

au,„ 2 ati

1=0 2

< C7/2

E

/

i=c,

(L-(1;Him))IN f

2 [

L2,,L2(c2)),,

alg

at'

2 iL 2(,;L 2(,))],„

when 11K1 — ti211[H—(m]N < 8. Th's completes the proof for the case of k = 0. Differentiating the equation (1.1a)—(1.1c) with respect to the time variable and repeating the above process give the desired result for k > 0. 0 N In the proof of Theorem 4.1 it is clear that if A : Y is Lipschitz continuous, then the map u : P C [Hm(12)]3 1,1 is also Lipschitz continuous. For K E P and ( = ((1,(2,(3) E [Bm (0)]3 let U1((1) be the solution of the equation

C [L-(r)]3 [L.(r)]N.

(4.2)

= 12, pUz((,)tt — div pA,,Ugi)t + v • a x,,(U,((,)) = g2, U1((2 ) = 0,

f2 x I, x I, f2 x (—co, 0],

for i = 1,2,3 where the source functions on the right-hand sides are given by 1

=

12 = grad((2 div u,), 13 = 2 div((3E(u„)), 91 = — P(DiA)K((i)ux,t — (1AKuK,t, g2 = — P(D2A)/A2)UK,t — (diV uk )v,

g3 = —p(D3 A)„((3)uk,t — 2(3v • E(uk).

47

Parameter Estimation of Elastic Media The corresponding weak formulations for Ui((i)'s are r(Ui ((i ), u) = — (P(DiA)K((i v)r — v) — ((1 AK Un,t Or, r(U2((2), v) -=- (P(D2A).((2)u„,t, v)r — ((2 div uk , div v), — 2((3E(u,), grad v). L(U3((3), v) = — (p(D3A)k((3)uk,t,

By Theorem 3.1 and Theorem 4.1 we have the following energy estimates for Ui((i)'s. We shall omit the proof. THEOREM 4.2. Let K E P and ( = ((1, (2, (3) E [IPNCI)]3 . Suppose that f E {wk+2,2 (i; L 2 (52))1 )) ] N and g E [wk+2,2 (i; L 2 (r,))‘,N for a nonnegative integer k. Then there exists a unique solution Ui ((i ) to the equation (4.2) in [wk+1,2(i; L2 (w)]N n [wk,2(i; (n)) for i = 1,2,3. Furthermore, Ui ((i ) satisfies k+2

f

E i=k+.5,1 ati [L2(I;L2(Q))IN

akui(o

at

at g

atI EL2(1,1,2(r))],

for i = 1, 2, 3 where C = C(S1,T, K., K*) is a positive constant. By the estimate in Theorem 4.2 Ui((i) belongs to [Wk±l ''(/; L2 (o))yv n [Wk,c° (i; Hi (so)r. We are now to show that the map u : P C [Hm(52)]3 U is Frechet differentiable with respect to each parameter. Indeed, we have Ui((i) = (Diu),((i) for K E P and ( = ((1, (2, (3) E [Hm (12)]3. For i = 1,2,3 we define Ri(k, (i) = Ui((i) —

—

where ei 's are the standard basis vectors in 1R3 such that ( = E31 (iei. Then it is enough to show that Ri(K,(i) = o(11(i II H-0-0 ) as (i —> 0. Let = p {()-1,+(,e, —

(DiA)K((. )u ,t}

for i = 1,2,3 and define f l = (idtt(u, +(1el, ti), = — grad((2 div d(u,K (2e2, /0), f 3 = —2 div K3E(d(u, K + (3e3, k))) , 91 =

(AK-1-(iei g2 = (0 diV(d(U,

g3 = 2(3e(d(u, K

,t

AKUK,i)

+ hl,

(2e2, K))) v + h2 , (3e3 , it)) . v + h3.

Then .1V(K,(i ) satisfies the following equation: (4.3)

— div cr),,,(Rz (K,(i)) = 2 , pit,R;(K,(i ) + v • cr,,,,,(Ri (K, (i)) = gi,

S2 x I, F x I,

= Rit (k (i)lt=o = 0,

Ri (K,

for i = 1,2,3. THEOREM 4.3. Let it = (p, A, p) E P and ( = ((1, (2, (3) E [Hm(10]3 . If .,N, then for all rl > 0 there exists f E [W4 2(I; L2 (0))iN and E [W4 2(I; L2 (rspi > 0 such that whenever < 6, g

(k, (i)1Iu 5_ C7/1101H— (0)

4 E at f

l=c5i1

Di g [L 2 (1;1,2 (0))]N

[1,2(1;L2(r))]N

)

48

DONG PYO CHI and JINSOO KIM

for i = 1,2,3 where C is a positive constant which is independent of (2 's, f and g. Proof. Let us consider the case when i = 1. The variational formulation is then to find (K,(i)(•,t) E [H1 (1)]" such that )), grad v) + (pA„14 (K, (1), v)r

(PRit(K, (i),v) + (4.4)

tc), v) + (6( 21,+( i eiun+(iel,t — At,uk,t),v)r = (Cidtt(u, K + + (p {(A,c+oe, — AK)uti+e,e,,t — (Di A) tc ((i )14,4 , v) r

for all v E [111 (M]" with the initial condition in (4.3). We add the inequality (3.4) with IP = (i) to the equation (4.4) with v = 1-4(K,(1)(•,t) and integrate the resulting inequality from 0 to t with respect to the time variable. Then by Korn's inequality we get


0 such that 2

3 1B11

1B31

C772116 112Hm(S-2)

l=1

attf [L 2(1;L 2(1-2))1K (a'

+111R1 (K, (1)(', t)iirip(s-or 2

g

2

ati [L2 (I;L2(P))1N

)

49

Parameter Estimation of Elastic Media

whenever 11(1ll < 62 . Similarly, we can deduce the fact that there exists 63 > 0 such that 2

4

1B21

1B41

C1/2116111/m (C2) 1=2

+

J0L

2

f at' lL2 (1;1,2 (Q))]N

(K,

YtT [L2 (I;L2 (1AN

)

(Q))N ds

whenever 63. Let us set 6 = mini 0 h(x) > C for almost all x E B (a , 5) n D Or

—h(x) > C for almost all x E B(a, 5) n D. Here B (a , 5) is the open ball with radius 15 centered at a. Note that (2.0) can be relaxed. For simplicity and to make the explanation clear we assume that yo = 1. The case when yo is not constant is treated in [151. Let F be an arbitrary nonempty open subset of aa Put D(F) = ff E H 112 (a12) I supp f C We define the localized Dirichlet-to-Neumann map: D(F) f We assume that both D and h are unknown. In this section we consider the problem of finding a procedure to reconstruct D. A reconstruction procedure of h is described in the next section. We denote by GO the standard fundamental solution of the Laplacian. Let c be a needle. Then one knows PROPOSITION 2.1. For each 0 < t < 1 there exists a sequence (vr ) n=1,... of harmonic functions in H1 (Q) such that supp (un lan) C F vn converges to G(. — c(t)) in 1-111,,c(12 \ {c(e)1 0
M if Ig(P)1 < M, P E an. (Here, g' means the tangential derivative on an.) (N2) {P E g(P) > 0} and {P E g(P) < 0} are nonempty connected subsets of an. The condition (N2) prohibits the solution u having a critical point in SI while (N1) essentially means that Vu has a lower bound on an. Thus (4.1) is obtained. Using unique continuation and the fact ui = (1 - 1)H + constant, the estimate (4.2) is proved with OW = I log tl -a (a > 0) in [25]. Thanks to the explicit expression of the solution like (3.4), the estimate, and hence stability, is improved to OW = t' (a > 0) in [13]. In [13] the stability result is extended to the class of perturbed disks. We obtain the following error estimates for perturbed disks. THEOREM 4.1. Suppose that D1 and D2 are e-perturbations of disks. If g

satisfies the conditions (NI) and (N2). IDIAD21 < C (e +11AD1 (9) -AD2(9)11.u,-(an)r

78

H. KANG AND J. K. SEO

for some constant C independent of D3 . In Theorem 4.1, e is the amount of the perturbation. A C2 domain D is eperturbation of B if 8D is given by ap

+ ew(x)v(x), x E aB

where (0B) < 1. Theorem 4.1 follows from the stability estimate for the class of disks and uniform stability for the direct problem P[D, g] under perturbation of domain D. We also proved THEOREM 4.2. Let D3 and c be as above. If AD,(g) = AD, (g) on as), then IDADo l 1. Notice that by (5.2) fas2

(h — AD(g))gda > 0.

It then follows from (5.1) that

ao (h — AD(9))9ciff < µ f I\7h12dx. f On the other hand, by (5.2), we have

Livhi2dx,,,[f iv(up—h)12dx+ f IVuDI2dx

]

< C2 f (h — AD(g))gda. (5.3) an We may argue in the same way when k < 1. It thus follows that there are constants Cl and C2 such that C1 fao (h — AD(g))gda
1 such that K1DI P f IV1112 dX where p depends on the prescribed Neumann data g. The following theorem is obtained in [4]. THEOREM 5.1. There exist constants C1 and C2 such that 1/p

C1

fan

(h — AD(g))gda < IDI < C2

Jan

(h — AD (g))gdo-

This theorem holds for any measurable set D. A Final Remark. Even if we dot not mention about them in this article, there have been several important numerical and more practical works to identify unknown objects. We mention one simple numerical algorithm based on the representation formula. From the current-voltage measurement (g, f), compute H = —Scig +Ds-2 f

in Rn \

Based on the observation (2.9), the algorithm is to find D so that 11H + SD,PD11L2(s) =0 where S is a subset of IV \ Si such that any harmonic function in IV \ f2 vanishing on S vanishes everywhere. This algorithm is proposed and implemented to identify disks in [24].

80

H. KANG AND J. K. SEO REFERENCES

[1] G. Alessandrini, Remark on a paper of Bellout and Friedman, Boll. Unione. Mat. Ita. (7) 3A (1989), 243-250. [2] G. Alessandrini, V. Isakov, and J. Powell, Local uniqueness in the inverse problem with one measurement, Trans. of Amer. Math. Soc., 347 (1995) 3031-3041. [3] G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object, to appear in SIAM J. of Math. Anal. [4] G. Alessandrini, E. Rosset, and J.K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, preprint. [5] H. Bellout and A. Friedman, Identification problem in potential theory, Archive Rat. Mech. Anal., 101 (1988) 143-160. [6] H. Bellout, A. Friedman, and V. Isakov, Inverse problem in potential theory, Trans. Amer. Math. Soc. 332 (1992), pp. 271-296. [7] B. Barcelo, E. Fabes, and J.K. Seo, The inverse conductivity problem with one measurement: uniqueness for convex polyhedra, Proc. Amer. Math. Soc. 122 (1994), pp. 183-189. [8] R. R. Coifman, A.McIntosh, Y. Meyer, L'integrale de Cauchy definit un operateur bourn& sur L2 pour courbes lipschitziennes, Ann. of Math., 116 (1982), 361-387. [9] D.J. Cedio-Fengya, S. Moskow, and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998), pp. 553-595. [10] G. David and J.-L. Journe, A boundedness criterion for generalized Calderon-Zygmund operators, Ann. of Math., 120 (1984), 371-397. [11] L. Escauriaza, E.B. Fabes, and G. Verchota, On a regularity theorem for Weak Solutions to Transmission Problems with Internal Lipschitz boundaries, Proceedings of A.M.S., 115 (1992), pp 1069-1076. [12] E.B. Fabes, M. Jodeit, and N.M. Riviere, Potential techniques for boundary value problems on Cl domains, Acta Math., 141 (1978), pp 165-186. [13] E. Fabes, H. Kang, and J.K. Seo, Inverse conductivity problem: error estimates and approximate identification for perturbed disks, to appear in SIAM J. of Math. Anal. [14] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J. 38 (1989), pp. 553-580. [15] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rat. Mech. Anal., 105 (1989), pp. 563-579. [16] G.B. Folland, Introduction to partial differential equations. Princeton University Press, Princeton, New Jersey, 1976. [17] V. Isakov, Uniqueness and stability in multi-dimensional inverse problems, Inverse Problems, 9 (1993), 579-621. [18] V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems 6 (1990), pp. 311-318. [19] H. Kang and J.K. Seo, Layer potential technique for the inverse conductivity problem, Inverse Problems 12 (1996), pp. 267-278. [20] , On stability of a transmission problem, J. Korean Math. Soc., (1997), pp. 267-278. [21] , Inverse conductivity problem with one measurement: uniqueness for balls in 110, to appear in SIAM J. of Applied Math. [22] , Identification of domains with near-extreme conductivity: global stability and error estimates, preprint. [23] , Uniqueness and non-uniqueness in the inverse conductivity problem with one measurement, preprint. [24] H. Kang, J.K. Seo, and D. Sheen, Numerical identification of discontinuous conductivity coefficients, Inverse Problem 13 (1997), pp. 113-123. [25] , Inverse conductivity problem with one measurement: Stability and estimations of size, SIAM J. of Math. Anal., 28 (1997), pp 1389-1405. [26] J. Powell, On a small perturbation in the two dimensional inverse conductivity problem, Jour. of Math. Anal. Appl., 174 (1993), pp 292-304. [27] J.K. Seo, A uniqueness result on inverse conductivity problem with two measurements, J. of Fourier analysis and applications, 2(3) (1996), pp 227-235. [28] J. Sylvester and G. Uhlman, The Dirichlet to Neumann map and applications, Inverse Problems in Partial Differential Equations, SIAM, Philadelphia, 1990,197-221. [29] G.C. Verchota. Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains. J. of Functional Analysis, 59 (1984), 572-611.

A Moment Method on Inverse Problems for the Heat Equation MISHIO KAWASHITAt, YAROSLAV KURYLEVI

AND

HIDEO SOGAT

Abstract. In this paper we consider an inverse problem for the heat equation in a bounded domain. The uniqueness and reconstruction are studied in terms of some bilinear form on a product set of harmonic polynomials. This form is represented by the Dirichlet-Neumann map R which is the observation data. Key words. Inverse problem, Heat equation, Harmonic moments, reconstruction of a coefficient AMS subject classifications. 35R30, 35K05, 80A23, 41A27

1. Introduction. Let f2 be a connected and bounded domain in Rn (n > 2) with CI ( l > 2) boundary r, and consider the mixed problem for the heat equation in (0, +co) x f2, (P(x)at — A)u(t, x) = 0 u(t, x') = X(t)p(x') (1.1) on (0, +oo) x F, u(0, x) = 0 on Q. Here the density p(x) is a CI function on 11 satisfying (1.2)

0 < p1 < p(x) < P2(< +oo), and X(t) is a (arbitrary) fixed C°° function satisfying 0 < X(t) < 1 in R, x(t) = 1 for t > 1 and X (t) = 0 for t < 1/2. The function p(x') in (1.1) is the boundary value of a harmonic polynomial p(x) (i.e. Ap = 0). By HP' we denote the set of all harmonic polynomials of degree < m ( m = 0,1, 2, • • • ). For any p, q E HP'(-=--- l4„°=0 HP') we put (1.3)

lii,(t; p, q) = f0 P(x)u' (t, x)q(x) dx ,

M "p(p, q) = f0 p(x)p(x)q(x) dx ,

where u = uP is the solution of (1.1). In this note, we explain some properties of 0). Namely, we use LEMMA 4.1. Let p > 1 and m be any positive even integer, and set

/2

Nrirr nlin E

67(X) = (

f_..2

x 121k

k=0

•

Then for any p(x) E 0+u (r2) (where 1 is an integer with 0 < 1 < m/2 and 0 < a < 1) we have II p(x) —

— Y)P(Y) dy Mc' (fi,,_1/ 2) {11-°12 + (C" 1,1)mm-m12±1 }

C

where C, C' are independent of p, p and m, and 11 = { x E Sil dist (x, r) > e }. This lemma is derived from the estimation of each of the following Il ti 13:

ai P(x) — a: f FAn (x

Y)P(Y) dy

= f iny)(a: p(x) — a2 D p(x — y)) dy + f az Sµ (x — y)D p(y) dy R. 5 R. VI + f (a: Oct':'(x — y)

0:5,7r(x — OP(Y) dY

EI1+72+I3i where Dp E Cl+'(Rn) is a continuous extension of p E We fix a basis foi,",1,--.1,...,N(n) in HP' so that {pr} C fp:71+11. Lemma 2.3 means that any xa (I a I < m ) is expressed in the form N(m)

xc, =

> cf,„Nx)pr(x).

The function (57(x — y) in Lemma 4.1 is decomposed into a sum of polynomials x" y3:

'5Am (x y) =

E Ia+$l 0, among all admissible displacements ui(x; with the constraints cr,,n, = Si on rd. Here we regard J : H1/2(r,d)2 w Ft+ = [0, +oo), and the sums are taken for repeated indices i, j = 1, 2. The strain energy added in Eq. (12) as a regularizer guarantees unique existence of the minimum of the functional J(w). With a suitable choice of positive real numbers av for v = 0, 1, 2, • • •, we shall consider the minimizing process; w(v+i) = cow) — av jo(wm) (2.6) where the functional gradient J°(w) E H1/2(rid)2 is defined from the first variation; (2.7)

+ aw) — J(w) = < J°(w), bw > + o(II Ow II)

with a real-valued functional o( II 5w II) of higher order than II ,;5(a II as it tends to zero with the (L2)2 —norm on r id . To seek a concrete expression of J°(w), we notice that

J (co + &a) — J(w) = f d {[ui(x; + 8w) — fci (x)]2 — (x; co) — ft,(x)12 } dr +77 sl

=

rd

fo-ii (x; + bco)Eii (x; + 5w) — crii(x; co)Eij (x; co)}

[ui (x; w + 6w) + ui (x; w) — 2fti(x)] [ui (x; w + Oco) — ui (x; co)] d]

+71 1. fa- (x; w + (5w)Eij (x; w + co) — aid (x; w + (5(.4E0 (x; w) + vii (x; w + 6w)Ei3 (x; co) — crii (x; w)E- • (x; u.,)} d12 = f

d

[2ui (x; w) + o(11 644) II) — 2ui(x)] oui(x; w)dr

+71I faii (x; = f

d

+ (5(.46E0 (x; co) + 6o-i •( ; )Eij (x; w)} dC/

2 [ui(x; c,./) —

(x)1 bui (x; co)dr

+77 f faii(x; w)(5.60(x; co) + acro (x; co) =

clf 2 +

2 [ui (x; w) — 'Mx)] bui (x; w)dc rd

+7.1

sl

2crij (x; co)(5.Eii (x; co)dS1 + 0(11 •54.0 II)

= f 2 [ui(x; w) — d +71

(x)] oui (x;w)dr

2ai •(x; u. ) 196'11 (x; co)dft +

Here we have put (2.8)

bui (x; co) = ui (x; w + bw) — ui (x; w) ,

Ow II) •

Ow II)

NUMERICAL SOLUTIONS OF THE CAUCHY PROBLEM

125

and correspondingly we introduced the definition for 5e, and bat). Moreover, we used the relation; Crijkij = (2/2E0 + AOijekk)kij = 4E06E0 = (20E0 Aoijo610E0 = (50-060 .

AEkkOEll

We notice that the stresses baij induced by the displacements 5u, satisfy Nat =0 ax, ba-un, = 0 (Su, = (5(..oz

(2.9)

(2.10) (2.11)

in S-2 , on rd , on rid .

Equation Eq. (17) follows the constraints cri,n, = Si on rd imposed in the admissible space. We now introduce the dual displacement Cul (x), it2(x)) E H1(12)2 and the corresponding dual stresses ("7,, being a solution of the system of equations; ao-, —0 ux j

(2.12)

in SI ,

subject to the boundary conditions; (2.13) (2.14)

atj ni = 2 tu, (x; w) — uz (x)] iti = 0 on rid •

2718; on rd ,

Using the Gauss divergence theorem, we know that fs-1

— f 13-zi buidS2 =6-2,72,Juedr f

abu d ax

Here we can see that a5u,=aui as; as; As the matter of fact, from the symmetry of aij we see abui 1 abui axi -5T 3 3 ) = 2 (eri3 axi 1 0574 abui 1 aoui 0,5ui = er" ( axj axi = xi = (3-0 (5Eij = (2kceij + )bij tkk )(560 = 2eiji.uSeij + ekkAoEtt = ei; (206i; + AmEtt) aui au; „ axi )ocrij = ei 50-0 = 2 1,axj 1 aui aui aui 1, aui Oai ) = k —oa• + —00-••) = axj axj 3' 2 axi 2 axj aui = as;

aoui a_ x j =

1 aSui aoui

126

K. ONISHI AND Q. WANG

from which we get

of, a& Su2dS2 = rti n3 (Su I dr — f —.r50-zidO si az; L ax; Jr1-• = f i3r,inibuic/T — r

f fti oatini dr + f

ilt a5aii

2 ax;

d9 .

Therefore, from Eqs. (19), (17), (21), and (16) we obtain 0=

f

rd

erijnjSuidf +

erijnjau,dr rid

Consequently, from Eqs. (20), (10), (18), the traction condition in Eq. (11), and using the divergence theorem again we know that

J(ca + 8w) — J(w) = f ef on3Suidr — 277 f Siouidr + f 2o-oniSuidr + rd

rd

=— rid

mei

•

(Reiff' +

2o- • 6widr o(11 6co rid

21Si)6coidr +

= f

.5c.o 11)

(5w II) •

rid

Now we know the explicit form (2.15)

Jif(w) = —a•ti n3 + 21151

on rid •

Using this result, we can summarize an algorithm for the minimization as follows: Algorithm I

Given (40) . For v = 0, 1, 2, • • •, until satisfied, do: acr(v) Solve u xi = 0 with o- iv)72.3 1 r, = Si, u!') Ir.a = (.4v) to find u2") (x) on rd and .92(v) (x) on rid. a-d (v) c") (x. ca) —I-4(x)] + 2i) S, , fii(v) Solve c.tn x3 = 0 with ertjc") - d = 2[u(v) , to find J°(w(v) ) i.e. J:(co(v) ) =- —Slu) +27S2") on rid • Update w("-1) = co(') — az,..1°(o.)(0 ).

=0

2.1.2. Traction Approach. Instead of identifying the boundary displacements W2) on rid in subsection 2.1.1, we shall consider in this subsection identification of boundary traction r = (T1, T2) on rid. We express u,(x) = ui(x; r) to denote

w =

the dependence of the solution u, on T. Our objective is to find a proper Ti, which minimizes the following functional (2.16)

K(r) := f [S,(x; 7- ) — Si (z)] 2c1r + rd

f ai3 E c1S1

among all admissible tractions S,(x; r) with the constraints u, = ft, on I'd. Here we regard K : 11112(r,02 r 1—* R+ . With a suitable choice of positive real numbers a„ for v = 1,2, • • •, we shall consider the minimizing process; (2.17)

T(v+1) = .7-(v) — auic(r(0)

127

NUMERICAL SOLUTIONS OF THE CAUCHY PROBLEM

where K'(r) E H'/2(rid)2 is defined from the first variation (2.18)

K(r + Sr) — K(T) = < K' (r), Sr > +o (11 ST ID •

To seek a concrete expression of K'(r) in a similar way as to ..r(w), we notice that

K (r + Sr) — K (r) = f 2[Si(x; T) — St (x)]5S,(x; r)dr rd

+

~sz

thin 2crti(x;T) oxi (x;r)df2+ 0

Sr II)

where we put variations in boundary traction by

SS,(x; r) = Si (x; T ST) — St (x; r) , and Sui(x; r) are corresponding variations in the displacement. Using the relation aid

19(514 aui = ---05150

ax

ax

and by the integration by parts, we have

K (r + 5r) — K(T) = f 2[Si(x;r) — Si (x)]5Si(x; r)dr rd

+27, fui bv n dF — 271 f ui 56cr"cif! + 0 (11 Sr n

II) •

The stresses Sri; induced by the displacements oui satisfy akij =

ax;

0

bui = 0

5Si = Sri

in Si, on rd , on rid •

We now introduce the dual system (2.19)

8x;

in S2 ,

=o

ui = 2[Si(x; r) — Si (x)1 + 2rjui Si = 0 on rid •

(2.20) (2.21)

on

rd ,

As before we can see that

0 = f oxj

ouidn = Jr &on buidr — f aouIdC2 = f

= f S-ibuidF — f

oxi

dr + f fija, (5a" dS2 = — f axi rd

which yields the relation;

frd

itibSidr = —

rid

.

of, n taxi

— f — f

rid

df2

1.4(5ridr ,

128

K. ONISHI AND Q. WANG

Consequently we know that K(r + or) — K(r)

f

utos,dr + 271/ u,Ort dr + o (II or r.d

fr.d

+ 2nui)br,dr + o or

ID •

Therefore we obtain K'(r) in the explicit form (2.22)

K: (r) =

+ 27/u, .

Using this result, we can summarize an algorithm for the minimization in the traction approach as follows:

Algorithm II Given T(°) . For v = 0, 1, 2, • • until satisfied, do: ao-() Solve a xi " = 0 with u2(v) Ir d = 0.1;)n.711' d = to find S2(v)(x) on rd and u!') (x) on rid.

(V)

ae(1)

Solve u, n Ird = 2[e(x;r) — Si(x)] + 2r/vi , x = 0 with to find K:(1-(0) = +2nu!v) on rid . Update r(v+1) = r(v) — ai,K°(co(0).

=0

2.2. A NUMERICAL PROCESS. 2.2.1. Choice of ak. We shall discuss a way of choice of the sequence {a1,}. To this end, we employ the Armijo criterion [5] in mathematical programming, so that the sequence {a,} satisfies J(co(u+1) ) < J(w(0 ). Namely, the steps a, are controlled in such a way that J(w(v)

J°(w(v) ) 11 2

ay./°(4.0(v) )) < J(co(v) )

with a constant l (0 < 1 < 1/2). Here we denote J°(w) 112= < J°(w), J°(w) >=

f

rid

1 .7°(‘'. )1 2dr •

Controlling the step size a, Given parameters 0 < 6 < 1/2, 0 < T < 1 ( say, e = 0.1,7- = 0.5 ), and E = 10 — 4 . If II P(4.0 )11< e, then stop. else/ 0 := 1. For m = 0, 1, 2, • • •, do: If J(co(v) — Sm.1°(w(u) )) < JP(v) ) el3m J°(w(") ) 112, then a„:= 13m . else 13m+i := . 2.2.2. Boundary Element Method. For each step v in the displacement ap(x) and t')(x) on proach, we merely require boundary values u!') (x) on rd, rid to implement the algorithm. For each step v in the traction approach, we merely require boundary values ,S,'') (x) on rd, li!v) (X) and fti(y) (x) on rid to implement the

NUMERICAL SOLUTIONS OF THE CAUCHY PROBLEM

129

corresponding algorithm. On these reasons, the boundary element method is expected to fit for an approximate method of solution to get those values for displacements and tractions. The basic tool in the application of the direct boundary element method is a boundary integral equation derived in the following: We start with the equilibrium equation Eq. (10). We consider the corresponding weak form; ,2-71: (x)dct(x) asi _

=0

Buz au, 1 84 = CI -- the test functions 4(x). Let a . = - — + — . By using axi " 2 (Oxi Ox: a ax integration by parts twice yields

Jr

ate n

— f uicri*inidr +

Put Si = aui ni and S: = (2.23)

ui

acr? c/C2 = O. forxi

3. Then we have

Si4dF - uiS:dr + Jut 2 a„-1:J df/ = .

As the test function, we take u: (x) = u*ki (x, t) satisfying

Oa:,

(2.24)

-,6(x 4

ax3

k = 1, 2 .

The fundamental solution uZ z (x, t) is understood as the i-th component of the displacement at the point x when unit point load is exerted at the point in the direction of the ordinate xk in the elastic continuum extending to infinity. For plane strain, we have

11 u:3 (x, t) = 87(1 v)G { (3 - 4v) In (- ) 6

-

with r = -

SZ3 (x, =t)

r

Or Or

aZYx;

ar_xi — 6 , and the corresponding traction is given by r

ax, —1

4741 -

v)r [1(1 - 2060 +

ar ar ar

5-771, - (1 - 2v) C

ar ar - y;.Tri,)] . axt ni

Using 43 and SZ), we can obtain from Eqs. (30) and (31) the integral representation of the displacement u,(t) at the internal point t of I/ in the form (2.25)

ut (t) = f 43 (x, t)S (x)dr (x) -

Cuj (x)df (x)

On the other hand, at the point t on the boundary, the following boundary integral equation holds. (2.26)

co (t)ui (t) + f S:3 (x, t) (x)dr (x) = fr uZi (x,

i(x)clf (x) ,

130

K. ONISHI AND Q. WANG

where the coefficients cu depend only on the geometry of the boundary. They can be expressed as

(2.27)

=

f s:;(x,Cdr(z) •

1 Particularly when the boundary F is smooth, we know c„, = 2 . Moreover, if the point t is interior to the domain C2, then we know c2j = ou . Therefore, the boundary integral equation and the integral representation as well are written in the form

(2.28)

fr

(x, t) { ui (x) — u (t)} dF(x) = fr u:i(x, t)S i(x)dr(x)

The boundary element method using linear interpolation functions is used for the numerical solution of the plane elastostatics problems. The boundary integral equation is discretized by the method of point collocations.

2.2.3. An Example in Plane Elastostatics. A Cauchy problem for plane stress state in a square domain f/ = (0,3) x (0, 3) is illustrated in Fig. 4. A part of the boundary consisting of three sides BC, CO, OA constitutes rd, on which both displacements (fzi, u2 ) and tractions (Si, S2 ) are prescribed as follows.

(ui , R2) -= (x1,0) ,

R2) = (xi, 0) (wi

012) =-- (x1,0) ,

1 ) on (Si, S2) = (0, 1.4i (S1, S2) = (

(S1, S2)

E 0) 1 — l/2

(0,

1

vE

y2 )

on

BC , CO ,

on OA.

Boundary element mesh is also presented in Figure 4. The domain was divided into cells for convenience. By the displacement approach, we identified unknown displacements (wi,w2) on the rest of the boundary AB = I'2& We took q = 0 in this example because exact

131

NUMERICAL SOLUTIONS OF THE CAUCHY PROBLEM

data are available. Initial guess is set as co(°) = (0,0).

(31, 32) = (0, 14,7) (u1, u2) = (xi, 0) C

rd

(S1) S2)

(14.1,1/2)

= (--f=2 E/, 0)

(r/1,7-./2)

=

rd

(0, o)

0

X2

xl

3) x (0,3)

rd

rd

=

CO2)

•

A

•

(6 x 6 mesh)

(51,32)=(0,_f3) (.1,..2)=(x1,0)

Figure 4. Cauchy problem in plane elastostatics (left) and linear boundary elements (right). Displacement identified by the boundary element method is shown in Figure 5, where the true displacement is also presented for reference. The numerical identification process was stable. The identified displacement on rid after 19 number of iterations is in good agreement with the true displacement.

Figure 5. True displacement (left), and the displacement identified (right).

132

K. ONISHI AND Q. WANG

3. CONCLUDING REMARKS. We considered initial value problems of the Laplace equation, and the Navier equations in elastostatics as well, regarded as boundary inverse problems. Using the method of the steepest descent for a functional to be minimized, each of our problems is led to an iterative process consisting of the solution of primary and dual problems. The finite element method and boundary element method are applied for numerical solution of the primary and dual problems. The effectiveness of our treatment was demonstrated by simple examples. It is concluded from our numerical experiments that the method proposed in this paper is stable and convergent.

REFERENCES [1] K. AMAYA AND S. Aoxl, Inverse analysis of Galvanic corrosion problems using a priori information, Proceedings of the 43rd National Congress on Theoretical and Applied Mechanics, Japan Academy Council, Tokyo, January (1994), pp. 481-482. [2] S. KUWAYAMA, S. KUBO, K. OHJI, AND T. TAKAHASHI, Analysis of boundary value inverse problems by using the relationship between magnification of errors and singular values, Proceedings of the 43rd National Congress on Theoretical and Applied Mechanics, Japan Academy Council, Tokyo, January (1994), pp. 515-518. [3] M. KUBO, Y. Iso, AND 0. TANAKA, Numerical analysis for the initial value problem for the Laplace equation, in Boundary Element Methods (Eds; M. Tanaka, Q. Du, and T. Honma), Elsevier Sci. Publ., (1993), pp. 337-344. [4] M. KUBO, L2-conditional stability estimate for the Cauchy problem for the Laplace equation, Journal of Inverse and Ill-posed Problems, 2, No.3 (1994), pp. 252-261. [5] L. ARMIJO, Minimization of functions having Lipschitz-continuous first partial derivatives, Pacific Journal of Mathematics, 16 (1966), pp. 1-3. [6] D. D. Ang, D. D. Trong, and M. Yamamoto, Unique continuation and identification of boundary of an elastic body, Dai Hoc Tong Hop Ho Chi Minh City University, preprint (1995), pp. 118. [7] W. YEIH, T. KOYA, AND T. MURA, An inverse problem in elasticity with partially overprescribed boundary conditions, Part I: Theoretical approach, Journal of Applied Mechanics, Transactions of the ASME, 60 (1993), pp. 595-600; and T. KOYA, W. YEIH, AND T. Mule, Part II: Numerical details, ibid., 60 (1993), pp. 601-606. [8] K. KOBAYASHI, K. ONISHI, AND Y. OHURA, On identifying Dirichlet condition for 2D Laplace equation by BEM, Engineering Analysis with Boundary Elements, 17 (1996), pp. 223-230. [9] Y. OHURA, K. KOBAYASHI, AND K. ONISHI, Identification of boundary displacements in plane elasticity by BEM, Engineering Analysis with Boundary Elements, 20, No.4 (1998), pp. 327335.

Inverse Source Problems in the Helmholtz equation Sabur ou Sait oht Abstract. In this paper, we shall consider the inverse source problems in the Helmholtz equation, but in the first part in this paper we shall introduce general methods for linear transforms in the framework of Hilbert spaces and new ideas for some general inverse problems in the linear systems based on the recent research note ([6]). The original parts in mathematics were written before the publication of [6] as an original work. Key words. Integral transform, reproducing kernel, Helmholtz equation, inverse source problem AMS subject classifications. 44A05, 46E22, 35R30

1.

Linear transforms of Hilbert spaces

In 1983, we published the very simple theorems in [5]. Certainly the results are very simple mathematically, however they appear to be extremely fundamental and widely applicable for general linear transforms in the framework of Hilbert spaces. Moreover, the results gave rise to several new ideas for linear transforms themselves. We shall formulate a 'linear transform' as follows: (1)

f (p) = I F(t)h(t,p)dm(t), p E E.

Here, the input F(t) (source) is a function on a set T, E is an arbitrary set, dm(t) is a a-finite positive measure on the dm measurable set T, and h(t, p) is a complex-valued function on T x E which determines the transform of the system. We shall assume that F(t) is a member of the Hilbert space L2 (T, dm) satisfying (2)

f IF(t)I2dm(t) < co.

The space L2(T, dm) whose norm gives an energy integral will be the most fundamental space as the input function space. In other spaces we shall modify them in order to comply with our situation. As a prototype case, we shall consider primarily or, as the first stage, the linear transform (1) in our situation. As a natural result of our basic assumption (2), we assume that for any fixed p E E, h(•, p) (3) for the existence of the integral in (1). 2.

E L2 (T, dm)

Identification of the images of linear transforms

We formulated linear transforms as the integral transforms (1) satisfying (2) and (3) in the framework of Hilbert spaces. In this general situation, we can identify the space of output functions f (p) and we can completely characterize the output functions f (p). t Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376-8515, Japan

133

134

S. SAITOH

In order to identify the image space of the integral transform (1), we consider the Hermitian form (4)

K (p, q) = fT h(t, q)h(t, p)dm(t) on E x E.

The kernel K(p, q) is apparently a positive matrix on E in the sense of n n

E E j=1 j'=1

>0

for any finite points {pi} of E and for any complex numbers {C, }. Then, following the fundamental theorem of Aronszajn-Moore, there exists a uniquely determined reproducing kernel Hilbert space HK comprising functions f(p) on E satisfying (5)

for any fixed

q E E, K(p, q) belongs to

HK

as a function in p,

and (6) for any q E E and for any

f E HK

(f(),K(' , q))11K = PO.

A kernel K(p, q) satisfying (5) and (6) is called a reproducing kernel for the Hilbert space HK. For various constructions of the reproducing kernel space HK from the positive matrix K (p, q), see ([6], Chapter 2, Section 5). Then, the point evaluation f (p) (p E E) is continuous on HK and, conversely, a functional Hilbert space comprising functions on E such that the point evaluation is continuous admits the uniquely determined reproducing kernel K (p, q) satisfying (5) and (6). Then, we obtain Proposition 1. The images f(p) of the integral transform (1) for F E L2(T, dm) are characterized as the members of the Hilbert space HK admitting the reproducing kernel K(p, q) defined by (4)-

3.

Relationship between the magnitudes of input and output functions

Our second theorem is Proposition 2. In the integral transform (1), we have the inequality Ili


a}. The kernel K0,, (r', r") is a positive matrix on {r > a} and so, there exists a uniquely determined Hilbert space Hic“, admitting the reproducing kernel Ka,j(r', r"). Furthermore, the images u(r') of (13) belong just to the Hilbert space HKa The expansion (18) implies that the images u(r') are expressible in the form oo

u(r') = k2 E(2n 1)CV (a)

(19)

n=0

E Em,n(n+

)!

m=0

x P,1.," (cos 0' )11;,,I) (kr')(A,T cos nice' + .13,1„" sin mcp') B,T} satisfying

for some constants { oo (20)

n

E(2n

1)C,(,'>(a)

n=0

\

E 67(n "71 m)!

m=0

{iArn2 + 11,112} < cc.

Conversely, any u(r') defined by (19) with (20) converges absolutely and uniformly on compact sets in {r' > al and belongs to the Hilbert space HK.,. Since the family in,n_o {PP (cos 9) cos Try, 17 (cos 0) sin imp}" is complete in the Hilbert space consisting of the functions f (9, (p) with finite norms

l.

77 27r o fo I

f

f (0, (,o)12 sin OdOchp} < oo

(see, for example, [2, p.24]), we have the representation of the norm 11u11HK„, in the form (21)

(2n + 1)00 (a) t rn (ri(n+ min )1)1{1A,T12 +1/4"12}.

11u113/K... = k2 n=0

m=0

Furthermore, we have the isometrical identity 47 7. a), the wave u(r') is expressible in terms of (25)

u(b, 0 , (p)

for any fixed b(b > a).

139

INVERSE SOURCE PROBLEM

We shall derive the inversion formula representing p* in terms of (25). Using the reproducing property of Kn,i(r,r') in we have u(r') = (u(r), = (u(r),

r/)) HK,,,, p iklr-ril e -ikle-ri

1

I

47r ria ir'

u(r') =

1 pi (r) — dr r2

for a function p1 = rep satisfying ipi(r\i2d2 )1 r fr>a

(32)

f

>a

ip(r)(2r2dr- < co.

We form the reproducing kernel (33)

Ka,o (r' , r") =_

1 I 47r

eik lr'-rle-ikir"-r1 dr

r>a

le

ri r2.

By using (15) and the orthogonality relations of (16) and (17), we have n ((n - m )! Ka, o(r), r") k2 E (2n + 1)C7C,°) (a) E Em n=0 m=0 n + m)!

(34)

xP,71 (cos 0))P,T (cos 0")h2) (kr')41) (kr") x {cos rricei cos mcp" + sin my,' sin me},

where, note that 00 7r C$,°) (a) := f jn(kr) 2dr < 2k a

j: fri+1.(kr)21 dr < oo,

(see, for example, [7, pp.377 and 405] and [8, p.399]). This series (34) converges absolutely and uniformly on compact subsets on {r > al. Hence, by arguments parallel to those of Theorem 1, we have Theorem 2. For the source function p* satisfying (29) and (30), we have the inverse formula, for any fixed b(0 < b < a), p* (ri)

1 (u(r), ri

- r11 ) HK.

Em m)! ( 0 (n + m)! 47rkr2 n=o CI(1°) (a)41) ) (" )m=

x P" (cos

7r f 27r

f

u(b, 0, (p)P,T (cos 0) cos m(cp - (,o1 ) sin OdOdcp.

8. Use of radiation patterns The behaviour at infinity of the solutions u(r) of (11) is r etkr u(r) = g( r ) 47rr [1 + 0(r -1)], where the radiation pattern g is given by (12) (see, for example, [2, p.20]). We shall give the solution of the inverse problem representing the source function p in terms of the radiation pattern g.

142

S. SAITOH

In order to examine the integral transform (12), we form the reproducing kernel K(1_1 , if ) = _ 1 i e-ik(:4,r)eik(f4,r)ar. . 'r' r" 47r 0 for all x E S2,

where •ymin is a given positive constant. Then, our problem can be stated in the following sense: Determine unknown Neumann data g E II — (rE) and coefficient function 'y E Lm(fi), satisfying V • (yVu) = 0 au (IBP) u = uB and ry a- =413 au g 7 an

in ft, on rE, on rE,

where n is a unit normal vector exterior to the boundary (Figure 1.1). In case of the inverse problem in electrocardiography, rE and rE are the body surface and the epicardium, respectively. The problem (IBP) is not uniquely solvable.

*Institute of Computer Science, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan (shirotaMmit o.ipc.ibarald.ac. jp). t Common Chairs, Gunma University. qnstitute of Computer Science, Ibaraki University. 155

156

K. SHIROTA, G. NAKAMURA AND K. ONISHI

p Eicardiurn FIG. 1.1.

E:

au

= 97

Tomo-electrocardiography

In this paper, we propose an algorithm for the numerical resolution of the problem (IBP). We introduce an object functional to be minimized, then the problem (IBP) is recast to a variational problem. We adopt the direct variational method using the steepest descent. Moreover we make use of the projected gradient method to update the coefficient function. We show that this algorithm can treat the unknown boundary values and C1'1 coefficient functions. For non-smooth functions, we confirm the effectiveness of our algorithm by numerical experiments. 2. NUMERICAL PROCEDURE. 2.1. Variational method. Let 02_ (n) be defined by C72,„(n) := 11' E Ci'l (C2) I 'Y(x)

"Ymin for all x E c/} .

Moreover, we assume that (2.1)

-y E

(n),

'us E Mrr

g E 114 (rE),

— 4 (r /3 )•

The unknown Neumann data_g and coefficient function y are determined by minimizing the functional J : (SI) x 117 (r E) R+ := [0, +oo), defined by J('Y, g) = f

E

Ig[7,

4/31 2 dr.

Here, the function g[-y, g] is defined by g[y, g] := -y

au[-y, g] an rB

where the function u[y, g] is the solution of the boundary value problem V • (-yVu) = 0 in SI, (PRP)

u = 11B an an

= 9

on r B, on r E.

IMPEDANCE COMPUTED TOMO-ELECTROCARDIOGRAPHY

157

However, it is difficult to find one of the minima of J directly, because two unknowns are involved. We notice that -y and g are mutually independent. To find the minimum 'y and g, we make use of the steepest descent method: For k = 0, 1, 2, ... , = "-rk ak,f-y("Tkl 9k), gk-F1=-- 9k OkJg(Yk> 9k).

(2.2)

Here J.-1 and J9 mean the first variations of J, defined by g) — J(ry, g) = J(7+ g), 5-y) + (116711Q) J(-y, g + 5g) — J(-y, g) = (J9(-y, g), bg) + o(1lgli rE ) , where

(to,

:= fc, tozP dC/,

(1, 9) := L E fg dr,

110in := (12 1(701 2 dC/)

IlfIlr. := (LE If12 dr)1.

To implement the steepest descent method, we must get concrete expressions of the first variations J. and J9. To begin with, we will obtain the first variation J.: We take any admissible g, and fix it. For -y C C,),Tm (12) and -y + 5-y e ca), we notice J(-y + dry, g) — J(-y, g) ={lq[y + O-r, g] — 41312 — 14[1', g] — 413121 dr r8

=

IrB {g[-y+ 5-y, 9] + gt-r, — 21131 x lq[-y + 51', 9] —

= =

g]}

dr

frB {2 (q[1', 9] — 4B) + oq,} .5q, dr frB 2 {g[-y, g] —4B}aq.„ dr + frB 15,1,12 dr,

where Sq., := q[y + 51', 9]— q[-y, Now, we take the function v which is the solution of the boundary value problem V • eyVv) = 0 (DUP) v = 2{g[v, 9] — B }

ry

=0

in C2, on rB, on rE.

158

K. SHIROTA, G. NAKAMURA AND K. ONISHI

From the definition of the functions u[-y, g] and v, we have C. • Vvdft J. fey + 8-y)Vu[-y + by, g]I —ey-yVu[-y, 8,y) au r.y

irBurE I st

= fr B

,yau

'

v dr

an t

•fey + (5-y)Vu[1, + (5-y, g] — -yVu[-y, g]l v c1S1

fq[-y + (5-y, g] — ey, g]} v dr

. 2 fey, g] — qB } (5q,df.

=1rE

On the other hand, we note that the most left-hand side

f(-r + 8-Y)N'Y + ,57, gl

Vvd9

(5-yVu[-y + (5-y, g] • Vvc/Q +

J

-yV(5u, • VvdS2,

g]. Since (5u7lr„, = 0 and the function v is the solution where 5u, := u[-y+o-y, g] of the boundary value problem (DUP), we notice si n -yVou, • Vv c/S/

=

frE = 0,

av av (5u,-y— dr' + f (5u,-y— dI' — an an rE

(5u, (V • (yVv)) (K2

so that fey + (5-y)Vu[1( + ö-y,, g] — -yVu,[-y, g]} • Vvcill

= f (5-yVu[-y + (5•y, g] • Vv (151

=

J

(5-yVu[-y, g] • Vv d12+

J

(5-yV(5u, . Vv d2

S2

0

Therefore,

frB 2 {q[-y, g] — 4B } 4, dr = f (51A7u [-y, g] • Vvc/f2+ (5-yVdu, • Vv sz st Hence, we obtain J(-y + (5-y, g) — J(-y, g)

(5-yVu[ry, g] • Vv c/C1+

J

(5-yV(5u, • Vv al+ La k5q712 dr•

Here we note that

Lo-yV 5u.., •• Vv c/C2 = 0(115-Y11n),

14712 dr = 0(11.5-Ylln)rp

IMPEDANCE COMPUTED TOMO-ELECTROCARDIOGRAPHY

159

Consequently we know that

J7(1', g) = Vu[y, g] • Vv. In a similar way, we can get the first variation J9: For g E Hi (rE) and (5g E Hi (FE), we have

J(-y, g + (59) — J(-y, g) = f 2 {q[-y, g] — .1,3 } (5q9 dr + f 14912 dr, rE rE, where kg := q[-y, g + bg] — q[-y, g]. We put Su, = u[-y, g + (5g] — u[-y, g]. Then, from Green's integral theorem,

0

= fn _ i

{V • (yVv)bu, — vV • (-yVbu,)} A/

Jr Bur El(' av '2• ) (5u9 v EY aaanug ) if dr.

Since Oug lrB = 0 and -y Ov = 0, an rE

av \ ( 08u, \ 1 li an ) jug v 'I'n )f

li,E{(- -

dr =

-

05ug ) dr.

frBurE v (-r On

, 08u9 O5u Moreover, since vIr E = 2 WY, 9] — qa I , -Y g = (5q, and -y = 89, we On rB an rE have °bug — ) dr =_ f 2 {4 [y, g] — q,3 15q, dr + f v69 dr. rE, rE A-BurE v Cry On Therefore, we obtain fr B 2 {4[y, g] — 4,3 } Sq, dr = — i v6gaT, rE and

J(-y, g + 89) — J(-y, g) = — i vbg dr + f 18q9 I 2 dr. rE rE Here we note that

LB 1 ,5q212 dr = 0 (116911r.) • Hence, we obtain

J9(-Y, 9) = —vIr E . We assumed that the exact conductivity satisfies the condition (1.1). However, we cannot guarantee the updated conductivity -yk+i obtained by using (2.2) to satisfy the condition (1.1). In order to overcome situation, we adopt the projected gradient method. This method is the revised algorithm of the steepest descent method.

160

K. SHIROTA, G. NAKAMURA AND K. ONISHI Let P : L2 (C2)

C7;L(F1) be a projection, defined by -Y(x), if ^l(x) (P7) (x) = { 'Ymin, otherwise.

Then, the conductivity is updated by the following steps: 1. := P ("Yk J-y(7k 90) and sk = 'Yk ik • 2. -yk+i = aksk (0 < Cek < 1). This updated conductivity -yk.1-1 satisfies the condition (1.1), if the previous conductivity -yk satisfies it. 2.2. Numerical algorithm. To use the steepest descent method, we must choose the step size ak and Ok. We employ the Armijo criterion: Armijo criterion 1. If is sufficiently small, then stop; else go to the next step. 2. no := 1. 3. For m = 0,1,2, ..., until satisfied, do: If J (-yk — rm.,: else 7/m+1

gk), gk) J(-rk,gk)—&i. f J-y(rk, gic)skcISI, then ak := Tihn•

1 Here 0 < 1 < — and 0 < T < 1 are given constants. For choosing the step size Ok, we 2 also use the Armijo criterion. In order to solve the inverse problem (IBP), we summarize the algorithm as follows: Numerical Algorithm (F2) and initial Neumann data 1. Pick an initial coefficient function -yo E C1,1 lrtnin go E HI(I'E). 2. For k = 0, 1, 2, . , (a) Solve the boundary value problem in Q, on "B,

V • (7kVuk) = 0 Uk = UB

auk 'yk— an to find Vuk and nk

on r E

=gk

auk

• an r B (b) Solve the boundary value problem V • (7kVvk) = 0 vk = 2(qk

auk 7k— an

=0

413)

in Q, on r B, On FE

to find the first variations 47k,gk) = Vuk • Vvk and Jg(-rk, sk) —vkh-E •

=

IMPEDANCE COMPUTED TOMO-ELECTROCARDIOGRAPHY (c) (d) (e) (f)

161

"ik := P ("Yk — 1ey(70) and Sk = l'ic — '1k • Update the conductivity: 'yk-i-i = "Yk — aksk • Update the Neumann data: gk+1 = 9k — Okfg h,k, 9k). If J(ryk+i, 9k+1) is sufficient small, then stop; else k := k + 1.

3. NUMERICAL EXPERIMENTS. In this section, we show a numerical experiment about our algorithm. In order to solve numerically the two boundary value problems in our algorithm, we make use of the finite element method with linear triangular elements.

3.1. Smooth coefficient function case. We consider the following two-dimensional example: The domain Q is an annulus with the inner radius 1 and outer radius 3. Suppose that the exact coefficient function 7 and Neumann data g are set by

-y = cos (2r sin 0) + 1.5, g = — cos° {cos (2 sin 0) + 1.5} ,

where (r, 9) is the polar coordinates (Figure 3.2). The domain is divided into triangular mesh as shown in Figure 3.3.

FIG. 3.2. Example

FIG.3.3. Finite elements

The initial estimates are set as -yo = 1.0 and go = 0.5g. After 25 times of iterations, we have J = 0.90 x 10-7. Figure 3.4 shows the calculated Neumann data 925 on FE, which are in good agreement with the exact one. The distribution of the exact coefficient function is shown in Figure 3.5. Figure 3.6 shows the calculated coefficient function -y25. This calculated coefficient function is in good agreement with the exact one except for the coefficient function near the inner boundary.

162

K. SHIROTA, G. NAKAMURA AND K. ONISHI 3.00 2.00 1.00

5

G.

0.00 -1.00 -2.00

Central angle

FIG. 3.4. Calculated Neumann data

FIG. 3.5. Exact coefficient function

FIG. 3.6. Calculated coefficient function

In the previous experiment, the initial guess as to the Neumann data was good enough. The initial estimates are set anew as -yo = 1.0 and go = 0.0. After 30 times of iterations, we have J = 0.18 x 10-5. Figure 3.7 shows the calculated Neumann data. The estimated Neumann data are fairly good. Even if the number of iterations is further increased in this example, the calculated result does not improve. Figure 3.8 shows the calculated coefficient function, which is relatively good. 3.00 2.00 1.00 0.00 -1.00 -2.00 -3.00 0.00

1.05 2.09 3.14 4.19 5.24 6.28 Central angle

FIG. 3.7. Calculated Neumann data

IMPEDANCE COMPUTED TOMO-ELECTROCARDIOGRAPHY

163

O

O

-2

2

0

FIG. 3.8. Calculated coefficient function 3.2. Non-smooth coefficient function case. In this section, we attempt a numerical experiment in case that the coefficient function is not a continuous function. We recall that the coefficient function is often discontinuous in practical applications. In this case, we can not mathematically guarantee the existence of the first variations. Let C/ be the same domain as in the section 3.1. The exact coefficient function and potential in the domain S2 are set as {0 . 5 (1 < r < 2, 0 < 0 < 27r) 5.0 (2 3, v > 72 2 - 1. Take the transform P for (2.8)

L2 or

jv:1)

L2

170

TAKASHI TAKIGUCHI

Let P4°43) is the Jacobi polynomial of degree n and V,;,),(x) := Wu (x)Ix11 P t, 3 ' I+ 1) (21xI 2 — I XI (1;.„

E

(8 , y) :=

d„o„

(A9)V,,r (A 9 1 y)

LEA j+/:even a 2i := 77 2 Cr'

(21+ 7/ — 2) N(n, l)

1

N(n-1,j)x j=O, j+I:even

D(2m + 4v + 2 — n,21 — 2 + n, 2n — 4v) x x D(2rn + n + 1,2j +n — 3,4v — 2n + 2) x x D(21+ n —1,2j + n — 3, 2), X

where YA is the basis function for the spherical harmonics of degree 1, fl,;„ is the n —1 dimensional counterpart for V,na , so is i; for A and

D(4a,40, 4-y) :=

r(a + /13)r(a — F(cr+,3+7)F(a+,3 —7)'

dn,A , := 71Crii-1(0) rrn

. +Immo +7.- 2'2 +1) ri +1)' +v -i3 +3(m-

A9 : Rn Rn be an orthogonal transform such that A9 (0,• • • , 0,1) = 8 and clP(Ao) be defined by

E dNAoy,p).

YA(Aew)

LEA

Then Vr';, A

(2.9)

Um vA

UVLII'HULIra:")

m> 0, 0 < 1 < m, m+1: even

is the singular system of P. The singular values are estimated as

(2.10)

min crna = 0(m-1/2 ).

Remark. We apply this theorem for v = 2 '2 . This case, P* g(x) = f

n -1

g(8, Eox)(1 — I.E0 x12)-1 de

p# ow31 )(4 where P* is the adjoint transform in (2.8). 3. Reconstruction. In this section, we first review the known results on reconstruction of the potential from the data obtained by observing the scattering process, making use of which, we give our reconstruction formulas. In multidimensional case, the inverse problem was first studied by L.D. Faddeev (cf. [4] and reference therein). His study was succeeded by R.G. Newton (cf. [9]). Today, there exist various types of inversion formulas, which we introduce in this section.

171

INVERSE SCATTERING

Consider the Schrodinger equation (A + k2 )0 = Vzi)

(3.1) in R3 and its scattering solution (3.2)

7/1

(k, 9, z) = etkO.x +

ik r

r

A(k 0) +

1 — ) as r oc, ° (r

where r = jxl, i= r, k > 0 and E S2 . The first inversion formula is the application of Marchenko method (cf. [9] and reference therein).

v(x)=0•v{ „.2 f oo (3.3)

E

kdk f A(— k , , fleike (k, ,x)+ s2 ,cl ub (x)(y.nb(-9)e,„0„ — yrb(9),—c„0.1},

n,b

where 'On are orthonormal eigenfunctions of the Schrodinger equation with the eigenvalue —K2 and (3.4)

Y;(9) = f V (x)e".'4,(x)dx. R3

The author is not sure that Y.,t; are obtained by observation, but anyway this reconstruction formula is unstable, that is, if any of the data Y,', A, ubn contains errors then the potential given by (3.3) may be far from the original one. The second one is the Moses-Prosser System. In this system the potential is obtained by inverting the integral equation (3.5)

(5(p — 4)1 f (p + 1) a+ iko ,q). a+ (Ikl,p, (4) = k2 _ p2 ie + (27)3 R3 k2 p2 iE

For the definition of and the assumption on the potential, confer [9] and references therein. Note that the inversion of (3.5) is ill-posed. For these two reconstruction formulas, the potential is assumed to be continuous and rapidly decreasing. The third one is established by Y.Saito. For a potential satisfying IY(x)I < > 1, it holds that C(1 + (3.6)

lim k2 f

k —'cc

S2 x S2

A(k, 9, 0')e'k("' ).'

= —27

, V (x),, dy. fn.,x—Yr

He also gave the estimate for the rate of convergence. For these, confer [10] and reference therein. Note that the procedure inverting the integral transform (3.6) is ill-posed. Confer [10] for the reconstruction formula. None of the above inversion procedure being stable, we need a regularization method to construct the approximated potential from the data obtained by observation which contains errors. In order to establish a regularization method, we consider inversion formulas other than the above ones, which we establish in the following. Several years ago, V.Enss and R.Weder gave another proof of the uniqueness of the reconstruction of the potential, applying the methods of integral geometry. In their theory, they showed the following proposition.

172

TAKASHI TAKIGUCHI

PROPOSITION 3.1. (cf. Theorem 2.4 in [3]) Suppose V E V satisfy the assumption that for some 0 < p 0 (3.14)

suppV C {Ixl < K}.

r il L2(R.) =1 that Then we have uniformly for 1p E L2(.1V) with Ilib

(3.15)

+ 00

(ilv le-im"(S

= (f

V (x + Tii)(bodr,0) + o(1),

as Iv' -> oo. Therefore we have

(3.16)

lim Iv1-400

- /)0, = f

+00

V(x + 7-V)d.odr

in 1,2 (Rn ).

The proof of this proposition is also obtained by modifying the proof of Proposition 3.1. This proposition plays an important role to establish a regularization method in the fourth section. In the rest of this section, by applying Propositions 3.2 and 3.3, we construct our reconstruction formulas for the potential other than the above ones. PROPOSITION 3.4. Let V E V satisfy (3.11) then PV E S'(T). Proof. Let (3.17)

V = Vi + 172,

where V1 := F(Ix' < K)V and V2 := F(IXI > K)V. We have V1 E L2 (Rn) and suppVi is compact, hence there holds that PVI E L2 (T). By (2.11) there holds PV2 E L"(T). Therefore this proposition follows. El Propositions 2.2 and 3.4 enable us to establish the reconstruction formulas for the potential. THEOREM 3.5. Let V E V satisfy (3.11). Take (by as in Proposition 3.1. Then we obtain for a > 1- n that

(3.18)

V=

1 1 i'P#P-1 (- lim ilvle-imv•x(S -1-)0v ) 271Sn-21 Iv1-00

in S' (Rn ). Proof. By Proposition 3.2 we have

+00 lim ilv[e'(S - I)cv = f V(x + TV)00di-

IvI-* 00

in S'(R") (cf. (3.13)). It is not difficult to assert that f +cc V(x + TV)cbodr is measurable as a function of x with V as a parameter. Since 00 is analytic and its zero points are countable, 1 lim """x (S (Po

has sense for almost every x. Therefore we can prove that f +: v(x + Ti-.)d7- is measurable as a function of x with V as a parameter. Furthermore, by Proposition 2.6, we have f +07 V(x + TV)dr E S'(T) as a distribution of (x, v) E T. Therefore the theorem follows by Proposition 2.2. q When the potential is compactly supported we obtain (3.18) in L2(1e).

174

TAKASHI TAKIGUCHI

By virtue of Propositions 2.3 and 3.3 we obtain the following. THEOREM 3.6. The same assumption as Theorem 3.5 and assume (3.14). Then there holds (3.19) in

V=

1 1 P# I-1—( lim 271sn -21 Ivl

x(S - .000

L2 (R") .

Our main purpose in this article is to give regularization methods for the reconstruction of the potential. In this section, we have introduced several reconstruction methods, any of which is ill-posed. Therefore we need to establish some regularization method to construct an approximated potential from the data obtained by observation which contain errors. For this purpose we regularize (3.19) in the next section. 4. Approximation. In this section, we assume that V is compactly supported, for simplicity, let suppV C {ix' < p},

(4.1)

where p < 1. This case V E L2 (R"). In the previous section, we have established the reconstruction formulas for potential scattering by making use of the reconstruction formula for the x-ray transform. Our reconstruction formulas require the limit of high energy state, however, in practical experiments, it is impossible to give the velocity v with Iv' = oo. The only we can do is to give the velocity v with Ivi large and obtain the approximated limit state. This case we obtain the data near to f +00 V(x + 7- 7)00dr (4.2) J- 00 in L2(Rn) by virtue of Proposition 3.3. Also since we obtain the data (S - I)Ov by observation, they necessarily contains some errors. These reasons imply that in practice we only obtain the data near to (4.2) but not true ones. As is well known, the reconstruction formula for the x-ray transform is ill-posed as a map of L2(T) to L2 (Rn) small errors in (4.2) may cause fatal instability in our reconstruction formulas for potential scattering. Therefore, regularization is of significant importance for practical applications. We mean by "regularization" to construct some approximated solution for our problem out of the obtained data which are near to the true ones. Here, in this section, we elaborate to construct some regularized solutions for potential scattering. As we show in the following, we obtain the data near to PV in L2(T) (Proposition 4.2 below). Therefore for our regularization we apply regularization for the x-ray transform, in which the singular value decomposition (SVD) and the characterization of the image P({f E L2 (.1r)I suppf C K}) play important roles. Note that our regularization is different from the usual way for the x-ray transform since we cannot oo, which we mention below. estimate the errors in the approximation in lv I First, we prepare the L2 space with weight. DEFINITION 4.1. For a function g defined on T belongs to L2(T, W„ ) if and only if (4.3)

= f

foi lg(9,y)12(1-

< oo.

175

INVERSE SCATTERING

We define L2 (1r, W,;1) and L2(Z,w,7 1 ) in the same way, that is,

— fR,‘ If(x)12 (1 — ix12 )2 -vdx < 00,

&ofR IhP,$)12 (1 — s2 )1-.6/8 < co.

= f

Assume that we obtain the data (4.4)

F(V,(bo,x,v) =- ilvleimv.s(S — -1)0v(x) + e(x, v)

for large v by observation where e(x, v) implies the errors. If for fixed Id (of course, it is large enough) we obtain the estimate uniformly in "V (4.5)

:V (x + rir).1)0 (x)dr — F (V, cbo, x, ir)1

f

L2 (R')
0, µ > 0 and we have 2µ A + 2µ Z(0) = Z(0) — A + 3µ 111/./. A + 2µ •

The system of isotropic elasticity equations is a special group for which (2.6) has multiple eigenvalues ±.11 17 and the eigenvectors do not span C4 .

Applications to the inverse boundary value problems. Let us consider the problem of determining elastic parameters, C13ki, by making displacement and traction measurements at the boundary. These measurements made at the boundary are encoded in the Dirichlet to Neumann map. Here the Dirichlet to Neumann map on the boundary F is a C°°(r) to C°°(I) map : ulr tic, where t is given by (2.8) with (ni, n2, n3) being the outer normal to F. When we consider the system of anisotropic elasticity equations in the bounded region, the Dirichlet to Neumann map is a classical pseudodifferential operator of order 1 on the boundary and the surface impedance

SURFACE IMPEDANCE TENSOR AND BOUNDARY VALUE PROBLEM

185

tensor is essentially the principal symbol of the Dirichlet to Neumann map ([N-U1,2]). Hence in the problem of determining Cold from the Dirichlet to Neumann map, it is crucial whether or not the surface impedance tensor can determine Cokl. For the twodimensional anisotropic elasticity in the above first example, it is in general impossible to determine Coki from the symbol of the Dirichlet to Neumann map even if Coki is constant all through the region (cf. [N-Ti]). In fact, from (2.10), Z has only three independent components. The rotaional invariance of Z restricts the information the symbol of the Dirichlet to Neumann map has about the coefficients Coki. However, for the transversely isotropic elasticity in a three-dimensional space, the reconstruction of Cola from the symbol of the Dirichlet to Neumann map is possible at any point on the boundary ([N-T-1.1]). 3. Boundary value problem. We consider the Dirichlet problem of the second order elliptic system in the infinite plane region bounded internally by an elliptic hole. Using the theory of functions of a complex variable, we construct the solution when any Dirichlet data is given on the elliptic hole boundary. Next, we give the formula for the corresponding Neumann data on that boundary. In this formula, we see that the surface impedance tensor Z(0) in Section 2 gives a relationship between any Dirichlet data and the corrresponding Neumann data on the boundary. In a two-dimensional space 1R2, suppose that the boundary F of an elliptic hole is given by F: and let it be the region outside

X 2i X22

+

=1 (a > b),

r, = > 1. :a2 + b2

We consider the following Dirichlet problem: 2

(3.1)

a X) = 0, E Cijkl axia aXiUk (

x = (x1, x2) E c, i = 1,2,

j,k,1=1 (3.2) (3.3)

ulr = g E C l+a(F), au, (1 u is bounded in it, — = o -) as R = \ix?+ axi R

oo, i, j = 1,2,

where the constant coefficients Cokt (1 < i, j, k,1 < 2) satisfy the strong ellipticity condition (2.2) and c1+-(r) denotes the space of Holder continuous functions defined on F with exponent 1 + a (0 < a). To construct the solution of this problem, we make use of the forms (2.3) and (2.9), where a and p are those in Section 2 with {m, n} = fel, e2}. Now let us identify the point (x 1, x2) in 1R2 with the point z = x 1+ix2, (i = ‘,/.) in C. Using a conformal map from the region {m•x-Fpn•x:xE ft} to the region lz I > 1, we convert the problem of finding the scalar functions f in (2.3) and (2.9) to the problem of finding harmonic functions with prescribed data on the unit circle. The uniqueness of the solution to (3.1) — (3.3) follows from the strong ellipticity condition (2.2). Hence, we have

186

K. TANUMA LEMMA.

We can construct the unique solution to the Dirichlet problem (3.1) —

(3.3). Put z = xi -I- ix2

and

m(z) =

a2

z+

a— b 1 . 2 z

Then m(z) maps the unit circle C : Izi = 1 conformally onto I', and the region lz I > 1 onto 11. Let a = eie (0 < 0 < 27) denote the point on C. Computing the Neumann data for the solution constructed in the above, we have THEOREM 2. Let t(m(a)) denote the corresponding Neumann data at the point m(a) on F, that is, 2

t(m(a)) = (

Ec j,k,l=1

., 814 ‘..., 1jkilij, OX/

2

E j,k,1=1

„._,, aUk T U2jki , Vi) , 0X/

where (v1, v2) is the unit outer normal to F at the point m(a). Then t(m(a)) = Re (

(3.4)

a Z(0) U(rt)) . i rn (a)i

Here Re stands for the real part, Z(0) is the surface impedance tensor (2.7) at = 0 and U(a) = h(a)

h(s) — p.v. —ds, 7ri s —a I81=1 1

h(a) =

cd g(rn(ele)),

where p.v. means the principal value of the singular integral. Remarks. 1. In (3.4), the coefficients Ciiki are involved only in Z(0), and Z(0) is independent of m(a) E F. This fact is consistent with the rotational invariance of Z(0) in Theorem 1. 2. This formula can be regarded as a global and exact representation of the Dirichlet to Neumann map when the region is an infinite region with an elliptic hole. On the other hand, when the region is a bounded domain with a smooth boundary, the Dirichlet to Neumann map cannot be expressed only in terms of the surface impedance tensor (cf. [W-Y]). Note that the symbol of the Dirichlet to Neumann map in [N-T1], [N-U1,2], whose asymptotic expansion can be written only by using the surface impedance tensor and its derivatives, is a (micro)local representation of the Dirichlet to Neumann map modulo smoothing operators. 3. For the system of anisotropic elasticity equations, [T-Y] obtained the formula for the displacements on the elliptic hole boundary when the tractions are prescribed on that boundary.

REFERENCES [B-L1] D. M. BARNETT AND J. LOTHE, Synthesis of the sextic and the integral formalism for dislocations, Greens functions, and surface waves in anisotropic elastic solids, Phys. Norv., 7 (1973), pp. 13-19. [B-L2] , Free surface (Rayleigh) waves in anisotropic elastic half spaces: the surface impedance method, Proc. R. Soc. Lond., A 402 (1985), pp. 135-152. [C-S] P.CHADWICK AND G.D.SMITH, Foundations of the theory of surface waves in anisotropic elastic materials : in Advances in Applied Mechanics, C.H.Yih ed. vol 17, Academic Press, New York, 1977, pp. 303-376. [G-Z] A. E. GREEN AND W. ZERNA, Theoretical Elasticity, Oxford, 1954.

SURFACE IMPEDANCE TENSOR AND BOUNDARY VALUE PROBLEM

187

[L-B] J. LOTHE AND D. M. BARNETT, On the existence of surface-wave solutions for anisotropic elastic half-spaces with free surface, J. Appl. Phys., 47 (1976), pp. 428-433. [N] G.NAKAMURA, Existence and propagation of Rayleigh waves and pulses : in Modern Theory of Anisotropic Elasticity and Applications, J.J.Wu, T.C.T.Ting, D.M.Barnett, ed., SIAM proceedings, SIAM, Philadelphia, 1991, pp. 215-231. [N-TI] G. NAKAMURA AND K. TANUMA, A Nonuniqueness theorem for an inverse boundary value problem in elasticity, SIAM J. Appl. Math., 56 (1996), pp. 602-610. [N-T2] G. NAKAMURA AND K. TANUMA, A formula for the fundamental solution of anisotropic elasticity, Q. JI Mech. appl. Math., 50 (1997), pp. 179-194. [N-T-U] G. NAKAMURA, K. TANUMA AND G. UHLMANN, Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math. In press. [N-U1] G. NAKAMURA AND G. UHLMANN, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), pp. 263-279. [N-U2] , A layer stripping algorithm in elastic impedance tomography : in Inverse problems in Wave propagation, G. Chavent, G. Papanicolaou, P. Sacks and W. Symes eds., the IMA volumes in Mathemaatics and its applications vol 90, Springer, 1997, pp. 375-384. (Si] A. N. STROH, Dislocations and cracks in anisotropic elasticity, Phil. Mag., 3 (1958), pp. 625-646. [S2] , Steady state problems in anisotropic elasticity, J. Math. Phys., 41 (1962), pp. 77103. [Tal] K. TANUMA, Surface impedance tensors of transversely isotropic elastic materials, Q. JI Mech. appl. Math., 49 (1996), pp. 29-48. [Ta2] , Surface impedance tensors and boundary value problems for second order elliptic systems in the plane, submitted. [Til] T. C. T. TING, Effects of change of refernce coordinates on the stress analyses of anisotropic elastic materials, Int. J. Solids Structures, 18 (1982), pp. 139-152. ITi21 T. C. T. TING, Anisotropic Elasticity, Oxford, 1997. [Ti-Y] T. C. T. TING AND G. YAN, The anisotropic elastic solid with an elliptic hole or rigid inclusion, Int. J. Solids Structures, 27 (1991), pp. 1879-1894. [W-Y] M. Z. WANG AND G. P. YAN, Boundary value problems of holomorphic vector functions and applications to anisotropic elasticity, Quart. Appl. Math., 55 (1997), pp. 231-241.

ASYMPTOTICS FOR THE SPECTRAL AND WEYL FUNCTIONS OF THE OPERATOR-VALUED STURM-LIOUVILLE PROBLEM IGOR TROOSHIN Abstract. Proximate asymptotics for the spectral and Weyl functions of the operator-valued Sturm-Liouville problem are given. Key words. spectral function, Weyl function AMS subject classifications. 34B20,34E10,34K25,34L40

Let H be a separable Hilbert space (dim H < oo) with an inner product (-, .) and r(H) be a linear normed space of bounded linear operators in H equipped with operator norm • • Let us denote by C(n) ([0, co), £(H)) the linear space of n-times continuosly differentiable functions y : [0, oo) r(H) . Here and afterwards we understand continuity, differentiability and integrability in the sence of operator norm ill We consider Sturm-Liouville operator 1(y) = —y" + q(x)y (0 < x < oo) where q(x) E C(') ([0, co), r(H)) (n > 0) is a bounded selfadjoint operator in Hilbert space H for every fixed x E [0, co). We denote by c(x, fi) and s(x, /) the solutions of Sturm-Liouville equation (0.1)

l(y) = zy

satisfying the initial conditions c(0, V7z) = .91(0, V7 z) = E, c' (0, 0 . Here E is an identity operator in Hilbert space H. Equation ( 0.1 ) has an operator solution

= s(0,

=

F(x, z) = c(x, N,5) + s(x, V7z)m(z), such that I I F(x , z)I I E L2(0, 00) for any nonreal z, the operator function m(z) is an analytical function in the upper and lower half-planes of the z-plane and m(f) = m*(z) for any nonreal z. This fact is a generalization of the classical theorem of H. Weyl and proof in the operator valued case was given by M.L.Gorbachuk [1] The function m(z) is called the Weyl function and the investigation of its asymptotic behavior for oo is the aim of this present paper. Asymptotics of Weyl functions in the scalar case were studied by many authors (see, f.e., [2]-[4]) (In this paper we use the approach of A.Boutet De Monvel and V.Marchenko [5]) Although in most cases asymptotics were obtained in some sectors of the open upper (lower) halfplane of the z-plane. Here we are trying to obtain the asymptotics of the Weyl functions as close to the real axis of the z-plane as possible. *Precision Mechanics and Control Institute, Saratov, Russia (lisuecrisscross.cora). 189

190

IGOR TROOSHIN

This investigation is closely connected to the investigation of asymptotic behavior of antiderivatives for the spectral function of the Sturm-Liouville problem ( 0.1) with boundary condition y'(0) = 0. The operator function p(µ) is called the spectral function for problem ( 0.1 ) with a boundary condition y'(0) = 0 if it is a left semicontinuous and nondecreasing (i.e. + At.t) - p(µ) is a positive operator in H for any .6,µ > 0) operator function such that p(1.1) is a selfadjoint operator for any fixed E (-oo, oo) and there is a Parseval equality

f (x)g* (x)dx = f lc( f , Vii)dp(i.t)c* (g, N/Ti) for every pair of finite piecewise continious operator-valued functions f(x), g(x). Here c(f , fit) = f (x)c(x, ffi)dx . The existence of operator spectral functions, which can be nonunique, was proven by Rofe-Beketov [6]. It was also shown that for any 0 < h < oo there is an constant C(h) > 0 such that for anyµ < 0 there is an estimate

MPH < C(h)e-hV71. (We set p(-oo) = 0 ). The Weyl function and spectral function of the problem (0.1) , y'(0) = 0 are connected by the formula (0.2)

m-1(z)=f

dP(P) ict

00 z

(This fact is well known in the scalar case fact and was proven by M.L.Gorbachuk in the case of an operator-valued problem [1].) The use of this connection allows us to prove the following theorems. Let us denote by Nri the branch which takes on positive values on the upper side of the cut traced along the positive semiaxis. THEOREM 0.1. Let N be any natural number such that N < n. Then the Weyl function has uniform asymptotics

(0.3)

m(z) =

+

E (2i.(0/71k) + aN-F1(z) k

k =1

V '1

where o1 (x)

= q(x), 0.2(x) = -4 (x), cra (x) = q" (x) - 42 (x)

k-1 Grk+1(X) = -a;, (x)

E ak- ( )63(x), k = 3,...,N -1. j=1

Asymptotics for the Spectral and Weyl Functions

Ila N+1(z)11= 0(z-4), 1z1

191

00

) in any domain Imz > EIRez11+9 0 = If the potential q(x) has compact support, then asymptotics (0.3) take place in the whole z-plane for N = n. We also obtained asymptotics of the spectral functions and their antiderivatives. THEOREM 0.2. The antiderivatives of the spectral function p(p) of the boundaryvalue problem ( 0.1), V(0) = 0 have asymptotics (0.4)

1

- t i\n , () t = C

_ n+ 2 +

n cspn-

+ on+1,n(p)

n.

aP

Cs =

(-1)8+1 2 —(9+2) (n + 1)!B(n +1— s/2, s/2 +1) 88+2

8=1

where

with B(p,q) = rr( rp)±r(qq)) and 2 115n+i,n(P)11= 00;i,

oo.

where /(31 = 2E, /33 = — 4a1( 0), /34 = —4o-2 (0),

i)—(n+2))

11,3n+3(Vrz)11 = 0((—

and the rest of 0k could be found recurrently from the formula n-1-2

0k

n

1 (8+1) RE

Ec E ( 2i ,5)k + on-1-3(ii) = s=1 VZ k=3

k=1

aim) s k + o(ivi-(.9±-))]

Similarly to the proof of Theorem 0.2, we can obtain asymptotics for spectral functions of the boundary problem (0.1), y(0) = 0. THEOREM 0.3. The antiderivatives of the spectral function p(p) of the boundaryvalue problem (0.1), y(0) = 0 have asymptotics

(p n!

where

dp(t) =

n

„_ f+, + Eds, sn+i,n(p) s=1

192

IGOR TROOSHIN

C8 =

(n

(-1) 8-12-8 1)!B(n + 2 - 8/2, 0)680), 0"-1(0) = 0(µ 2+1),µ co.

While proving these main theorems we introduce the supplementary Sturm-Liouville equation (0.5)

la (y) -=- -y" + qa (x)y = A2y, 0 < x < co

with compactly supported potential qa(x) = q(x)h(x - a), a > 1, where h(x) is a real-valued infinitely differentiable on the real axis function such that h(x) = 1 for -co < x < -1 and h(x) = 0 for 0 < x < co . ma (z) denotes later the Weyl function of equation (0.5) and pa (µ) denotes the spectral function of the problem (0.5), y'(0) = 0. This paper consists of five sections. First section is devoted to some prelimenary estimates. Second and third sections are devoted to investigation of asymptotic behaviour of Weyl function and spectral function of supplementary problem with compactly supported potential qa (x). We prove the Theorem 0.2 in the fourth section and Theorem 0.1 in the fifth section. The Weyl functions and Weyl solutions are the proper spectral data for the respective inverse problems which are arising in the study of the Cauchy problem for nonlinear evolution PDE's whose initial data are nondecaying at infinity (see, f.e., [9]). 1. Preliminary Estimates. A. We will now investigate the supplementary Sturm-Liouville equation (0.5) with compactly supported potential qa(x). The proof of the Lemma 1.1 follows the proof of [8], pp.50-57. LEMMA 1.1. There is a solution g(x, A) to equation (0.5) such that g(x, A) = ei A'E x > a and g(x, A) = e' A'u(x, A), 0 < x < a, where u ( x,

un+ (x, A)

= Pn(x, A) + (2/A)n+i P,i (x, A) = E, n = 0, n

P,(x, A) = E +

uk(x) ,n>1 (2/AY' k=1

Asymptotics for the Spectral and Weyl Functions

193

a a u1 (x) = — f q(t)dt, uk-Fi(x) — — f la (uk(t))dt, k = 1, ...,n — 1,

' +1 (x, A)II = and function un+i(x, A) has a uniform estimatellun+i(x, )011= 0(1), Ilun oo. o(A), I lu'n'1_1 (x, A)I I = o(A2 ) on the upper halfplane Ina> 0, 0 < x < a, IAI Proof. Let us denote by g(x, A) the solution of the following Cauchy problem ia(Y) = — y"+ Ya(x)y = A2 y, 0 < x < co

y(a, A) = e' AaE, y(a, A) = iAe1A a.E. Thus, from the definition of qa (x) it follows that g(x, A) = ez AsE, x > a. If we denote u(x, A) = e' Axg(x, A), then u(x, A) is the solution of the following initial problem (1.1)

—u"(x, A) — 2iAu'(x, A) + qa (x)u(x, A) = 0, 0 < x < a

(1.2)

u(a, A) = E, 12 / (a, A) = 0.

Let us seek that solution u(x, A) in the form (1.3)

u(x, A) = uo (x) +

u„ (x) un+i (x, A) ui (x) + + + 2iA (2iA)n (2iA)n+1

Substituting the right hand side of formula (1.3) into equation (1.1) and setting the coefficient of the powers (2iA) —k (k = —1,0, 1, ..., n — 1) equal to zero, we get the system of equations u'o (x) = 0,

= —4_1 (x) + qa (x)uk_ 1 (x), k = 1,...,n

" (x) — qa(x)un(x)]. n +1 (x A) — 2iAuni+1(x, A) + qa (x)uni_ 1 (x, A) = 2iA[un Let us take a

Uo(X) =

E,

= — I la(uk(t))dt, k = 0,..., n,

' A xvn+i(x,A), where na+i(x, A) is the solution to the equation

194

IGOR TROOSHIN

(1.4)

to (v„+i (x, A)) — A2v„+i(x, A) = —2iAeiAsUni+1 (X)

satisfying the initial conditions vrt+i (a, A) = +1 (a, A) = 0. Thus u(x, A) is the solution to equation (1.1) with initial conditions (1.2),because u8 (a) =

(a) =

= (-1)s- 1 u(is) (a) = (-1)s- 1 el) (a) = 0, s = 1, ...,n+ 1.

Let us now estimate un+1 (x, A) and its derivatives on the halfplane ImA > 0. The method of variation of constants leads to the following representation of the solution vn+1 (x, A) of equation (1.4): sinA(i — t ) [2iAeiAtuin+1(t) + qa (t)vn+i (t, A)]dt. A

vr,±1(x, A) = It follows that

u„-F1 (x, A)

a

= ur,±1 (x) + f e2jA( t-x),, L.+1 n (t)dt— .

f a 1 — e21)*—x)

(1.5)

ix

2iA

qa.(t)un+i(t, A)dt

Let us denote M(A) = maxo 0, 0 < x < a, IAI oo. Therefore, from equation (1.1) it follows that A)ii = o() 2 ) uniformly on the same domain. Lemma 1.1 is proved. El B. We prove in this subsection some supplementary Lemmas, which will be used in the proof of Theorem 0.1. LEMMA 1.2. There is an following estimate on any angular domain e < arg z < 71" — E, 0 <
sin fizi.

196

IGOR TROOSHIN

CN =

2 (sine) (n+2) (1 + N)fr`+1 n +2

0 LEMMA

o
cIR.z11+9 , e > ,

f

o

cc

(1 +

iz

— ii i m+2 (501)0

= 0(1ZI —(1±9(n+1)) )

oo

for Izi

for any bounded function 504, such that lb(p)I = o(1) for lid oo . Proof. If we denote C(x) = supp>x 16(11)1 then we obtain an estimate for Izi > 2

.(1+ 1

Jo

IAD 2

IZ — ktin+2

(1 + /2) 3- dp

o(p)ditil

0

C(0) fz l

111n+2

Izl

ro

C( 2 )

43 2 lz - Pi n+

It is clear that Iz - µl > 4[ for litj < It follows immediately that

(1 f o

A

in +2 dP 5 (

1;1

) -(n+2) f 2 (1± 0


0, y > 0 (the case x < 0 is treated on the same way). After change of variables µ = x + Ty we obtain

/1.1

IZ

—11[0+2

•< -

(x + 7-y) 21 ,,n+1 (1 + 7-2)1+1"r

In the domain IRzi > ;s-z we have an estimate r°° (1 + T)i < yn-1-1 _co (1 + 72)1+1d r _ p1 rn+2 d Finally on the domain IRzi > `s•z > fiRzli+8 we obtain an estimate

f

00

J4 L

< C x-(1+9(n+1)) < Clzi-(1443(n+1)) — plm+2

Asymptotics for the Spectral and Weyl Functions

197

2. Asymptotics for the Weyl Function in the Case of Potential with Compact Support. In this section we investigate the Weyl function of the supplementary Sturm-Liouville equation (0.5) with a compactly supported potential qa (x). Let us denote by V—z the branch which takes on positive values on the upper side of the cut traced along the positive semiaxis. LEMMA 2.1. The Weyl function ma (z) of the problem (0.5) in the case of compactly supported potential qa (x) has uniform asymptotics (2.1)

ma (z) = i \rzE+

(0) 2_, (2iak.V7 z)k + un+i(fi)

k=1

where al (x)

), a2 (x) = — ( x ) ,

k -1

ak+i (x) = --ak` (x)—

o-k_, (

x),(x), k =1,...,n — 1.

) Ilan+i (N5)11 -= o((/ )-n

on the whole z-plane . Proof. We proved in Lemma 1.1 , that there is a solution to equation (0.5) such that lig(x, N5)I I E L2( 0, CX)) for z on the whole z-plane and g(x, /) -+-

0 < x < a,

zx u(x,

where z) un+1(x Ni(2i N5)n+1

u(x, /) = Pn(x , v')

n = 0,

Pn(x,‘5) =

Pn(x, -\5) = E +r-‘ u") L-4 k=1

a

u1 (x) =--

(2i Vi)k a

71 >1

f q(t)dt, uk+i(x) = f la(uk(t))dt, k = 1,

n — 1,

and the function un±i (x, \5) has a uniform estimate I lun±i (x, ii)11= 0(1), (x, .A11 = o(rz), lju'ai+1(x, = o(z) on the whole z-plane .

198

IGOR TROOSHIN

We can choose positive numbers R and C such that for Izi > R there are inverse operators u-1(x, and Pn-1(x, N5) and 11u-1(x, fi)11 < C, 11Pn-i (x, -Vrill< / C. Let us denote m(x, z) = g' (x, Vi)g-1(x, V7z). Then we can represent the function rn(x, z) in the form rn(x, z) [ei fix u(x, \rz)]I e-iVixu-1(x, Nrz) where o-(x, Then

+ (x, Nrz)

= u'(x,

(j(Z ' N5)

= [Pni (x N.5)

n+( V7Z (2i1,An-1-1) p n(X' ‘5)

Un+i 1-1 (2iNti)n-F1

= Pn' (x, -V7z)P.71 (x, V4) + crn+1(x, where (x A.5) = [u'n+1(x, A.5) - Pni (x, V7z)Pri 1 (x, N5)un-f-i(x,

(x Nrz)• (2i Vi)n+1

We can see from this relation that 'n+1(x, Vz) = fun'±i (x, -Vz) - Pn" (x, VZ)P,.,T 1 (x, Vi)un±i (x, Nrz)±

+11'n' (x, 1/i)PrT1 (xl 15)]2 un+1(xl 1.5)u-1(x, (x , Nri)un+ (x , N/-)T Z" (2i)n+1

-P;i (x, 1

(2i.v7z)n+1 fUn'+1 (x)1,5)

Pni (x, ‘- ,5)P;j1 (x, Nrz)un-1-1(x, V 4)}x

xu-1(x, Vi)u1(x, Nrz)u - (x, Thus, it follows easily that Ilan+i (x, N5)11 = o(z),

+1 (x, Nrz)11

o(z-

Let us now note that we can expand P,c(x, Vi)P,T 1 (x, /) on the series of (2ifi) -n for lz1 > R

Asymptotics for the Spectral and Weyl Functions

i)P,, 1 (x, N5) = E

1

k =1

199

ak (x)

( 2iviyc •

In other words we get the representation

(2.2)

(2.3)

a(x,

a'(x,

=

\5) =

E paik:rz;k

°«5)-n),

k=1

(x) o((f)-1n-11 ).

E (2crik.v )k

k=1

It follows from equation (1.1) and relations u'(x, /) = a(x, fi)u(x, + a2(x, /))u(x, /) that a(x, V7z) satisfies the equation

(a' (x , (2.4)

+ a2 +

u"(x,

- qa = 0.

Substituting the right hand side of formulas (2.2) and (2.3) into equation (2.4) and setting the coefficient of the powers (2ifi)-k , k = -1, 0, 1, n - 1 equal to zero, we get the system of equations

a1(x) = qa(x), a2(x) = k -1

a k+1 (X) =

-aik (x) - E crk _3 (x) (x), k = 2, ..., n - 1. j=i

In the case of compactly supported potential, the Weyl solution is unique. But 11g(x, ,/ )II E L2(0, co) for z on the whole z-plane. It follows from this fact that there is an operator-valued function C(z) (invertible for sufficientely large IzI ) such that

F(x, z) = g(x, fz-)C (z) where F(x, z) is the Weyl solution. Now we can turn to the Weyl function:

ma (z) = F' (0, z)F-1 (0, z) = (0, Nrz)C (z)[g(0, fz)C (z)]-1 =

= 9'(0, Nrz) g-1 (0, /) = m(0, z) Thus, the Weyl function m(z) can be represented in the form

=

200

IGOR TROOSHIN

ma (z) =

+Q(0,

Lemma is proved El Corollary: For sufficiently large zI there is an inverse function m,7,1 (z) which has the following asymptotic estimates on the z-plane n+2

(2.5)

E+E + On +3 (N5) V7Z k=3 (2i V7Z )k Pk

ma1(z) =

where = -40-2(0),

03 = —40.1 (0),

110,1+3 6[ 2)11 = f)(

N/i

—(n+2)

and the rest of 0, could be found recurrently from the formula n+2

X—`

k=3

(2i li)k On-F3(Vi)

=

E( s=1

n

1

tm

)(s+1)RE (2ivi)k )s + 0(1,51— (s+n))] vi k=i

3. Asymptotics for the Antiderivatives of Spectral Function in the Case of Potential with Compact Support. In this section we prove Theorem 0.2 for the supplementary Sturm-Liouville problem (0.5), y'(0) = 0 with a compactly supported potential qa (x). Let us denote C, as a circle of radius µ with the centre on the origin of the z-plane It follows from equality (0.2) that - z)n f dpa(t) dz z-t 27rin! f

f(it - z)n (z )d z= „ 27rin! m a

=

f _ „„

µ - z)n 0d*Pa(t) 2 R-(in!(z

f

=

f

- on dpa(t) n!

where integration on Cp is anti-clock-wise. It follows from the fact that in the case of compact supported potential qa (x) the spectral function pa (t) is a constant operator on (-co, -K] for some K > 0 that 1P

J_„,3

(11

.

t)n

n!

dpa(t)

f (11, zr 27rin. 4

changing variables z = A2 we obtain

m-1(z)dz.

201

Asymptotics for the Spectral and Weyl Functions

l (it — n! J

dPa(t)=

fc.,,

(it .A2)n rn;:1 (A2)AdA ,

where C, = {IAI = 0 < argA < 7r}. It follows from Corollary of Lemma 2.1, that p

f oo

n.

dpa (t) =

n+2

1 (P irzn!

13k

A2)n

A)k —' (2iA k=3

+ On+3(A)]AdA

where

On+3

= 0(A-(n+2))

on the upper halfplane Im A > 0. It is easy to estimate for tz —> co

11(5n+10.1)11 = 11 f

(,, A2)71 con/2) • 1 0,1+3 (A)AdAl I = 0(f 1A'-1IdA) = ‘f.' 7rzn! c,

Let us now calculate the main terms of representation for antiderivatives of the spectral function. 0 < q < 7r On the contour C, we can represent A = It follows that A2 =

_

ezio )

= —2ipei g5 sin0.

Consequently A2 ) k =

k pk ei k (sin(/))k

.

For 1 < j < n + 2 (01 = 2E); we can now calculate that f

(kt A2 r /3i AdA = Rin! (2iA)3

f = 7rzn! 0 n( -2i)ntin (-1)'(2i)n7rn!•

(sinOr

p

f

(203

je2i(kdO =

e 45(n-j+2 )(sinondo.

Let us now change the variables of integration a = z —

202

IGOR TROOSHIN

e2 rk(n — j+2) (SillOrdcb = -2

[cosa(n + 2 — j) — isina(n + 2 — j)1(cosa)nda

and let us note that

f

(cosa)nsina(n + 2 — j)da = 0.

Thus (cosa)ncosa(n + 2 — j)da

7r (n + 1)2n+IB(n + 2 — j/2, j/2)

and consequently

fc

(itirin! A2)n (2i"33APAdA =

(-1)3-12—i

(n + 1)!B (n + 2 — j/2, j/2)

Theorem 0.2 in the case of compactly supported potential qa(x) is proved. 4. Asymptotics for the Antiderivatives of the Spectral Function. In this section we give a proof of Theorem 0.2. Rofe-Beketov [6] proved that for every pair of finite piecewise continuous operatorvalued functions f (x), g(x) there is a Parseval equality

f (x)g* (x)dx = f

f , Vii)dp(p)c* (g,

where c( f , Vit) = f0 f(x)c(x, ji)dx. Let us consider two boundary-value problems (4.1)

1(y) = —y" + q(x)y = zy, y'(0) = 0,

(4.2)

la(y) = —y" + qa (x)y = zy, y'(0) = 0.

where qa (x) = q(x)h(x — a), a > 1, h(x) is a real-valued infinitely differentiable on the real axis function such that h(x) = 1 for —co < x < —1 and h(x) = 0 for 0 < x < co. We note that c(x, = ca (x, /A) for x < a — 1 and for the function (x) = 0, x a — 1 we have c(f , ffi) = ca( f , Vit) . Thus

rco

c( f Vii)*(1,2) — dPa(it)ic* (9,

=0

203

Asymptotics for the Spectral and Weyl Functions

for any piecewise continuous operator-valued functions

(x), g(x) such that

f (x) = g(x) = 0 for x > a — 1 . Let us denote

p(A) = k=1

sin(2— A) 2— k A •

The function cos(Ax)p(eA) is an even function of exponential type Ix' + e and it belongs to the space L2 (—oo, oo) . It follows from this fact (see, for example, [8] p.101-102) that there is scalar function L E L2(0, oo) such that f,(t) = 0 for t > IXI ±f and

p(EA) cos Ax = f f-e (t) cos Atdt. We now take

f, (t) = fE (t)E — f

lx1+€

1,(7-)L(7,t)dr,

where L(x, t) is a kernel of inverse operator-transformation:

cos AxE = c(x, A) — f x L(x, t)c(t, A)dt. It is clear that for so chosen function

.txE, c(f,, 01) = p( N/FIE) cos .11 EE if g,(t) = E (E Also c(gc , =s for c < t < oo . Therefore, if Ix' + c < a — 1, then

f

P(1/Tic) cos 'fix

sin /c VF-16

L(7- ,t)dT) for 0 < t < e and g,(t) = 0

[dP(P) — dPa (it)] = 0

Let f (x) be any infinitely differentiable scalar function which is equal to zero when lx1 > a — 1 — b for 6 > E Then we have

f :o f (x) f I p(/ TO cos VItx sin VTIE [dp(it) — dpa (µ)]dx = 00 VT/E oo = f

co

ci (Vii)P(VT/e) sin Vilf [dP(P) VT/6

dPa( 1)] = 0

204

IGOR TROOSHIN

where c f (p) =

f (x) cos pxdx.

Let us denote

G(x) = f cos ffix[dp(µ) - dPa(11)]• It follows from exponential decay of spectral functions for p G(x) is infinitely differentiable and f 00

00

_cc

G (x) f (x)dx = f o+

of (VTOPP(ii)

-oo (see [6] ) that

- d Pa (1)1

We set 21,11 (P(ii2) — P(o+)), aa(p) = 21111 (pa(p2 ) — pa(o+)),

cr(m) Then we have

f

f (p)[do-(p) -

a(2)] = f G (x) f (x)dx

The functions a(p) and on (p) are odd and 00

E f (p) = f f (x)e-'Pr dx =- c f (p) - i f f (x) sin pxdx. 00

00

Thus 100 j_ °G C f

(p)[da(p) - daa (p)] = f

E f (p)[da(p)

- dcra (p)].

Let us now note that (5 is arbitrary. It means that we proved that for all scalar infinitely differentiable functions f (x) such that f (x) = 0, lx1 > a - 1 there are the equalities (4.3)

. Ef

. co (µ)do- (µ) = f E f (p)daa (p) + f G(x) f (x)dx

We have already proved in previous section (setting p(-oo) = 0 ) that there is the following asymptotics : pa (p) =

2 77

E + o(1), p

DO ,

205

Asymptotics for the Spectral and Weyl Functions

i.e. , 1, 1 pa (0+) + PM, IµI -> oo. aa(A) = 1/-61E - sgnp2 Therefore from the above lirn Ica (o + Ao) - aa(c)II = 1,1->co

(4.4)

for any real Ao • Repeating the proof of Marchenko's Tauberian theorem [7] we obtain from formula ( 4.3) and estimate (4.4) that (4.5) f P (— )klo- (A) - dcra (A)] =

N

where P(A) = (1 - A2 )n,

E ( 171!i)

m

G(m)(0)p(m)(0)N,

00.11r(N)11 < C(a

1)-('+1),

Let us now note that

f (4.6)

.)\ 22 ) n[da(A) - claa(A)] = A-)[cla (A) - a (0)] -= f NN (1 - 1+. P(--N

=

f

- tr[dp(t) - dp„.(t)],

where we put µ = N 2 t A2 . We are now going to calculate the second term of the formula ( 4.5); so :

G(m) (x) = f (cosfiix)( m) [dp(µ) - d NW]. Thus, from the fact that (cosVitx)(m) l x=0 = 0 for odd in and (cosffix) (m) lx=o = (-p) 2 for even in it follows that G( m) (0) = 0 for odd m and

G(m) (0) = (-1) +1 f o t [dp(t) - dpa (t)] for even m.

206

IGOR TROOSHIN

Let us also note that for lx1 < 1 we have P(x) = (1 — x2)n = Ersi_o C 4(— x2) 8 = ns=o CV -1)8 x2s and P(2s) (0) = (-1)8(2s)IC Therefore

E 0 00 •

11 77/-1 (z) -"c1(z)11=o(lzl-(z-1-0(n+1)))

on the domain IRzl> caz > ciRz11+8. Compare estimates (5.2) and (5.3) with asymptotics (2.5) for ncl (z) we obtain an asymptotics

(z) =

N+2 .N5

Z.., (2iNk k=3

0(IZI-(1+0(n+1.))).

Asymptotics (0.3) for m(z) follows from this formula. Theorem 0.1 is proved. Acknowledgments. I would like to cordially thank Prof. V.A.Marchenko for his valuable advice and encouraging this work. REFERENCES [1] M. L. GORBACHUK, On spectral functions of a second-order differential equation with operator coefficients, Ukr. Mat. Zh., 18, No2 (1966), pp. 3-26. [2] W. N. EVERITT, S. G. HALVORSEN, On the asymptotic form of the Titchmarsh-Weyl mcoefficient, Applicable Analysis, 8 (1978), pp. 153-169. [3] F. V. ATKINSON, On the location of the Weyl circles, Proc. of the Roy. Soc. of Edinburgh, 88A (1981), pp. 345-356. [4] B. J. HARRIS, The asymptotic form of the Titchmarsh-Weyl m-function associated with a nondefinite, linear, second order differential equation, Mathematika, 43 (1996), pp. 209-222. [5] A. BOUTET DE MONVEL AND V. MARCHENKO, Asymptotic formulas for spectral and Weyl functions of Sturm-Liouville operators with smooth coefficients, in Operator Theory, Advances and Applications, Vol. 98, Birkhauser Verlag Basel/Switzerland, 1997, pp. 102-117. [6] F. S. ROFE-BEKETOV, Expansions in eigenfunctions of infinite systems of di fferential equations in non-self-adjoint and self-adjoint cases, Mat. Sbornik, 51, No3 (1960), pp. 293-342. [7] V. A. MARCHENKO, The Tauberian theorems in the spectral analysis of differential operators,Izv. Akad. Nauk SSSR, Ser. Math, 19 (1955), pp. 381-422. [8] V. A. MARCHENKO, Sturm-Liouville operators and applications, Birkhauser Verlag, 1986. [9] V. A. MARCHENKO, The Cauchy problem for the KDV equation with non-decreasing initial data, in V.E.Zaharov (ed.), What is Integrability?, Springer-Verlag, New York, 1990, pp. 273-318.

EXACT CONTROLLABILITY METHOD AND MULTIDIMENSIONAL LINEAR INVERSE PROBLEMS MASAHIRO YAMAMOTO'

Abstract. For fixed p, we consider the solution u(f) to

(x, t) + Au(x,t) = f (x)p(x,t) (x E CZ, t > 0) u(x, 0) = ui (x, 0) = 0 (x E 0),

B,u(x,t)= o (x E 012, t > 0 : 1 < j < m),

where u' = =ems, C W (r > 1) is a bounded domain with smooth boundary, A is a uniformly symmetric elliptic differential operator of 2m order with t-independent smooth coefficients, B3 (1 < j < m) are boundary differential operators such that the system {A, B3 } 0 as 1 oo. That is, n

(3.11)

E inw(f) - c,w(f)iii.(r.(0,7.)) —> 0 ,=1

as 1 --> co. On the other hand, by Lemma 2 and (3.10), we have Ciu(ft )(x,t) = fo p(s)Cjw(h)(x,t — s)ds (1 j 0 which is independent of yo, fa. Furthermore if we choose = 0, then limn ,0 — foil F2 = 0. However parameters a = a(8) such that lim o this choice of regularizing parameters does not guarantee any concrete convergence rates of regularized solutions. If we can pose a suitable additional assumption on fo, then we can derive a concrete rate under suitable choices of regularizing parameters. That is, by Corollary 3.1.3 (p.35) in [8] we see : If 10 E R(G*) and a = a(6) = M45 for an arbitrarily given constant M4 > 0, then

Ilia - /04'2 = 0(4 as 6 0. Thus it is significant to search for a sufficiently wide subset of R(G`), and we show THEOREM 2. Under Assumptions A-C and (3.1), G.L2 (r x (0,T))11 J R.0 10(v); v E 1-1j,(0,T; L2 (r))n}. The right hand side is a reachable set of a solution 0(v) to (2.2) - (2.4) by Ho (0, T; L2 (11)n-boundary control v and the Hilbert Uniqueness Method for exact controllability is applicable. Thus by this theorem and Corollary 3.1.3 (p.35) in [8], we reach

218

6>

M. YAMAMOTO THEOREM 3 (yS-CONVERGENCE OF REGULARIZED SOLUTIONS). Let yo = Gf0 , a 0 and our 0 and Ilya — lioilL2(r x (o,T)). < S. For a regularizing parameter >

available data yo with noise-level 6, let fg, be a unique minimizer of OW — YSII 2L2(1' )(OJT" + allf112F2 over f E F2 . If fo E Rio and we choose a = M4 5 with A 4.4 > 0, then lif« — foliF2 = 0(V43)

0.

ash

Proof of Theorem 2. We define an operator L : F2 -- L2(r x (0, T))n by (4.4)

Lf = (Ciw(f),-,Caw(i)),

where w(f) is the solution to (3.7) - (3.9). By the definition of F, the operator L is bounded from F2 to L2(f x (0, T))". By Lemma 2, we decompose G as (4.5)

G f = K1,2 L f

(f E F2 ),

where we regard K as an operator from L2(0, T; L2 (r))" to itself and we denote it by K = K L2. Therefore we obtain G* = L*K12, so that we have 1Z(G*) = {L*v;v E R.(KI2)}.

On the other hand, for n E (TA , ...,77,0

E

L2(0, T; L2(11)n, we directly see

T (K1211)(X , t) = f p(4- — t)71(x, Ock (x E F, 0 < t < T), t

so that 12.(1{12 ) = {v obtain

E H1(0, T; L2(r))n;v(•,T) =

(4.6)

IZ(G*) D {L*v;v E H01 (0, T; L2(r))"}.

0} by (3.1). Consequently we

Thus, for the proof, we have only to determine L*. For this, we need LEMMA 3. Let v = (vi ,..., vo) E cnr x (0, T))" and f E F2, we have (L f,v),t.,2(r x(0,T))^ = < 11(v)(' ) 0)1 f > F',F2 •

Here < •, • >F,F2 is the duality pairing between l'; and F2. Proof of Lemma 3. First let us assume that f E C00 (1-2). Then by Theorems 3.1 (pp.103-104 : Vol. II), 2.1 (pp.95-96 : Vol. II) and 8.2 (p.275 : Vol.I) in [29], W(v)

MULTIDIMENSIONAL INVERSE PROBLEMS

219

and w(f) are so regular that that the following calculations are justified.

fo L_A,(v)(.,t)w(f)(x,t)dxdt =

(f0 OM" (X ,

f)(x t)dt) dx

(by (2.2)) =

t)w( f)(x , Off, —

sa

f

11)(v)' (x,t)w(f)' (x,t)dt) dx

(by integration by parts) = —J Z 7,1)(01 (X , t)w(fr (x,t)dtdx

(by 0(v)i (x,T) = w( f )(x , 0) = 0)

= =

I OM ,t)w(f)" (x,t)dtdx + f IP(v)(x , 0) f (x)dx

ff2 o (by integration by parts and o(v)(x, T) = 0, w(f)1(x, 0) = f (x)) T

f 'OM (X t)Aw(f)(x,t)dxdt + f z13(v)(x , 0) f (x)dx,

— fo (by (3.7)),

namely,

Aw( f)(x,t)113(v)(x,t) JoT f.( =f

— AlP(v)(x, t)w(f)(x,t)) dxdt

(v)(x , 0) f (x)dx

Applying the Green formula (e.g. Section 2.4 of Chapter 2 (Vol.I) in [29], cf. p.112 in Lions [281), we obtain

Em 107.OR CjW(f)(X,t)Bii,b(V)(X,t) — BjW(f)(X,t)CjIP(v)(x,t)dSxdt

3=1

= f IP(v)(x , 0) f (x)dx.

By 13,w( f)(x , t) = 0 (x E aD, 0 < t < T : 1 < j < m) and ./3,0(v)(x,t) = 0 (xEafi, 0 0 are Lame constants which are independent of x and t. We choose I' and T such that

r D r+(xo)

(6.4) and (6.5)

where F+(x0) and R0 are defined by (5.1). Then F2 can be constructed similarly to the manner in Section 2 and we can show that L2(1/)^ D F2 (Section 1 of Chapitre IV in [28]), so that Theorem 1 is applicable. THEOREM 6. On the assumptions (3.1), (6.4) and (6.5), there exists a constant M = M(p, 12,r, T) > 0 such that aua(f) n

11' (0,T ;L2

(T))^

+ II

ni(f

)11 IP (0 ,T;L2 (n)

I

•

For inverse problems for the elasticity equation, we refer to Bukhgeim and Kardakov [4], Grasselli and Yamamoto [7]. As for exact controllability, we further refer to Nicaise [31], Telega and Bielski [44]. II. Inverse source problem for Maxwell's equations. Let (E(I), H (I)) be the solution to

(6.6)

(x, t) — curl H(x,t)= p(t)I(x) (x E fl, t > 0)

(6.7)

iiHi(x,t)+ curl E(x, t) = 0

(6.8)

div E(x,t) = div H(x,t) = 0 v(x) x E(x,t) = 0

(6.9) (6.10)

(x Eft, t > 0) (x E 11, t > 0)

(x E 00, t > 0)

E(x,0) = H(x,0) = 0

(x E ft)

for I E 00)3. If f2 C R3 is a star-shaped bounded domain with respect to some xo E IV and an is smooth, then we can show (Lagnese [21], [22]) that there exists T0 > 0 depending only on f2, e and µ for which F2 can be constructed similarly to the manner in Section 2, and that 012)3 3 F2. Thus we have THEOREM 7. On the assumption (3.1), if T > To, then

for I E H2 (0)3 satisfying div I(x) = 0 (x E ft) and I(x) x v(x) = 0 (x E

an).

Remark. For the exact controllability for Maxwell's equations, see Nalin [30], Russell [40], Zhou [55]. As for inverse problems for Maxwell's equations, we refer to Romanov and Kabanikhin [39], Yamamoto [51], [52].

MULTIDIMENSIONAL INVERSE PROBLEMS

223

7. Reduction of the general inverse source problem to an equation of second kind.

We discuss the initial - boundary value problem (1.1) - (1.3) with p = p(x, t) satisfying

LT (7.1)

(7.2) (7.3) (7.4)

p'(.,t)0(.,t)dt

M5110iicao,T];FD, TG E CU°, Ti;FD

IlfP(.,0)11F2 < m511111F2,

f E F2

p E H1(0, T; L'(CI)) m511/11F2,

f E F2.

Here M5 > 0 is independent of 0 and f . Throughout this section, we pose Assumptions A and B. Remark. If we can characterize F2, for example, as F2 = L2(52) (cf. Examples in Section 5), then the restrictions (7.1) - (7.4) are equivalent to

(7.1')

p E H1(0,T; L'(1)),

P(', 0) E L'(11).

For p depending also on x, we discuss Inverse Source Problem. The inverse source problem is seemingly far from formulations by equation of second kind, or by fixed point theorems. However through the Hilbert Uniqueness Method and a duality argument, we can reduce the inverse source problem to an equation of second kind from which we can find f as a fixed point. This reduction step is very useful towards final results for the general inverse source problem. For similar linear inverse problems with singular data such as Dirac delta functions in multidimensional cases and similar ones with smooth data in one-dimensional cases, we can reduce the problems to a Volterra equation of second kind (e.g. Chapter 2 in Lavrentiev, Romanov and Vasiliev [26], Chapter 2 and Section 3 of Chapter 4 in [38]). However in multidimensional cases with not necessarily singular data, any general way for such reduction seems not published. The purpose of this section is to explain the reduction procedure within our framework of the Hilbert Uniqueness Method. Therefore here we do not intend complete researches on uniqueness and stability in actual linear inverse source problems. L2(r x (0, TM is defined in Theorem We recall that a linear operator g : 0 in Section 2 and satisfies (2.8). We define a linear operator S in F2 by (7.5)

(S01)(x) = f T pl (x,t)0(9(01))(x,t)dt

(01 E FD.

Then we have THEOREM 8. Under Assumptions A and B, (7.1) - (7..4);

(1) S : FZ --+ F2 is a bounded linear operator. (2) Let v E H1(0,T;L2(r))n . Then f E F2 satisfies (7.6)

g* (v1 — (Ciu(f)',....,Cnu(1) 1 )) = 0

224

M. YAMAMOTO

if and only if f E F2 satisfies (7.7)

p(•, 0)f + S* f = g*v'.

Here St is the adjoint of S : FZ --+ F2, and g* is the one of a bounded linear operator g : L2(r x (0, T))n. The operator equation (7.7) is our desired one which is an operator equation of the second kind if p(-, 0) 0 0 on O. That is, if p(x,0) 00,

x E St,

then (7.7) is an equation of the second order: (7.7')

f+

11 St f = p(,O) p(.,0)

By the following corollary, we see that it is sufficient to consider (7.7)' for reconstructing f . COROLLARY 2. In Theorem 8, if f E F2 satisfies (7.6')

(Ciu(f), ••••,Cnu(i))= v

for v E H1(0, T; L2(r))n , then f solves (7.7). Remark. In general, 7Z(g) is not dense in L2(r x (o,T))n, so that g* is not injective. Thus in Theorem 8, we can not replace (7.6) by (7.6)'. Next we have to study the unique solvability of the equation (7.7)'. By the contraction mapping principle, we can readily see COROLLARY 3. Let p' (• , •) Pe ' 0) L1 (0,T,Le'(Q))

be sufficiently small and let v = (Ciu(i), ••••,Cnv(f)) solution of (7.7)' by iteration.

Then f is given as a unique

Remark. We do not give direct expression of 5* in the general case. In special cases, we can express S. For example, in Example I in Section 5, let r = 1 (i.e., the spatial dimension is 1), S2 = (0,1), r = {0} (one end point) and T = 2. Then we can construct the control operator g : L2(0,1) L2(0,2) by consideration of the dependency domain of the one-dimensional wave equation and D'Alembert's formula (e.g. Chapter 0 in Komornik [19]). Proof of (1). By Assumption B, (2.8) and (7.1) - (7.4),

=

f0

5 M5 110 (9(01))1IC([0,7 ];FD

tott)0(9(01))(•,t)dt

. M1M511g(01)IlL2 (Tx(0,T))"

Fz

AiNsikkiliF7

which implies the part of (1) of this theorem. Proof of (2). First we show

MULTIDIMENSIONAL INVERSE PROBLEMS

225

Under Assumptions A and B, (7.1) - (7.4), for x (0, nr and f E F2, we have

LEMMA 4 (DUALITY EQUALITY).

E

any v =,

< OM(' 0),,0)

> F;,F2

+(f T

( ,011(v)('

t)dt, F2 F2

(7.8)

f T Ciu(f)1(x,t)v,(x,t)dSz dt. r

3=1 0

(S), Proof of Lemma 4. First assuming that v E cnr x (0, 7') r and f E we see by Theorems 3.1 (pp.103 - 104: Vol.II), 2.1 (pp.95 - 96 : Vol.II) and 8.2 (p.275 : Vol.I) in [29] that W(v) and u(f) are so regular that we can calculate o (1111(f)"(x,t)0(v)(x,t)dx)dt

f as follows.

oT f

(fo u(f)"(x,t)0(v)(x,t)dx)dt

= fo ([u(f)1 (x,t)0(v)(x,t)lit:g'

— f u(fr (x,t)0(vr(x,t)dt)dx

(by integration by parts) oT u(f

=- f (f

t)IP(v)' (x,t)dt)dx

(by W(v)(x, T) = u(fr (x, 0) =- 0)

L(-[u(f)(x,t)0(v)'(x,t)]ta: +

u(f)(x,t)Y)(v)"(x,t)dt)dx

(by integration by parts)

= L(fo T u(f)(x,t)0(v)"(x,t)dt)dx (by W(v)'(x,T) = u(f)(x,0) = 0). Therefore using (1.1) and (2.2), we have iP(y)(x,t)Au(f)(x,t) — u(f)(x,t),40(v)(x,t)dx)dt CU,. f

f(x)(1 p(x,t)0(v)(x,t)dt)dx. o

Applying the Green formula (e.g. p.112 of Lions [28]) and taking into consideration

M. YAMAMOTO

226

the boundary conditions of u(f) and 1,b(v), we reach

In f (x)(1, p(x,t)W(v)(x,t)dt)dx =t Ciu(f)(x,t)v3 T (x,t)dSx dt r 3=1 o for v E cnr satisfies

x (0, T))n.

Since v E C,r(F x (0, T))n, the time derivative tF =

111"(x, t)

AW(x, t) = 0

(x E CI, 0 < t < T) (x E (2)

W(x,T) = le(x,T) = 0

(x E F, 0 0 in St.

Then, as in Example I in Section 5, we see that F2 = L2(52) (e.g. Komornik [19]). In addition to (7.1') we assume (7.19)

p,

E H2 (0, T; L'(10).

P('0) Then by the argument in the proof of Lemma 5.5 in Puel and Yamamoto [35], we can prove COROLLARY 4. Under the assumptions (5.4), (5.5), (7.1)', (7.18) and (7.19), the operator S* : L2(11) L2(S2) is compact. Therefore the equation (7.7)' is a Predholm equation of the second kind in L2(12). In Corollary 4, for the unique solvability, it suffices to verify that f + p !0) S* f = 0 implies f = 0.

230

M. YAMAMOTO

Acknowledgement: This work is partially supported by Sanwa Systems Development Co., Ltd. (Tokyo, Japan). The beginning part of this paper has been written during the author's stay at Freie Universitet Berlin and he thanks Professor Dr. Rudolf Gorenflo (Freie Universitet Berlin) for beneficial discussions.

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