INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 21 (2005) 1953–1973 doi:10.1088/0266-5611/21/6/010

A variational approach to an elastic inverse problem

B M Brown1, M Jais1 and I W Knowles2

1 School of Computer Science, Cardiff University, Cardiff CF24 3AA, UK2 Department of Mathematics, University of Alabama at Birmingham, Birmingham,AL 35294, USA

E-mail: m.jais@cs.cf.ac.uk

Received 14 April 2005, in final form 4 October 2005Published 28 October 2005Online at stacks.iop.org/IP/21/1953

AbstractWe present a variational approach to the seismic inverse problem of determiningthe coefficients C and ρ of the hyperbolic system of partial differential equations∑j,k,l

∂

∂xj

(Ci,j,k,l(x)

∂

∂xl

uk(x, t)

)= ρ(x)

∂2

∂t2ui, 1 � i � n,

from traction and displacement data measured on the surface. A crucial pointof our approach will be a transformation of the above system to an ellipticsystem of partial differential equations

−∑

k

∇ · (Ci,k∇uk(x, s)) + ρs2ui(x, s) = 0, 1 � i � n.

Thus, we transform the inverse problem for a hyperbolic system to an inverseproblem for an elliptic system. We give a definition of the direct and inverseseismic problem, where we distinguish between the isotropic and anisotropiccases. Further, we develop the theoretical results that we need for a successfulrecovery procedure of the coefficients C and ρ in the isotropic case. Ourapproach consists of a minimization procedure based on a conjugate gradientdescent algorithm. Finally, we present various numerical results that show theeffectiveness of our approach.

1. Introduction

In every elastic media �, the stress τ and the strain e are linked by the relation

τi,j =∑k,l

Ci,j,k,lek,l , (1)

where C is the fourth-order elasticity tensor. Because of the symmetry of the strain and stresstensors, the three-dimensional elasticity tensor has at most 36 independent entries and in thecase of perfect elasticity only 21 independent entries. If the properties of the elastic solidvary with direction, then the elasticity tensor has indeed up to 36 or 21 independent entries:

0266-5611/05/061953+21$30.00 © 2005 IOP Publishing Ltd Printed in the UK 1953

1954 B M Brown et al

this is the so-called anisotropic case; while if the properties of the solid do not vary with thedirection, then the elasticity tensor is called isotropic and has only two independent entries:the so-called Lame parameters λ and µ. The elements of C are in general not continuous ordifferentiable but bounded. In this work, we confine ourselves to the isotropic case where theelements of the elasticity tensor satisfy

Ci,j,k,l = λδi,j δk,l + µ(δi,lδj,k + δi,kδj,l). (2)

The seismic wave equations are defined by

ρ∂2ui

∂t2=

∑j,k,l

∂j (Ci,j,k,l∂luk), 1 � i � n, (3)

over � × (0,∞) with initial conditions for u(x, 0) ∈ L2(0, T ;H 1(�)) and ∂tu(x, 0) ∈L2(� × (0, T )) and either Dirichlet boundary conditions

u(x, t)|∂� = �(x, t) ∈ L2(0, T ;H 1/2(∂�)), (4)

when we have measurements of the displacement on the boundary, or Neumann boundaryconditions ∑

j,k,l

Ci,j,k,l∂jukνl = τi(x, t) ∈ L2(0, T ;H−1/2(∂�)), 1 � i � n, (5)

when we have measurements of the traction on the boundary. The infinite time interval is amatter of convenience since a typical seismic event takes place over a fixed time period andthus the signal may eventually taken to be zero. This is the direct problem for the seismicwave equation. An important condition on the unique solvability of the seismic wave equationin the isotropic case is the strong convexity condition

µ > 0, 2µ + λ > 0. (6)

This can be guaranteed if the associated Poisson ratio

λ

2(λ + µ)(7)

takes values only in the interval [0, 0.5]. In this paper, we shall be concerned with the seismicinverse problem in the isotropic case, that consists of recovering the Lame coefficients and thedensity from measurements of displacement–traction pairs on the boundary.

2. Formulation of the problem

Let � be an open, simply connected subset of Rn with C1,1 boundary. We will often consider

sub-matrices Ci,k, 1 � i, k � n, of the elasticity tensor C, defined by

(Ci,k)j,l = Ci,j,k,l , 1 � j, l � n.

In accordance with (2) and (6), we make the following assumptions on C and ρ:

C ∈ L∞(�)n×n×n×n, ρ ∈ L∞(�), Ci,k = CTk,i , (8)

〈CX,X〉Rn×n � 0, X ∈ R

n×n, (9)

‖C‖ � α > 0, (10)

ρ(x) � β > 0, ∀x ∈ �, (11)

where α and β are constants in R and 〈X, Y 〉Rn×n = ∑

i,j Xi,jYi,j . Under these assumptions,the seismic direct problem has a unique solution and depends continuously on the given data

A variational approach to an elastic inverse problem 1955

(see for example [16]). However, the seismic direct problem can only be solved if one knowsthe coefficients C and ρ in (3). The seismic inverse problem can be formulated as follows:

Problem 2.1 (the seismic inverse problem). Given u(x, 0), ∂tu(x, 0) in � × {0} andindependent measurements of displacement–traction pairs (�m, τm),m = 1, 2, . . . , on∂� × (0, T ), recover the elasticity tensor C and the density ρ in �.

Our approach to solving problem 2.1 is inspired by a method developed by Knowlesfor elliptic problems (see for example [10, 11]). Briefly, this consists of defining a suitablefunctional F(E, �) which has the property that F(E, �) � 0 and F(E, �) = 0 ⇐⇒(E, �) = (C, ρ). The numerical algorithm commences with an arbitrary guess for (C, ρ),then a gradient descent method is applied until F(E, �) = 0. Of course, there are manyfunctionals that have this property but the one defined in [12] is inspired by much experiencein solving elliptic problems and seems to have no spurious local minima. Since (3) is not anelliptic equation but a hyperbolic equation, we have to transform it first, in order to apply ourmethod. Therefore, we apply a Laplace transformation

ui(x, s) =∫ ∞

0e−stui(x, t) dt, 1 � i � n, (12)

to (3). The transformed equation is then given by

−∑

k

∇ · (Ci,k∇uk) + ρs2ui = 0, 1 � i � n. (13)

The boundary conditions (4) and (5) then have the form

u(x, s)|∂� = �(x, s) ∈ H 1/2(∂�)n (14)

in the Dirichlet case or

(∂Cu(x, s))i |∂� =∑

k

〈Ci,k∇xuk(x, s)|∂�, ν〉Rn = τi (x, s) ∈ H−1/2(∂�), 1 � i � n,

(15)

in the Neumann case.Being independent of t, the tensor C and the density ρ have not changed under the

transformation and we can restrict ourselves, in what follows, to the transformed equation.For this reason, we will write u,� and τ instead of u, � and τ , respectively. The boundaryproblem we want to work with is then given by

−∑

k

∇ · (Ci,k∇uk) + ρs2ui = 0, 1 � i � n, (16)

and either

u(x, s)|∂� = �(x, s) (17)

in the Dirichlet case or

(∂Cu(x, s))i |∂� =∑

k

〈Ci,k∇xuk(x, s)|∂�, ν〉Rn = τi(x, s), 1 � i � n, (18)

in the Neumann case. Equation (16) has the following properties:

(i) Since C and ρ do not depend on s, we do not have to solve (16) for all s ∈ C and we canrestrict ourselves to one (or finitely many) real s.

(ii) All the occurring variables are real-valued.

1956 B M Brown et al

(iii) The partial differential equation (16) is strongly elliptic, since C, ρ and s2 are all positive,and therefore the corresponding Dirichlet and Neumann problems have exactly onesolution.

We make the following definition.

Definition 2.2. In the case of Dirichlet boundary conditions, we define the differential operatorAC,ρ : H 1

0 (�)n → H−1(�)n by

(AC,ρu)i = −∑

k

∇ · (Ci,k∇uk) + s2ρui, 1 � i � n. (19)

In the case of Neumann boundary conditions, we define the operator AC,ρ : H 1(�)n →H−1(�)n by

(AC,ρu)i = −∑

k

∇ · (Ci,k∇uk) + s2ρui, 1 � i � n. (20)

Since the system of partial differential equations (16) is strongly elliptic, there exists foreach � ∈ H 1/2(∂�)n a unique solution u(x) that satisfies (16) and the Dirichlet boundarycondition (17). We can therefore define a Dirichlet–Neumann map �C,ρ : H 1/2(∂�)n →H−1/2(∂�)n by

(�C,ρ(�))i =∑

k

〈Ci,k∇uk|∂�, ν〉Rn , 1 � i � n, (21)

where u satisfies (16) and u|∂� = �. The seismic inverse problem can then be formulated asfollows:

Problem 2.3. Given the Dirichlet–Neumann map �C,ρ : H 1/2(∂�)n → H−1/2(∂�)n,determine C and ρ in �.

Remark. Knowledge of the Dirichlet–Neumann map assumes that the waves are created byapplying displacements �m on the surface. In our computations in section 3, we will assumethat the waves are created by tractions τm. Thus, τm is noise-free, but �m might contain noise.

In the isotropic case, we have the following uniqueness result due to Nakamura andUhlmann (see [18]). (See [21] and [18].)

Theorem 2.4 (uniqueness of the inverse problem). Let n � 3. Let (λi, µi, ρi) ∈ C∞(�) ×C∞(�) × C1(�), i = 1, 2, satisfy the strong convexity condition µ > 0, λ + 2µ > 0, ρ > 0.Assume �(λ1,µ1,ρ1) = �(λ2,µ2,ρ2). Then, (λ1, µ1, ρ1) = (λ2, µ2, ρ2) in �.

Therefore, under the above assumptions, any solution of the inverse problem can beuniquely recovered from the Dirichlet–Neumann map. The smoothness assumptions on λ,µ and ρ in the above theorem are of a technical nature, since the proof makes extensive useof pseudodifferential operators. We remark that this result does not hold anymore, in theanisotropic case, where the entries of C can vary with direction (see [15]). However, one ofus (MJ) has recently shown [5] that the support of the coefficients may be recovered even inthis case by an extension of Kirsch’s factorization method (see [7]).

To solve the seismic inverse problem, we want to define a functional G(E, �), on somedomain DG, that has a unique global minimum for (E, �) = (C, ρ). We now discuss thisconcept and start by introducing some notation.

A variational approach to an elastic inverse problem 1957

Notation 2.5 (subindices and superindices). We will use the subindices i, j, k, l to denotecomponents of vectors, the subindices m, z, r to denote components of infinite or finitesequences, the superindices E and C to denote dependence on tensors and the superindices �

and ρ to denote dependence on density functions. For example, we denote the kth componentof the solution u of the boundary value problem,

−∑

k

∇ · (Ci,k∇uk) + ρs2ui = 0, 1 � i � n, u|∂� = g,

by uC,ρ

k . The only exceptions to this rule will be our notation of the Dirichlet–Neumann mapand of differential operators, where we keep the standard notation of �C,ρ and AC,ρ insteadof �C,ρ and AC,ρ .

Knowledge of the Dirichlet–Neumann map �C,ρ : H 1/2(∂�)n → H−1/2(∂�)n

corresponds to knowledge of �C,ρ�m for every �m ∈ H 1/2(∂�)n, where �m, m = 1, 2, . . . ,

is a known basis of H 1/2(∂�)n.For each pair of (E, �) we can define solutions u

E,�m ,m = 1, 2, . . . , and u

E,�m ,

m = 1, 2, . . . , where uE,�m satisfies (16) and the Dirichlet boundary condition

uE,�m

∣∣∂�

= �m, (22)

and uE,�m satisfies (16) and the Neumann boundary condition

∂EuE,�m

∣∣∂�

= �C�m. (23)

With the help of the above definitions, we can now define a functional G that will prove tohave the desired properties we need for a successful descent procedure. We define the domainof the functional G as follows:

DG = {(E, �)| E satisfies (8)–(10), E − C|∂� = 0, E ∈ L∞(�)n×n×n×n

and E ∈ H 1/2+ε(�)n×n×n×n in a neighbourhood of ∂� for some ε > 0, � satisfies (11)}.Remark. The domain DG implies that we know the tensor C and thus the Lame coefficients λ

and µ on the boundary. This is certainly a restriction; however, in most physical applicationsthis is justified.

Definition 2.6 (the functional G). We define the functional G(E, �) on DG by

G(E, �) =∞∑

m=1

γm

∫�

∑i,k

Ei,k∇(u

E,�

m,k − uE,�

m,k

) · ∇(u

E,�

m,i − uE,�

m,i

)+

∑i

s2�(u

E,�

m,i − uE,�

m,i

)2dx,

(24)

where all γm � 0, m = 1, 2, . . . , are chosen in a way such that the series converges with atleast one γm > 0.

Remark. In an implementation with real data, we have only finitely many Dirichlet–Neumannpairs (�m,�C,ρ�m), 1 � m � M , and thus we can choose γm = 1 for 1 � m � M .

In the following, we will write u and u instead of uE,� and uE,� when it is clear to whichE and � we are referring. An obvious property of G is the following.

Theorem 2.7. G(E, �) � 0 and if uniqueness holds for the seismic inverse problem, we alsohave

G(E, �) = 0 ⇐⇒ (E, �) = (C, ρ).

1958 B M Brown et al

Proof. The fact that G(E, �) � 0 follows from (9) and (11). If G(E, �) = 0, thenu

E,�m = u

E,�m . Since u

E,�m and u

E,�m satisfy the same strongly elliptic partial differential

equation, they can only be equal if they satisfy the same boundary conditions. Therefore,�E,��m|∂� = ∂E,�u

E,�m

∣∣∂�

= ∂E,�uE,�m

∣∣∂�

= �C,ρ�m|∂�,∀m, which means that �E,� =�C,ρ and therefore by uniqueness (E, �) = (C, ρ). �

Before we can prove more properties of the functional G, we need some intermediateresults.

Lemma 2.8. If H |∂� = 0, (E, �) ∈ DG, (E + H, �) ∈ DG and � is fixed, the following holds:

lim‖H‖→0

‖uE+H − uE‖H 1(�)n = 0,

where uE+H and uE solve (16) for C = E + H and C = E, respectively.Analogously, we have if (E, �) ∈ DG, (E, � + h) ∈ DG and E is fixed:

lim‖h‖→0

‖u�+h − u�‖H 1(�)n = 0,

where u�+h and u� solve (16) for ρ = � + h and ρ = �, respectively.

Proof. The proof of lemma 2.8 is standard and therefore omitted. �

We also need the Frechet differentiability of the solution u as a function of E and as afunction of ρ.

Lemma 2.9. The Frechet derivative of u(E) ∈ H 1(�)n as a function of the tensor E is givenby

u′(E)H = A−1E (∇ · (H∇ · uE)).

Here we have omitted the dependence on the density �, since we do not allow it to vary.The Frechet derivative of u(�) with respect to the density � is given by

u′(�)h = A−1E (−s2hu�).

Here we have omitted the dependence on the tensor E, since here we always use the same E.

Proof. The proof is again a standard proof and therefore omitted. The interested reader canfind it in [4]. �

Another result that we shall need is the following.

Lemma 2.10. For any tensor κ ∈ L∞(�)n×n×n×n with κ ∈ H 1/2+ε(N)n×n×n×n in aneighbourhood N of ∂� and κ|∂� = 0, the following inequality holds:

‖∇ · (κ∇ · uE+H )‖H−1(�)n � K

for some real constant K, that does not depend on H where E,E + H ∈ DG and ‖H‖L∞ < ε

for a fixed ε > 0. (We are only interested in H with small norm since we always demand that‖H‖ → 0 in the results to come.) Here ∇· stands for the element-wise operation of ∇.

Proof. We define a functional F : H−1(�)n → R by

F(�) =∫

�

∑i,k

κi,k∇uE+Hk · ∇�i dx,

which satisfies

|F(�)| � K‖�‖H 10 (�)n ,

A variational approach to an elastic inverse problem 1959

where K does not depend on H, because of lemma 2.8 and since ‖H‖L∞ < ε. Therefore,F ∈ H−1(�)n. From this we can conclude

‖F‖ = ‖∇ · (κ∇ · uE+H )‖H−1(�)n � K.

This completes the proof. �

We shall now calculate the Gateaux derivative of the functional G. The formula for theGateaux derivative of G(E, �) = ∑

m g(E, �,�m), where

g(E, �,�m) =∫

�

∑i,k

Ei,k∇(u

E,�

m,k − uE,�

m,k

) · ∇(u

E,�

m,i − uE,�

m,i

)+

∑i

s2�(u

E,�

m,i − uE,�

m,i

)2dx,

can obviously be obtained by computing the Gateaux derivatives of g(E, �,�m), m =1, 2, . . . . Since the dependence on m affects only the boundary conditions of um and um,we can suppress the dependence on m in the following and concentrate on a generic functionalg(E, �) which will represent any functional g(E, �,�m).

Definition 2.11. We define the generic functional g(E, �) by

g(E, �) =∫

�

∑i,k

Ei,k∇(u

E,�

k − uE,�

k

) · ∇(u

E,�

i − uE,�

i

)+

∑i

s2�(u

E,�

i − uE,�

i

)2dx. (25)

Therefore, g(E, �) represents an arbitrary addend of G(E, �).

The last five lemmas enable us to prove one of the main theoretical results needed in ourapproach to solving the seismic inverse problem.

Theorem 2.12 (Gateaux derivative of G). For (E, �), (E + H, � + h) ∈ DG, the Gateauxderivative G′(E, �)(H, h) of the functional G is given by

G′(E, �)(H, h) =∞∑

m=1

γm

∫�

∑i,k

Hi,k

(∇uE,�

m,k · ∇uE,�

m,i − ∇uE,�

m,k · ∇uE,�

m,i

)+

∑i

s2h((

uE,�

m,i

)2 − (u

E,�

m,k

)2)dx. (26)

Proof. As we pointed out above, it is sufficient to prove this for the functional g(E, �). As itwill be clear throughout this proof to which (E, �) we are referring to, we omit the superscriptsof u and u in this proof. Now we take r ∈ R and differentiate the expression

−∑

k

∇ · ((Ei,k + rHi,k)∇uE+rH,�+rh)k + s2(� + rh)uE+rH,�+rh

i = 0, 1 � i � n,

with respect to r. Since we know from lemma 2.9 that u and u are differentiable with respectto r, we can calculate wk = ∂uk

∂r

∣∣r=0 and wk = ∂uk

∂r

∣∣r=0. The functions w and w then satisfy

the following equation:∑k

∇ · (Ei,k∇wk) = s2�wi + s2hui −∑

k

∇ · (Hi,k∇uk). (27)

From (22) we get

wi |∂� = 0, 1 � i � n, (28)

1960 B M Brown et al

since � does not depend on r, and from (23) we get∑k

〈Ei,k∇wk, ν〉Rn |∂� = 0, (29)

since again �C� does not depend on r and also since H |∂� = 0. Now we can calculate

g′(E, �)(H, h) = ∂

∂rg(E + rH, � + rh)|r=0

=∫

�

∑i,k

(Hi,k∇(uk − uk) · ∇(ui − ui)) +∑i,k

(Ei,k∇(wk − wk) · ∇(ui − ui))

+∑i,k

(Ei,k∇(uk − uk) · ∇(wi − wi))

+ 2s2�∑

i

(ui − ui)(wi − wi) + s2h(ui − ui)2 dx.

Consider now the function T defined by

T :=∫

�

∑i,k

(Ei,k∇(wk − wk) · ∇(ui − ui)) dx.

We get

T =∫

�

∑i,k

(Ei,k∇(wk − wk) · ∇(ui − ui)) dx

=∫

�

∑i,k

(Ei,k∇wk · ∇ui − Ei,k∇wk · ∇ui − Ei,k∇wk · ∇ui + Ei,k∇wk · ∇ui) dx

=∫

∂�

∑i,k

wk

⟨ET

i,k∇ui, ν⟩R

n dS −∫

�

∑i,k

wk∇ · (ET

i,k∇ui

)dx

−∫

∂�

∑i,k

wk

⟨ET

i,k∇ui , ν⟩R

n dS +∫

�

∑i,k

wk∇ · (ET

i,k∇ui

)dx

−∫

∂�

∑i,k

ui〈Ei,k∇wk, ν〉Rn dS +

∫�

∑i,k

ui∇ · (Ei,k∇wk) dx

+∫

∂�

∑i,k

ui〈Ei,k∇wk, ν〉Rn dS −

∫�

∑i,k

ui∇ · (Ei,k∇wk) dx.

Now we apply (28) and (29) to deduce that the boundary integrals are zero and substituteexpressions (16) and (27) giving

T = −∫

�

∑k

wks2�uk dx +

∫�

∑k

wks2�uk dx

+∫

�

∑i

ui(s2�wi + s2hui) −

∑i,k

ui∇ · (Hi,k∇uk) dx

−∫

�

∑i

ui (s2�wi + s2hui) −

∑i,k

ui∇ · (Hi,k∇uk) dx.

We can do the same for

S :=∫

�

∑i,k

(Ei,k∇(uk − uk) · ∇(wi − wi)) dx,

A variational approach to an elastic inverse problem 1961

giving

S =∫

�

∑i,k

(Ei,k∇(uk − uk) · ∇(wi − wi)) dx

=∫

�

∑i,k

(Ei,k∇uk · ∇wi − Ei,k∇uk · ∇wi − Ei,k∇uk · ∇wi + Ei,k∇uk · ∇wi) dx

=∫

∂�

∑i,k

wi〈Ei,k∇uk, ν〉Rn dS −

∫�

∑i,k

wi∇ · (Ei,k∇uk) dx

−∫

∂�

∑i,k

wi〈Ei,k∇uk, ν〉Rn dS +

∫�

∑i,k

wi∇ · (Ei,k∇uk) dx

−∫

∂�

∑i,k

uk

⟨ET

i,k∇wi, ν⟩R

n dS +∫

�

∑i,k

uk∇ · (ET

i,k∇wi

)dx

+∫

∂�

∑i,k

uk

⟨ET

i,k∇wi, ν⟩R

n dS −∫

�

∑i,k

uk∇ · (ET

i,k∇wi

)dx.

Again we apply (16) and (27)–(29) to get

S = −∫

�

∑i

wi s2� ui dx +

∫�

∑i

wis2�ui

+∫

�

∑k

uk(s2�wk + s2huk) −

∑i,k

uk∇ · (HT

i,k∇ui

)dx

−∫

�

∑k

uk(s2�wk + s2huk) −

∑i,k

uk∇ · (HT

i,k∇ui

)dx.

If we now use these expressions for T and S, we get

g′(E, �)(H, h)

=∫

�

∑i,k

(Hi,k∇(uk − uk) · ∇(ui − ui)) −∑i,k

ui∇ · (Hi,k∇uk) +∑i,k

ui∇ · (Hi,k∇uk)

−∑i,k

uk∇ · (HT

i,k∇ui

)+

∑i,k

uk∇ · (HT

i,k∇ui

)dx

+∫

�

s2�∑

i

2(−wiui + wiui + uiwi − ui wi) + 2wiui − 2wiui − 2uiwi + 2ui wi︸ ︷︷ ︸=0

dx

+∫

�

∑i

2s2huiui − 2s2hu2i + s2h(ui − ui)

2 dx.

As we can see in the previous expression, the second part of the last integral as well as thewhole second line from the bottom vanishes. If we now make a further integration by partsand note that H |∂� = 0, we get

g′(E, �)(H, h) =∫

�

∑i,k

(Hi,k∇(uk − uk) · ∇(ui − ui)) +∑i,k

Hi,k∇uk · ∇ui

−∑i,k

Hi,k∇uk · ∇ui +∑i,k

Hi,k∇uk · ∇ui −∑i,k

Hi,k∇uk · ∇ui

1962 B M Brown et al

+∑

i

s2h(u2

i − u2i

)dx =

∫�

∑i,k

Hi,k(∇uk · ∇ui − ∇uk · ∇ui)

+∑

i

s2h(u2

i − u2i

).

Therefore, the Gateaux derivative of G is given by (26). �Before we show that the Gateaux derivative is also a Frechet derivative, we calculate the

second Gateaux derivative.

Theorem 2.13 (second Gateaux derivative of G). If H |∂� = 0 and L|∂� = 0, then the secondGateaux derivative is given by

G′′(E, �)[(L, l), (H, h)] =∞∑

m=1

γm

∫�

2(⟨(

A−1E,�dm(L, l)

), dm(H, h)

⟩R

n

− ⟨(A−1

E,� dm(L, l)), dm(H, h)

⟩R

n

)dx,

where

dm(L, l)i =∑

k

∇ · (Li,k∇u

E,�

m,k

) − s2luE,�

m,i , 1 � i,� n,

and

dm(L, l)i =∑

k

∇ · (Li,k∇u

E,�

m,k

) − s2luE,�

m,i , 1 � i � n.

dm(H, h) and dm(H, h) are defined analogously.

Proof. We use the fact that

−AE,�uE,� + AE+L,�+lu

E+L,�+l = 0

to get(AE,�

(uE+L,�+l − uE,�

))i=

∑k

∇ · (Li,k∇u

E+L,�+l

k

) − s2l(u

E+L,�+l

i

), 1 � i � n,

(30)

and conclude

(uE+L,�+l − uE,�)i = (A−1

E,�(∇ · (Li,k∇ · uE+L,�+l) − s2l(uE+L,�+l)))i, 1 � i � n.

(31)

Again we will restrict ourselves to the generic functional g(E, �) being a representative ofany addend of G(E, �). Then,

�g′ := g′(E + L, � + l)(H, h) − g′(E, �)(H, h)

=∫

�

∑i,k

Hi,k

(∇uE+L,�+l

k · ∇uE+L,�+l

i − ∇uE,�

k · ∇uE,�

i

− ∇uE+L,�+l

k · ∇uE+L,�+l

i + ∇uE,�

k · ∇uE,�

i

)+

∑i

s2h((

uE+L,�+l

i

)2 − (u

E,�

i

)2 − (u

E+L,�+l

i

)2+

(u

E,�

i

)2)dx

=∫

�

∑i,k

Hi,k

(∇uE,�

k · ∇(u

E+L,�+l

i − uE,�

i

)+ u

E+L,�+l

k · ∇(u

E+L,�+l

i − uE,�

i

)− ∇u

E,�

k · ∇(u

E+L,�+l

i − uE,�

i

) − uE+L,�+l

k · ∇(u

E+L,�+l

i − uE,�

i

))+

∑i

s2h((

uE+L,�+l

i

)2 − (u

E,�

i

)2 − (u

E+L,�+l

i

)2+

(u

E,�

i

)2) dx,

A variational approach to an elastic inverse problem 1963

since ∫�

∑i,k

Hi,k∇uE+L,�+l

k · uE,�

i =∫

�

∑i,k

Hi,k∇uE,�

k · uE+L,�+l

i .

An integration by parts, (31) and the formula a2 − b2 = (a − b)(a + b) give

�g′ =∫

�

∑i,k

−(u

E+L,�+l

i − uE,�

i

)∇ · (Hi,k∇

(u

E+L,�+l

k + uE,�

k

))+

(u

E+L,�+l

i − uE,�

i

)∇ · (Hi,k∇

(u

E+L,�+l

k + uE,�

k

))+

∑i

(u

E+L,�+l

i − uE,�

i

)s2h

(u

E+L,�+l

i + uE,�

i

) − (uE+L,�+l − u

E,�

i

)s2h

(u

E+L,�+l

i + uE,�

i

)dx

=∫

�

∑i

− (u

E+L,�+l

i − uE,�

i

)(∑k

∇ · (Hi,k∇(u

E+L,�+l

k + uE,�

k

))− s2h(u

E+L,�+l

i + uE,�

i

))

+∑

i

(u

E+L,�+l

i − uE,�

i

)(∑k

∇ · (Hi,k∇(u

E+L,�+l

k + uE,�

k

)) − s2h(u

E+L,�+l

i + uE,�

i

))dx.

Now we divide by ‖L‖L∞ and ‖l‖L∞ to get

�g′ =∫

�

∑i

−uE+L,�+l

i − uE,�

i

‖L‖L∞‖l‖L∞

(∑k

∇ · (Hi,k∇

(u

E+L,�+l

k + uE,�

k

)) − s2h(u

E+L,�+l

i + uE,�

i

))

+∑

i

uE+L,�+l

i − uE,�

i

‖L‖L∞‖l‖L∞

(∑k

∇ · (Hi,k∇

(u

E+L,�+l

k + uE,�

k

)) − s2h(u

E+L,�+l

i + uE,�

i

))dx.

The result now follows from (30), (31) and lemmas 2.8, 2.10 and 2.9. This completes theproof. �

Since we calculated a uniform limit in the above proof of the second Gateaux derivative,we can conclude

Corollary 2.14. The functional G is Frechet differentiable.

The form of the second Frechet derivative of G does not enable us to conclude that Gis convex (and G is probably not convex). However, if we want to apply a steepest descentprocedure to the functional G, we have to make sure that G has not more than one localminimum, since otherwise we could get trapped throughout our minimization procedure inone of these minima and would end up with wrong results. We do not give a proof for aunique local minimum here; however, numerical experiments indicate that the functional G isessentially convex, i.e. G′(E, �)[H,h] = 0,∀(H, h), implies G(E, �) = 0. This assumptionis also supported by the fact that a similar scalar functional for the EIT problem has thisproperty (see [9]). Now we summarize our results. We have defined a functional G, whichhas both a unique global for exactly the tensor C and the density function ρ which we wantto recover. We have further reason to believe that this is the only local minimum as well.Therefore, the functional G satisfies all the conditions for a successful minimization procedureby steepest descent.

The only remaining theoretical question for our recovery procedure is whether theDirichlet–Neumann maps �E,� converge to �C,ρ as the functional G tends to zero. Since thetensor C and the density ρ are uniquely determined by the Dirichlet–Neumann map, this is avery crucial condition on the functional G (see our later discussion on an appropriate stopping

1964 B M Brown et al

criterion). However, the answer to this question is positive. To show this, we first definethe appropriate operator norm on L(H 1/2(∂�)n,H−1/2(∂�)n), for which we want to showconvergence.

Definition 2.15. We define the operator norm ‖·‖γ on L(H 1/2(∂�)n,H−1/2(∂�)n) as

‖A‖γ = sup�:�=∑

βm�m,|β|�γ

‖A�‖H−1/2(∂�)n .

Theorem 2.16. Suppose we have a sequence (Er, �r)r∈N in DG. If then G(Er, �r) → 0 asr → ∞, then also∥∥�Er,�r

− �C,ρ

∥∥γ

→ 0.

Proof. We write

G(Er, �r) =∞∑

m=1

γm

∫�

∑i,k

Eri,k∇(

uEr ,�r

m,k − uEr ,�r

m,k

) · ∇(u

Er ,�r

m,i − uEr ,�r

m,i

)+

∑i

s2ρ(u

Er ,�r

m,i − uEr ,�r

m,i

)2dx

as

G(Er, �r) =∞∑

m=1

γm

∫�

⟨Er∇ · (

uEr ,�r

m − uEr ,�r

m

),∇ · (

uEr ,�r

m − uEr ,�r

m

)⟩R

n×n

+∑

i

s2�(Er ,�r

m,i− u

Er ,�r

m,i

)2dx,

where ∇· operates element-wise; it is clear, since all �r and ‖Er‖ are strictly larger than zeroand E is positive definite, that∞∑

m=1

γm

∥∥∇ · (uEr ,�r

m − uEr ,�r

m

)∥∥2L2(�)n

+∥∥uEr ,�r

m − uEr ,�r

m

∥∥2(L2)n

→ 0 as r → ∞.

This means that∞∑

m=1

γm

∥∥uEr ,�r

m − uEr ,�r

m

∥∥2H 1(�)n

→ 0 as r → ∞. (32)

Now we know from the trace theorem that there exists a unique, continuous and right-invertibletrace operator T from H 1(�)n onto H 1/2(∂�)n. Therefore, we get∥∥T

(uEr ,�r

m − uEr ,�r

m

)∥∥H 1/2(∂�)n

� C∥∥uEr ,�r

m − uEr ,�r

m

∥∥2H 1(�)n

(33)

for some constant C, where

T(uEr ,�r

m − uEr ,�r

m

) = (I − �−1

Er ,�r�C,ρ

)�m, (34)

since by (23) we have(�Er,�r

(uEr ,�r

m

))i=

∑k

⟨Ei,k∇u

Er ,�r

m,k , ν⟩R

n = (�C,ρ(�))i, 1 � i � n.

Therefore, we can conclude that∞∑

m=1

γm

∥∥(I − �−1

Er ,�r�C,ρ

)�m

∥∥2H 1/2(∂�)n

→ 0 as r → ∞.

A variational approach to an elastic inverse problem 1965

Since the sequence (Er, �r) converges, it is bounded. Consequently,∥∥�Er,�r

∥∥γ

are uniformlybounded and from the identity

�Er,�r− �C,ρ = �Er,�r

(I − �−1

Er ,�r�C,ρ

),

we get∞∑

m=1

γm

∥∥(�Er,�r− �C,ρ)�m

∥∥2H−1/2(∂�)n

→ 0.

The result now follows from the definition of the norm ‖·‖γ . �

The last theorem ends the discussion of the theoretical results needed by our algorithm.In the next section, we will discuss the numerical implementation of the algorithm.

3. The algorithm

To minimize the functional G we apply a variant of the conjugate gradient method, the Polak–Ribiere scheme, to the functional G. To do this we start with starting coefficients (λ0, µ0, ρ0)

and update them by the following procedure:(hλ

0, hµ

0 , hρ

0

) = (gλ

0 , gµ

0 , gρ

0

) = −∇NG(λ0, µ0, ρ0),

where −∇NG is an appropriate gradient of G (see definition 3.1). At (λi, µi, ρi) we applya line search routine to minimize G(λ,µ, ρ), resulting in G(λi+1, µi+1, ρi+1) and we set(gλ

i+1, gµ

i+1, gρ

i+1

) = −∇NG(λi+1, µi+1, ρi+1) and(hλ

i+1, hµ

i+1, hρ

i+1

) = γi

(gλ

i+1, gµ

i+1, gρ

i+1

),

where γi is defined as

γi =∑

δ=λ,µ,ρ

⟨gδ

i+1 − gδi , g

δi+1

⟩H 1∑

δ=λ,µ,ρ

⟨gδ

i , gδi

⟩H 1

.

As we can see, the first thing we have to do, to apply this scheme, is to calculate thegradient of G. From the formula of the Gateaux derivative, we see that the L2-gradient of G isgiven by the list

∇G =(∑

m

γmηm

∑m

γmκm

),

where the tensor ηm is defined as

ηm,i,j,k,l = ∂juE,�

m,i ∂luE,�

m,k − ∂j uE,�

m,i ∂l uE,�

m,k , 1 � i, j, k, l � n, m = 1, 2, . . . , (35)

and the function κm is given by

κm =∑

i

(u

E,�

m,i

)2 − (u

E,�

m,i

)2, m = 1, 2, . . . . (36)

Remark. We are discussing lists here and not vectors, since the elements of the lists arenot members of the same vector spaces. However, this simplifies the notation. With this listnotation, we can write the Gateaux derivative of G as

G′(E, �)(H, h) = 〈∇G, H 〉,where H is the list given by

H = (Hh),

1966 B M Brown et al

and by abuse of notation, the ‘artificial’ inner product 〈∇G, H 〉 is the sum over the appropriateinner products of the entries of ∇G.

One of the major error sources in steepest descent methods is that the updated functionalafter a descent step does not continue to lie in the domain of the functional anymore. In ourcase, this presents a major problem. The update direction of the tensor E must vanish on theboundary of �, since the tensors E ∈ DG have to satisfy the condition (E − C)|∂� = 0.However, the terms

∂juE,�

m,i ∂luE,�

m,k − ∂j uE,�

m,i ∂l uE,�

m,k , 1 � i, j, k, l � n, m = 1, 2, . . . ,

do not vanish on ∂� in general. We overcome this problem by using a Neuberger gradientϑ (see [19]) for the update direction of the tensor E. In this situation, we give the followingdefinition of the Neuberger gradient.

Definition 3.1. In our case, the definition of the Neuberger gradient is given by

− �ϑm,i,j,k,l + ϑm,i,j,k,l = ηm,i,j,k,l , (37)

ϑm,i,j,k,l |∂� = 0, 1 � i, j, k, l � n, m = 1, 2, . . . . (38)

We can easily see that the Neuberger gradient vanishes on the boundary. We also have toensure that it is a properly defined gradient and descent direction. By an integration by partsit is easily verified that, if we omit the dependence on �, the Neuberger gradient satisfies

G′(E)(H) = 〈ϑ,H 〉H 10 (�)n×n×n×n .

Remark. ϑ is not only a good decent direction, since it solves equation (37), but also is givenby ϑm,i,j,k,l = (� − I )−1(−ηm,i,j,k,l), and therefore it is a preconditioned version of η. Sincethe entries of ϑ belong to H 1

0 , we expect that it is easier to recover smooth functions with theNeuberger gradient than with the L2-gradient. However, it might be a slight disadvantage touse the Neuberger gradient to recover discontinuous functions. In the one-dimensional caseof the inverse spectral problem for the Sturm–Liouville equation, this is certainly the case (seethe paper by Brown et al [1]). We do not have enough experimental evidence in our case yet,but first experiments indicate that this might also be true in our case.

Another condition that is crucial for the definition of the domain DG is that the tensor Ehas to be positive definite and that the density ρ satisfies ρ > 0. Since we are interested in therecovery of the Lame parameters λ and µ, the condition of positive definiteness is equivalentto µ > 0 and 2µ + λ > 0 (see (6)). Therefore, we have to make sure that the updated values ofλ and µ satisfy this condition. This is one characteristic of the ill-posedness of our problem,since if E is not positive definite anymore, we lose the strong ellipticity of our system ofpartial differential equations (16) with (C, ρ) replaced by (E, �). As a consequence of this,we cannot guarantee anymore the existence and uniqueness of the solutions of (16) and ournumerical elliptic solver would become unstable and our whole recovery procedure wouldfail. We control this problem by cutting off the values of λ, µ and ρ after each iteration, ifthey are below a certain cut-off value. This is often justified on physical grounds by the usualpresence of earlier measurements of data, which allows one to establish a minimum for Lameparameters and by what we know about the Poisson ratio, which takes its values in the interval(0.22, 0.35) (cf (7)). Assuming that the density ρ has a positive lower bound is natural. Weare thus getting a better condition for our algorithm and making it well-posed. The slightdisadvantage of introducing a cut-off value is that our algorithm is not a real descent algorithm

A variational approach to an elastic inverse problem 1967

anymore, but an iterative algorithm and we are not descending that fast anymore. However,this is a small price to pay, if we get a stable minimization procedure for it. In our case, wechoose a cut-off value of 0.5 for λ, µ and ρ. The remaining condition specified in the definitionof DG is that the Lame parameters have to be elements of H 1/2+ε ∩ L∞. Since the Neubergergradient is an element of H 1(�), the updated Lame parameters must still be elements ofH 1/2+ε. That they are also elements of L∞ follows from a regularity estimate by Morrey (see[17] or [2, p 82]). The proof that ϑm,i,j,k,l is an element of L∞, for 1 � i, j, k, l < n, can befound in [12, pp 11–2].

3.1. The stopping criteria

An important issue is that of a stopping criterion for our algorithm. Since G tends to zero, onemight suggest to stop the algorithm if G is small enough. This is not a very good criterion sincewe have no guarantee that if G is below a certain value, then the recovered coefficients must begood approximations. Especially in the presence of noise, the minimal value of G need not bezero any longer and therefore the above criterion would certainly fail. Another criterion wouldbe to measure the norm of the L2-gradient and if it is small enough, to abort the algorithm sincethe functional G has only one local minimum. However, we cannot be sure that the gradientdoes not have a small norm away from the local minimum. However, we know that if G tendsto zero, the Dirichlet–Neumann maps also converge (see theorem 2.15). Since the coefficientsare uniquely determined by the Dirichlet–Neumann map, we can expect satisfactory results,if the difference between the Dirichlet–Neumann map for the recovered coefficients and theDirichlet–Neumann map for true coefficients is sufficiently small. Therefore, we suggest thefollowing stopping criteria:

(i) Check if the norm of the L2-gradient is below a certain value.(ii) If (i) is true, check whether the difference between the Dirichlet–Neumann maps is

sufficiently small.

3.2. The implementation

The given data consist of displacement–traction pairs (�m,�m), with �m = �C,ρ�m, m =1, 2, . . . ,M , where M is a finite number. Therefore, we have to change the definition of thefunctional G to

G(E, �) =M∑

m=1

γm

∫�

∑i,k

Ei,k∇(u

E,�

m,k − uE,�

m,k

) · ∇(u

E,�

m,i − uE,�

m,i

)+

∑i

s2�(u

E,�

m,i − uE,�

m,i

)2dx.

(39)

Since we are working with a finite sum now, we can set γm = 1, 1 � m � M . Although theproof for the uniqueness result 2.4 is not valid for n = 2, we do all our implementations forthe two-dimensional case and choose � = [0, 1] × [0, 1]. This is justified by the facts thatwe do not know of any counterexample for the two-dimensional case, and that for the inverseEIT problem the uniqueness result for the function p,

−∇ · (p∇u) = 0,

is still valid for n = 2 (see [6]). However, if we deal with the scalar equation

−∇ · (p∇u) + qu = 0,

then one Dirichlet–Neumann map does not uniquely identify p and q simultaneously (see forexample [3]). Another reason for implementing the method in two dimensions is that thecomputing resources for a three-dimensional implementation have not been available.

1968 B M Brown et al

All our computations were done on a single PC with a 2.4 GHz processor and 1 GBDDR RAM. Needless to say, this is by no means optimal and it would be desirable to do thecomputations on parallel processors as there is natural parallelism in the algorithm. This wasnot possible in our case, since we have not had the resources to do this.

We create our data by choosing the Neumann data as polynomials Pr,s = ∑r,s αs,rx

rys ,where the coefficients αs,r are constants and r, s � N , for some N ∈ N. Since we are dealingwith relatively nice Neumann data, it is sufficient to set M = 5 and N = 2. Thus, the functionswe use as traction data are of the form

α0,1x + α1,0y + α1,1xy + α0,2x2 + α2,0y

2.

The Dirichlet data are attained by solving equation (16). In two later examples (see figures 4and 5), we will use noisy and time-dependent data to show that our proposed method is stableand that applying a Laplace transformation is not ruining the data.

The most natural way of stabilizing our algorithm is to choose more than one Laplaceparameter s in our transformation of the hyperbolic system (3) to the elliptic system (16). Thisway we get more Dirichlet–Neumann maps—one for each s—that all uniquely identify theLame parameters λ and µ and the density ρ. In our computations, we use between 1 and 12Dirichlet–Neumann maps. It is probably better to use even more than 12 Dirichlet–Neumannmaps (we would like to try 20–40), but this is not feasible on a single PC (see also the table withvalues for the average time for one descent step). The choice of the correct Laplace parametersdepends on the domain � and the expected values of the functions in equation (16). In ourcase, we choose the Laplace parameters s in such way that 0.5 < s2ρ � 12. We divide� = [0, 1] × [0, 1] into a regular finite-element grid using 7200 triangles. The numericalderivatives arising from (39) are computed by Matlab’s ‘pdegrad’ function, which uses centraldifferences. This proved to be sufficient for most of our implementations. However, if thegiven data contain a lot of noise (see figure 4), better differentiation techniques (see for example[13]) are probably necessary. All integrations are done by Simpson’s quadrature rule. The lineminimization in each descent step is done by Matlab’s ‘fminbnd’ function, which is similar tothe function ‘Brent’ in [20, chapter 10].

Now we present some numerical results for the recovered Lame coefficients, λ and µ,and the density ρ. As we mentioned earlier, we always choose � = [0, 1] × [0, 1]. Wepresent implementations for the case of smooth, continuous or just bounded functions. Inour implementations, we vary the number Z of Dirichlet–Neumann maps that we use. Wealso distinguish between the cases λ = µ and λ �= µ. We start with an example with for anon-smooth λ. We set ρ(x) ≡ 1 and

λ1(x) = µ1(x) ={

2.0, if |x1| < 0.5 and |x2| < 0.5,

0.5, otherwise.(40)

We can see from figure 1 that the recovery procedure works well and one Dirichlet–Neumann map is sufficient to recover λ. We also did implementations for functions with moresmoothness. In the case of λ ∈ C∞(�), we were able to recover λ with an L1-error of just0.0073.

Next we consider the case λ �= µ. Again we set ρ(x) ≡ 1 and use four Dirichlet–Neumannmaps for our calculations.

λ2(x) ={

1.5, if |x1| < 0.4 and |x2| < 0.4,

0.5, otherwise,(41)

µ2(x) =

1.6, if 0.25 < x1 < 0.75 and 0.25 < x2 < 0.75,

1.6, if −0.75 < x1 < −0.25 and − 0.75 < x2 < −0.25,

0.5, otherwise.

A variational approach to an elastic inverse problem 1969

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10.5

1

1.5

2

True λ1 Computed λ1, L1-error = 0.0752

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 1.

True λ2 Computed λ2, L1-error = 0.1524

True µ2 Computed µ2, L1-error = 0.0868

0.5

0.5

0.500

1

1

1

1.5

−0.5−0.5

−1 −1

0.5

0.5

0.500

1

1

1

1.5

−0.5−0.5

−1 −1

1.8

1.6

1.4

1.2

1

0.8

0.6

0.41

10.5

0.500

−0.5−0.5

−1 −1

10.5

1

1.5

2

10.5

0.500

−0.5−0.5

−1 −1

Figure 2.

Although we get satisfactory results for λ and µ, the results indicate that it is easier torecover the Lame parameter µ than the Lame parameter λ. This is not an unexpected resultsince we have seen that the coefficients of the elasticity tensor C are given by

Ci,j,k,l = λδi,j δk,l + µ(δi,lδj,k + δi,kδj,l).

We see that µ appears more often than λ in the definition of C and that its influence on theelasticity tensor is bigger than of λ. Therefore, one can expect that it is easier to recover µ.

1970 B M Brown et al

True µ

2.4

2.8

2.6

2.2

2

1.6

1.8

1.4

1.2

11

10.5

0.500

−0.5−0.5

−1 −1

2.4

2.6

2.2

2

1.4

1.6

1.8

1.2

1

1

10.5

0.8

0.500

−0.5−0.5

−1 −1

True λ3 Computed λ3, L1-error = 0.0316

1

1

2

10.5

0.5

1.5

0.500

−0.5−0.5

−1 −1

1

2

1

10.5

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0.500

−0.5−0.5

−1 −1

Computed µ3, L1-error = 0.1463

True ρ

−1

1

1−0.5

0.5

1.5

0.500

0.5−0.5

1 −1

−1

1

1−0.5

0.4

1.1

1.2

1.3

1.4

0.5

0.6

0.7

0.8

0.9

0.500

0.5 −0.51 −1

Computed ρ3, L1-error = 0.0458

Figure 3.

Now we consider an example where all three coefficients λ, µ and ρ are unknown.

λ3(x) = e(x21 −1)(x2

2 −1) + 0.5, (42)

µ3(x) =

2.0, if |x1 + 0.5| < 0.25 and |x2 − 0.5| < 0.25,

1.0, if |x1| < 0.25 and |x2 + 0.5| < 0.25,

0.5, otherwise,

ρ3(x) = 1.5 − ‖x‖∞.

We can see from the figures below that we got quite satisfactory results for the Lamecoefficients, λ and µ, as well as for the density ρ. This is very encouraging, since it showsthat we can recover all the unknown parameters in the seismic wave equation simultaneously,

A variational approach to an elastic inverse problem 1971

1.5

0.5

1

1.5

0.5

1

2.5

2

2

1

10.5

0.500

−0.5−0.5

−1 −1

1

10.5

0.500

−0.5−0.5

−1 −1

True λ4 Computed λ4, 10% noise,L1-error = 0.2240

Figure 4.

from measurements on the boundary. It also shows that our method provides a decent recoverymethod in the isotropic case even for three unknown parameters. We also obtained satisfactoryresults for the sole recovery of ρ with known λ and µ.

Since real data are never noise-free, we want to consider an example with noise. Againwe implement the function

λ4(x) = µ4(x) ={

2.0, if |x1| < 0.5 and |x2| < 0.5,

0.5, otherwise,

and assume that ρ(x) ≡ 1. We apply noise of 10% to our Dirichlet data, but no noiseto the Neumann data. We can see from figure 4 that the algorithm remains stable underperturbation. The corresponding L1-error is about three times higher than for the unperturbedfunction. Considering that we have only used one Dirichlet–Neumann map, this result is stillacceptable. It is therefore desirable to try recovering perturbed Lame coefficients with moreDirichlet–Neumann maps. One might also get better results by using a better differentiationmethod than central differences, since differentiation is itself an inverse problem and the errorin the data has a huge effect on the calculated differences. Examples of regularization methodsfor numerical differentiation can be found in [8, 14].

Finally, we want to consider an example with time-dependent data to show that a Laplacetransformation does not ruin the above results. We try to recover the function

λ5(x) = µ5(x) = (x2 − 1) ∗ (y2 − 1) + 0.5

and assume that ρ(x) ≡ 1. For the time dependence, we apply a sine wave to our data. Thus,we use functions of the form

sin(t)(α0,1x + α1,0y + α1,1xy + α0,2x2 + α2,0y

2)

as traction data and apply a finite Laplace transformation over the interval (0, 5). Apart fromthe effects of the Laplace transformation, we also apply a noise of 1% to the Dirichlet data.After 300 iterations, we obtained the following results. As we can see from figure 5, theLaplace transformation does not have any crucial effect on the recovery of the coefficients.

Finally, we want to point out that although we can in theory use as many Dirichlet–Neumann maps as we want, we have to consider the much higher computing costs as canbe seen from table 1. We can see that the higher stability of our descent procedure comesalong with a corresponding higher computing cost. The number of iterations also dependshighly on the smoothness of the coefficients. For smooth coefficients, 200–300 iterations were

1972 B M Brown et al

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10.5

1

1.5

True λ5

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10.5

1

1.5

Computed λ5, 1% noise,

L1-error = 0.02

Figure 5.

(This figure is in colour only in the electronic version)

Table 1. Time for one descent step.

Number of Average timeDirichlet–Neumann for one descentmaps step (s)

1 3654 9666 1297

12 2117

sufficient. For discontinuous coefficients our method needs between 400 and 2000 iterations,which also depends on how many Dirichlet–Neumann maps are used. Since there is naturalparallelism in the algorithm, it seems desirable to run it on parallel machines to reduce thecomputing costs.

References

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[2] Giaquinta M 1993 Introduction to Regularity Theory for Nonlinear Elliptic Systems (Lectures in Mathematics,ETH Zurich) (Basle: Birkhauser)

[3] Isakov V 1998 Inverse Problems for Partial Differential Equations (Applied Mathematical Sciences vol 127)(New York: Springer)

[4] Jais M 2003 A variational approach to the seismic inverse problem MPhil Dissertation UWC http://www.cs.cf.ac.uk/user/M.Jais

[5] Jais M 2005 Unique identifiability of the common support of coefficients of a second order anisotropic ellipticsystem by the Dirichlet–Neumann map J. Comput. Appl. Math. at press

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