An Abstract Framework for Elliptic Inverse Problems - CiteSeerX

Mathematics and Mechanics of Solids OnlineFirst, published on March 11, 2008 as doi:10.1177/1081286507087150

An Abstract Framework for Elliptic Inverse Problems: Part 2. An Augmented Lagrangian Approach

M ARK S. G OCKENBACH

Department of Mathematical Sciences, 319 Fisher Hall, Michigan Technological University, 1400 Townsend Drive, Houghton MI 49931-1295, USA A KHTAR A. K HAN

Department of Mathematics and Computer Science, 1001 New Science Facility, Northern Michigan University, Marquette, MI-49855, USA (Received 4 May 20071 accepted 7 November 2007)

Abstract: The coefficient in a linear elliptic partial differential equation can be estimated from interior measurements of the solution. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem. All of these properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Moreover, the analysis allows for the use of total variation regularization, so rapidly-varying or even discontinuous coefficients can be estimated.

Key Words: Lamé coefficients, linear elasticity, inverse problem, augmented Lagrangian, regularization

1. INTRODUCTION The problem of identifying parameters in elliptic boundary value problems from interior measurements has applications in several fields, such as image processing, groundwater management, identifying cracks, and modeling of car shields (see [1, 2]). Many papers have been devoted to the theoretical and numerical investigation of these problems. Although most of the early work has imposed smoothness assumptions on coefficients to be recovered, recent applications, such as in image processing, have necessitated methods allowing the recovery of discontinuous parameters. Motivated by the desire to identify discontinuous coefficients in elliptic or parabolic problems, several authors have considered the parameter space as the space of functions of bounded variations. In this setting, since the parameter space is a non-

Mathematics and Mechanics of Solids 00: 000–000, 2008 1 1SAGE Publications 2008

DOI: 10.1177/1081286507087150

Los Angeles, London, New Delhi and Singapore Figures 1, 2 appear in color online: http://mms.sagepub.com

Copyright 2008 by SAGE Publications.

2 M. S. GOCKENBACH and A. A. KHAN

reflexive Banach space, the analysis becomes more involved. The reader is referred to [3] and [4] for some instances where BV seminorm regularization has been successfully employed and discontinuous coefficients have been estimated. This field is dynamic and several interesting approaches are in making. Recently, in Part 1 of this paper [5], we presented an abstract framework for elliptic inverse problems. We have applied our theoretical results to identify discontinuous Lamé parameters in linear elasticity. In this work we continue the development of the abstract framework. We begin by describing the general nature of the problems we address. Most of the work on parameter identification has been in the context of the standard scalar elliptic boundary value problem (BVP) (see [6]) 23 4 1q3u2 5

f

in 34

u 5 0 on 536

(1a) (1b)

where the problem is to estimate q from a measurement of u6 In this context, there have been mainly two approaches for attacking the corresponding inverse problem. The first approach reformulates the inverse problem as an optimization problem and then employs some suitable method for the solution. The second approach treats (1) as a hyperbolic first order partial differential equation in q6 Furthermore, the approach of reformulating (4) as an optimization problem is divided into two possibilities, namely either formulating the problem as an unconstrained optimization problem or treating it as a constrained optimization problem in which the PDE itself is the constraint. In [5], we began to study common features of elliptic inverse problems by establishing an abstract framework based on the output least-square approach, which formulates the inverse problem as an unconstrained minimization problem. In this paper we extend the abstract framework for solving elliptic inverse problem by employing an augmented Lagrangian method. In Ito and Kunisch [7], the augmented Lagrangian approach was used to give an effective scheme for the recovery of smooth coefficients. Recently, Chen and Zou [3] have extended the work of Ito and Kunisch to recover discontinuous coefficients. In the present paper, we propose and analyze an abstract augmented Lagrangian approach to recover smooth as well discontinuous coefficients. Some examples of PDEs with discontinuities can be found in [8, 9]. The contents of this paper are organized into four sections. We begin, in the next section, by formulating the inverse problem of an abstract elliptic BVP. Some particular cases are also discussed. Section 3 is devoted to formulating the constrained minimization problem as an equivalent saddle point problem. The solvability of the minimization problem and hence of the saddle point problem is also given in this section. Section 4 deals with the issue of discretizing the continuous problem. Complete convergence analysis is given in this section.

2. PROBLEM FORMULATION We begin with by defining an abstract framework for the elliptic inverse problem. Throughout this work, we suppose that V is a Hilbert space, B is a Banach space, and A is a subset

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 3 of B with nonempty interior. We assume that T : B 6 V 6 V 7 1 is a trilinear form with T 1a4 u4 72 symmetric in u4 74 and that there exist positive constants 8 and 9 such that T 1a4 u4 72 8 99a9 B 9u9V 979V T 1a4 u4 u2  89u92V

for all u4 7 V4

for all u V4

a A6

a A

(2) (3)

Finally, we assume that m is a fixed bounded linear functional on V . Then, by the Riesz representation theorem, the variational equation T 1a4 u4 72 5 m172 for all 7 V

(4)

has a unique solution u for each a A. Therefore the mapping F : A 7 V defined by the condition that u 5 F1a2 is the solution to (4) is well-defined. In this abstract setting, the inverse problem associated with the direct problem (4) is the following: Given a measurement of u4 say z4 estimate the coefficient a which together with u makes (4) true. The output least-squares approach to the inverse problem of identifying a from a measurement of u 5 F1a2 would seek to minimize the functional 1 a 7 9F1a2 2 z92 2 defined by an appropriate norm. For example, Falk [10] analyzed the case of the L 2 norm applied to the scalar problem (1). The H 1 norm is also a possibility. In [5], we proposed to minimize the functional a 7

1 T 1a4 F1a2 2 z4 F1a2 2 z24 2

(5)

which seems preferable to other possibilities because it is convex. In this work we aim to analyze an augmented Lagrangian approach based on a variant of the above functional. We remark that (5) was motivated by the independent proposals of Chen and Zou [3] and Knowles [11] of using the a-dependent energy norms in place of the L 2 or H 1 norm. The work of these authors was in the context of the scalar problem (1). 2.1. Some Special Cases

Let us now discuss some cases which can be recovered from our abstract formulation. We have mentioned that that most of the works for parameter identification problems are in the context of the elliptic (1). For example, papers such as [7, 10, 11, 12, 13, 14, 15] have examined this problem or variations of it. This problem is not only mathematically interesting, it also appears in numerous industrial applications such as reservoir and underground water resources [11]. The books by Banks-Kunisch [1] and by Engl-Hanke-Neubauer [2] are excellent references for a detailed exposition of the subject.

4 M. S. GOCKENBACH and A. A. KHAN

The variational form of (1) is 1 1 q3u 4 37 5 f7 3

for all 7 H01 1326

3

(6)

The existence and uniqueness of solutions to (6) follows from the Riesz representation theorem, given the usual assumptions on q: There exist positive constants k0 4 k1 such that k0 8 q 8 k1 on 3. We write V 5 H01 132, B 5 L 132, and A 5 q L 132 : k0 8 q 8 k1 , where k1 k0 0 are given. We notice that the trilinear form T : B 6 V 6 V 7 1 given by 1 T 1a4 u4 72 5

3

q3u 4 37

then satisfies (2) and (3), where 9 5 1 and 82 0 comes from Poincaré’s inequality. In this setting the functional m is given by m172 5 3 f 76 There has also been some work on extending various techniques from the papers cited above to inverse problems involving other elliptic BVPs, especially the system of linear, isotropic elasticity: 23 4 

5

in 34

f

(7a)

5 2 eu  tr 1eu 2 I4



(7b)

u 5 0 on 1 4

(7c)

on 2 6

(7d)

n 5 h

Here 53 5 1 2 is a partition of the boundary of 34 n is the outward-pointing unit normal to 534 and u 5

4 13 3u  3u T 6 2

The problem in this case is to estimate and , the Lamé moduli. In recent years, this problem has been studied in the context of applications in biomedical imaging. The following papers address this inverse problem: [16, 17, 18, 19, 20, 21, 22, 23, 24]. The variational form of (7) is 1 1 1 2 u 4 7  tr 1u 2 tr 17 2 5 f 47  h 4 7 for all 7 V4 (8) 3

3

2

5 6 3 42 where V 5 7 H 1 132 : 7 5 0 on 1 . The variational equation (8) can also be expressed in terms of a trilinear form by defining 1 T 14 u4 72 5

3

2 u 4 7  tr 1u 2 tr 17 2 4

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 5 where  5 1 4 2, and 1

1 m172 5

3

f 47 

2

h 4 76

We write B 5 1L 13222 and A 5  5 1 4 2 B : a0 8 4  8 a1  4 where a1

a0 0 are given. Then, for any  A, T 14 u4 72 5 m172 for all 7 V has a unique solution u. The trilinear form T satisfies (2) and (3) where the constant 8 comes from Korn’s inequality and 9 5 10 will work. Related problems that may be of interest include the matrix conductivity problem, in which the scalar function q in (1) is replaced by a matrix, and elasticity systems such as (7) but with a more general stress–strain law relating  and  (to replace (7c)).

3. THE SADDLE POINT PROBLEM By the Riesz representation theorem, there is an isomorphism E : V 7 V  defined by 1Eu2172 5 u4 7V

for every 7 V6

For each 1a4 u2 A 6 V4 T 1a4 u4 42 2 m142 defines an element of V  6 We define e1a4 u2 to be the pre-image under E of this element: e1a4 u24 7V 5 T 1a4 u4 72 2 m172 for every 7 V6 We now consider the following constrained optimization problem: min

1a472 A6V

J 1a4 72 5

1 T 1a4 7 2 z4 7 2 z2 2

subject to e1a4 72 5 06

(9a) (9b)

The only solutions to e1a4 72 5 0 are of the form 1a4 F1a224 so (9) is equivalent to the problem of minimizing (5). The above problem is in general ill-posed and the functional J 1a4 72 should be regularized. As already mentioned, in elliptic inverse problems, it is of particular interest to estimate coefficients that are discontinuous or at least rapidly-varying. For this reason, we want any theory we develop to apply the case of BV-regularization, which has recently been considered in [15] and [3]. We now briefly discuss the space BV132 and its norm. By definition, the total variation of f L 1 132 is

6 M. S. GOCKENBACH and A. A. KHAN 71 TV1 f 2 5 sup

3

8 3 1 42 f 13 4 g2 : g C0 132 4 g1x2 8 1 x 3 6

(10)

When f belongs to W 141 132, then it is easy to show (by integration by parts) that 1 TV1 f 2 5

3

3 f 6

If f L 1 132 satisfies TV1 f 2  , then f is said to have bounded variation, and BV132 is defined by BV132 5

9

f L 1 132 : TV1 f 2  6

The norm on BV132 is 9 f 9BV132 5 9 f 9 L 1 132  TV1 f 26 The functional TV142 is a seminorm on BV132 and is often called the BV-seminorm. The (square of the) H 1 132-seminorm, 1  f 2H 1 132 5

3

1 3f 43f 5

3

3 f 2 4

is often used as a regularization functional. By contrast with TV142,  4  H 1 132 imposes a large penalty on large gradients and is therefore appropriate for recovering mildy-varying coefficients. The BV-seminorm, on the other hand, has been used successfully to recover sharply-varying images in image processing, and, as mentioned above, has recently been applied to elliptic inverse problems. For details on the functions of bounded variations, we refer to the text by Giusti [25]. We use the following properties of BV132 and L 132: 1. L 132 is continuously embedded in L 1 132 (so 9 4 9 L 132 is stronger than 9 4 9 L 1 132 ). 2. BV132 is compactly embedded in L 1 132. To continue the development of the abstract theory, we therefore assume that: 1. The Banach space B (corresponding to L 132 in the scalar case) is continuously embedded in a larger Banach space L. 2. There is another Banach space X (corresponding to BV132 in the scalar case) that is compactly embedded in L. The norm on X is defined by 9 f 9 X 5 9 f 9L   f  X 4 where  4  X is a seminorm. 3. The subset A (the domain of the solution operator F) is a closed and bounded subset of B  X , and A is also closed in L.

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 7

In dealing with the scalar elliptic inverse problem, the Lebesgue dominated convergence theorem and the Cauchy–Schwartz inequality play a decisive role. In our abstract development, these two results motivate the following assumptions on the trilinear form T 144 44 42. We will assume that L is a space of real-valued functions and that T 1a4 u4 72 8 [T 1a4 u4 u2]12 [T 1a4 74 72]12

for all u4 7 V4 a B6

(11)

Moreover, we assume that if ak  is a bounded sequence in B, ak 7 0 in L, and u4 7 are fixed elements of V , then T 1ak 4 u4 72 7 0. We also need to assume that the following condition holds for both 9 4 9 B and 9 4 9 L : 9a9 8 9a9

for all a6

(12)

We finally assume that we have a convex and lower semicontinuous functional R : X 7 1 satisfying the following property: There exists  0 such that R1 f 2    f  X for all f X6

(13)

Then we consider the following constrained optimization problem: min

1a472 A6V

J 1a4 72 5

1 T 1a4 7 2 z4 7 2 z2   R1a2 2

subject to e1a4 72 5 06

(14a) (14b)

Here R serves the role of a regularizing operator and  0 is the regularization parameter. We begin with stating the following existence theorem. Theorem 3.1 The constrained minimization problem (14) has a solution. Proof. The proof can be found in [5].

1

For r  0 and the functional J 144 42 given in (14a), we define the augmented Lagrangian functional 1r : A 6 V 6 V 7 1 as follows: r 1r 1a4 74 2 5 J 1a4 72  e1a4 724   9e1a4 7292V 6 2

(15)

The following result, which is the main result of this section, establishes the equivalence between the saddle point problem and the constrained minimization problem (14). Theorem 3.2 For each r  04 a pair 1a  4 7  2 A 6 V is a solution of (14) if and only if there exists  V such that 1a  4 7  4  2 A 6 V 6 V is a saddle point of the augmented Lagrangian 1r 6 That is

1r 1a  4 7  4 2 8 1r 1a  4 7  4  2 8 1r 1a4 74  2 1a4 74 2 A 6 V 6 V6

(16)

8 M. S. GOCKENBACH and A. A. KHAN Proof. Let 1a  4 7  2 A 6 V be a solution of (14). That is, J 1a  4 7  2 8 J 1a4 72 for all 1a4 72 A 6 V

and e1a  4 7  2 5 06

Let us define a set S 5 1J 1a4 72 2 J 1a  4 7  2  s4 e1a4 722 1 6 V 

1a4 72 A 6 V4 s  06

(17)

In view of the optimality of 1a  4 7  24 we have S  122 6 0V 2 5  where 2 5 t 1 : t 0 and 0V is the zero element of V6 In fact, if there exists 1 p4 q2 S  122 6 0V 24 then q 5 e1a4 72 5 0 and J 1a4 72  J 1a  4 7  2 for some 1a4 72 A 6 V6 This, however, contradicts the optimality of 1a  4 7  26 In view of Lemma 3.1 and Lemma 3.2, given below, the set S4 defined in (17), is solid convex.1 Then a well-known separation theorem (see [26]) ensures the existence of functionals 1 p1 4 p2 2 1 6 V  4 not both zero, such that for all 1a4 72 A 6 V4 s  04 t 2 we have p1 1J 1a4 72 2 J 1a  4 7  2  s2  p2 1e1a4 722  p1 12t26 By setting 1a4 72 5 1a  4 7  2 in the above inequality, we obtain that p1  06 We claim that p1 06 Indeed, if p1 5 04 then for p 2 5 E 21 p2 V we have e1a4 724 p 2  5 [T 1a4 74 p 2 2 2 m1 p 2 2]  0  1a4 72 A 6 V6 Now, for any a A4 let 7 V be a solution to the following variational problem: T 1a4 74 2 5 m12 2  p 2 4  for all  V6 The above two inequalities imply that 9 p 2 9 5 9 p2 9 5 04 contradicting that 1 p1 4 p2 2 5 104 026 Thus p1 5 06 By choosing  5 p 2  p1 4 we obtain that J 1a  4 7  2 8 J 1a4 72  1e1a4 724  2 which proves the second inequality in (16). (The first inequality in (16) is trivial.) Conversely, if 1a  4 7  4  2 A 6 V 6 V is a saddle point of 1r 4 then it follows from the first inequality in (16) that e1a  4 7  2 5 0V and this, in view of the second inequality in (16), confirms that indeed 1a  4 7  2 is a minimizer to (14). The proof is complete. 1 In the following two results, we give some properties of the set S, defined in (17), which have been used in the above proof. Lemma 3.1 The set S has a nonempty topological interior. Proof. Given arbitrary s0 04 we claim that 1s0 4 02 int1S26 Indeed, let  0 be arbitrary and let 1s4 p2 satisfy s 2 s0   9 p9V 8 6 For this p V4 we consider

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 9 T 1a  4 74 2 5 m12   p4  for all  V6

(18)

For the pairs 1a  4 7  2 and 1a  4 724 where 7 V is a solution of (18), we have 2J 1a  4 7  2 2 2J 1a  4 72 5 T 1a  4 7  2 z4 7  2 z2 2 T 1a  4 7 2 z4 7 2 z2 5 T 1a  4 7  2 z4 7  2 z2 2 T 1a  4 7  2 z4 7 2 z2 2 T 1a  4 7 2 7  4 7 2 z2 5 T 1a  4 7  2 z4 7  2 72  T 1a  4 7 2 z4 7  2 72 5 T 1a  4 7  2 74 7   7 2 2z26 Consequently J 1a  4 7  2 2 J 1a  4 72 8 c1 97  2 79 97   7 2 2z96 In view of the optimality of 1a  4 7  2 and (18), we have 97  2 79 8 8 21 9 p9V 97  9 8 8 21 9m9V  Combining the above three inequalities yields J 1a  4 7  2 2 J 1a  4 72 8 c2 4

(19)

where c2 is a constant. Considering s 5 s  J 1a  4 7  2 2 J 1a  4 72  s0 2  2 c2  Choosing  sufficiently small, we obtain s  06 Therefore, with s 5 J 1a  4 72 2 J 1a4 7  2  s 4 we have 1s4 p2 A 6 V6 The proof is complete. 1 Lemma 3.2 For any fixed 1a  4 u  2 A 6 V , the set S 5 1J 1a4 u2 2 J 1a  4 u  2  s4 e1a4 u22 : 1a4 u2 A 6 V4 s  0 is a convex subset of 1 6 V . Proof. Consider any 1r0 4 7 0 24 1r1 4 7 1 2 S, where ri 5 J 1ai 4 u i 2 2 J 1a  4 u  2  si 4

i 5 04 14

10 M. S. GOCKENBACH and A. A. KHAN 7 i 5 e1ai 4 u i 24

i 5 04 16

We must show that 11 2 821r0 4 7 0 2  81r1 4 7 1 2 S for each 8 104 12. Consider first 11 2 827 0  87 1 5 11 2 82e1a0 4 u 0 2  8e1a1 4 u 1 26 We have 1e1ai 4 u i 24 72V 5 T 1ai 4 u i 4 72 2 m172 for all 7 V4 and therefore 111 2 82e1a0 4 u 0 2  8e1a1 4 u 1 24 72 5 11 2 82T 1a0 4 u 0 4 72  8T 1a1 4 u 1 4 72 2 m172 for all 7



V6

If there exists 1a8 4 u 8 2 A 6 V such that 11 2 82e1a0 4 u 0 2  8e1a1 4 u 1 2 5 e1a8 4 u 8 24 then T 1a8 4 u 8 4 72 2 m172 5 1e1a8 4 u 8 24 72V 5 111 2 82e1a0 4 u 0 2  8e1a1 4 u 1 24 72V 5 11 2 82T 1a0 4 u 0 4 72  8T 1a1 4 u 1 4 72 2 m172 for all 7 V must hold. We therefore define a8 5 11 2 82a0  8a1 and u 8 to be the unique solution to the following variational equation: T 1a8 4 u 8 4 72 5 11 2 82T 1a0 4 u 0 4 72  8T 1a1 4 u 1 4 72 for all 7 V6

(20)

We notice that, when 8 5 0, the solution to (20) is u 0 and similarly for 8 5 1, so the notation u 8 is consistent for 0 8 8 8 1. It now suffices to show that 11 2 82 1J 1a0 4 u 0 2 2 J 1a  4 u  2  s0 2  8 1J 1a1 4 u 1 2 2 J 1a  4 u  2  s1 2 5

J 1a8 4 u 8 2 2 J 1a  4 u  2  s8

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 11 for some s8  0. The previous equation is equivalent to 11 2 82J 1a0 4 u 0 2  8 J 1a1 4 u 1 2  11 2 82s0  8s1 5 J 1a8 4 u 8 2  s8 4 or s8 5 11 2 82J 1a0 4 u 0 2  8 J 1a1 4 u 1 2 2 J 1a8 4 u 8 2  11 2 82s0  8s1 6 Since s0 4 s1 can be any non-negative real numbers and s8 must be non-negative, it is necessary and sufficient to prove that J 1a8 4 u 8 2 8 11 2 82J 1a0 4 u 0 2  8 J 1a1 4 u 1 24 or, equivalently, T 1a8 4 u 8 2 z4 u 8 2 z2 8 11 2 82T 1a0 4 u 0 2 z4 u 0 2 z2  8T 1a1 4 u 1 2 z4 u 1 2 z26 We now define  : [04 1] 7 1 by 182 5

1 T 1a8 4 u 8 2 z4 u 8 2 z26 2

To complete the proof, it suffices to show that  is convex, which we do by proving that   182  0 for 8 104 12. We first notice that a 8 5 a1 2 a0 and T 1a8 4 u 8 4 72 5 11 2 82T 1a0 4 u 0 4 72  8T 1a1 4 u 1 4 72 for all 7 V  T 1a 8 4 u 8 4 72  T 1a8 4 u 8 4 72 5 T 1a1 4 u 1 4 72 2 T 1a0 4 u 0 4 72 for all 7 V6 This shows that u 8 is defined by the variational equation T 1a8 4 u 8 4 72 5 T 1a1 4 u 1 4 72 2 T 1a0 4 u 0 4 72 2 T 1a1 2 a0 4 u 8 4 72 for all 7 V6

(21)

We also notice that T 1a1 4 u 1 4 72 2 T 1a0 4 u 0 4 72 5 T 1a1 2 a0 4 u 8 4 72  T 1a8 4 u 8 4 72 for all 7 V4 which will be useful below. We now have

(22)

12 M. S. GOCKENBACH and A. A. KHAN

  182 5 5

1 T 1a 8 4 u 8 2 z4 u 8 2 z2  T 1a8 4 u 8 4 u 8 2 z2 2 1 T 1a1 2 a0 4 u 8 2 z4 u 8 2 z2  T 1a1 4 u 1 4 u 8 2 z2 2 T 1a0 4 u 0 4 u 8 2 z2 2

2 T 1a1 2 a0 4 u 8 4 u 8 2 z2 1 5 T 1a1 4 u 1 4 u 8 2 z2 2 T 1a0 4 u 0 4 u 8 2 z2 2 T 1a1 2 a0 4 u 8 4 u 8 2 z2 2 2

1 T 1a1 2 a0 4 z4 u 8 2 z2 2

1 5 T 1a1 4 u 1 4 u 8 2 z2 2 T 1a0 4 u 0 4 u 8 2 z2 2 T 1a1 2 a0 4 u 8  z4 u 8 2 z2 2 1 5 T 1a1 4 u 1 4 u 8 2 z2 2 T 1a0 4 u 0 4 u 8 2 z2 2 T 1a1 2 a0 4 u 8 4 u 8 2 2 

1 T 1a1 2 a0 4 z4 z26 2

It then follows that   182 5 T 1a1 4 u 1 4 u 8 2 2 T 1a0 4 u 0 4 u 8 2 2 T 1a1 2 a0 4 u 8 4 u 8 2 5 T 1a1 2 a0 4 u 8 4 u 8 2  T 1a8 4 u 8 4 u 8 2 2 T 1a1 2 a0 4 u 8 4 u 8 2 5 T 1a8 4 u 8 4 u 8 2  04 where we use (22). This completes the proof.

1

By employing the notion of the augmented Lagrangian, we have formulated the constrained minimization problem into an equivalent saddle problem. The next step is to solve this saddle point problem to obtain an estimate for the required coefficient. To attain this goal, we discretize the continuous saddle point problem and show that, in a certain sense, solutions of the discretized problems converge to a solution of the continuous problem.

4. FINITE-DIMENSIONAL APPROXIMATION To solve either the forward or inverse problem, discretization is required. In the examples we have in mind, finite element discretization will be used. For the sake of the abstract development, we assume that Vk  is a sequence of finite-dimensional subspaces of V and, for each k, Pk : V 7 Vk is a projection operator. We assume that Vk and Pk have the property that

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 13 for all 7 V4

97 2 Pk 79V 7 0 as k 7 6

(23)

We similarly assume that Bk  is a sequence of finite-dimensional subspaces of B and define Ak 5 Bk  A6 Moreover, we need to assume that



Ak 5 6

(24)

k51

The necessary approximation properties of the spaces Ak are described by the following assumption: For any a A, there exists a sequence ak  such that ak



ak

7 a

R1ak 2 7

Ak

for all k4 in

L4

R1a26

(25a) (25b) (25c)

The above condition is motivated by the properties of the BV seminorm and the approximation properties of the BV spaces (see [3, 25]). We define the discretized solution operator Fk : A 7 Vk by the condition that u 5 Fk 1a2 is the unique solution of the variational problem: T 1a4 u4 72 5 m172 for all 7 Vk 6 We also consider the finite-dimensional analogue ek of the operator E defined as follows: ek 1ak 4 7 k 24  5 T 1ak 4 7 k 4 2 2 m12 for all  Vk 6

(26)

The inverse problem is discretized by defining J 1k2 : Ak 6 Vk 7 1 by J01k2 1a4 u2 5

1 T 1a4 7 2 z4 7 2 z26 2

We consider min

1a472 Ak 6Vk

J 1k2 1a4 u2 5

J01k2 1a4 u2   R 1k2 1a2

subject to ek 1a4 72 5 06

(27a) (27b)

Here R 1k2 : Bk 7 1 is a finite-dimensional version of the regularizing operator which is assumed to be convex and lower semicontinuous and satisfy ak A k

for all k  Rk 1ak 2 2 R1ak 2 7 06

(28)

14 M. S. GOCKENBACH and A. A. KHAN For r  04 the discretized augmented Lagrangian L r1k2 : Ak 6 Vk 6 Vk 7 1 is defined as r L r1k2 1ak 4 7 k 4  k 2 5 J 1k2 1ak 4 7 k 2  ek 1ak 4 7 k 24  k   9ek 1ak 4 7 k 292V 6 2

(29)

We have the following finite-dimensional version of Theorem 3.2. Theorem 4.1 For each r  04 a pair 1ak 4 7 k 2 Ak 6 Vk is a solution of (27) if and only if there exists k V such that 1ak 4 7 k 4 k 2 Ak 6 Vk 6 Vk is a saddle point of the discretized augmented Lagrangian L r1k2 4 that is, for all 1ak 4 7 k 4 k 2 Ak 6 Vk 6 Vk 4 we have L r1k2 1ak 4 7 k 4 k 2 8 L r1k2 1ak 4 7 k 4 k 2 8 L r1k2 1ak 4 7 k 4 k 26 Proof. The proof is the same as for Theorem 3.2.

(30)

1

The following result is useful. Lemma 4.1 Suppose 1ak 4 7 k 2 Ak 6 Vk for all k and suppose ak 7 a in L 6 (a) If 7 k 7 7 in V, then e1ak 4 7 k 2 7 e1a4 72 in V6 (b) 7 k 7 7 weakly in V4 then ek 1ak 4 7 k 2 7 e1a4 72 weakly in V6 Proof. The proof of this result relies on the abstract assumptions imposed on the trilinear 1 form and can be found in [5]. The following is the main result of this section.

9 Theorem 4.2 For each r  04 for each subsequence of the saddle points 1ak 4 7 k 4 k 2 k 3 9

of L r1k2 there exists a subsequence, still denoted by 1ak 4 7 k 4 k 2 k 3 4 with ak 7 a  in L4 7 k 7 7  weakly in V and k 7  weakly in V such that 1a  4 7  4  2 is a saddle point of 1r 6 Proof. Assume that 1ak 4 7 k 4 k 2 Ak 6 Vk 6 Vk is a saddle point of L r1k2 4 that is, L r1k2 1ak 4 7 k  k 2 8 L r1k2 1ak 4 7 k  k 2 8 L r1k2 1ak 4 7 k  k 2 for all 1ak 4 7 k 4 k 2 Ak 6 Vk 6 Vk 6 The first inequality above implies that e1ak 4 7 k 2 5 0V and hence we have J 1ak 4 7 k 2 8 for every 1ak 4 7 k 2

r J 1ak 4 7 k 2  ek 1ak 4 7 k 24 k   9e1ak 4 7 k 292V 2 Ak 6 Vk 6

(31)

Since ak Ak 4 9ak 9 L is bounded uniformly by assumption, and from 97 k 9V 8 8 21 9m9V  it follows that 9ak 9 L  97 k 9V 8 C6 In order to show that  k  is also bounded, we proceed as follows. For any  04 choose 7 k Vk so that e1ak 4 7 k 2 5 2 k

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 15

which implies that 2 k 4 V 5 T 1ak 4 7 k 4 2 2 m12 for all  Vk 6

(32)

By choosing  5 2 k 4 we obtain 2 k 4 2 k V 5 T 1ak 4 7 k 4 2 k 2 2 m12 k 2 or T 1ak 4 7 k 4 k 2 2 m1 k 2 5 29 k 92V 6

(33)

By choosing ak 5 ak in (31), for all 7 k Vk 4 we have 1 r 1 T 1ak 4 7 k 2 z4 7 k 2 z2 8 T 1ak 4 7 k 2 z4 7 k 2 z2  T 1ak 4 7 k 4 k 2 2 m1 k 2  9e1ak 4 7 k 292V 6 2 2 2 The above inequality, in view of (33), implies that 1 1 T 1ak 4 7 k 2 z4 7 k 2 z2 8 T 1ak 4 7 k 2 z4 7 k 2 z2  2 2 Therefore

r 2 2 9 k 92V 2



r 2 2  9 k 92V 6 2

8

1 1 T 1ak 4 7 k 2 z4 7 k 2 z2 2 T 1ak 4 7 k 2 z4 7 k 2 z2 2 2

8

1 T 1ak 4 7 k 2 z4 7 k 2 z2 2

9  9a 9 B 97 k 2 z92V 2 k 3 4 8 99ak 9 B 97 k 92V  9z92V 6

8

Now we need to bound 7 k 6 Once again, by choosing  5 7 k in (32) we obtain that T 1ak 4 7 k 4 7 k 2 5 m17 k 2 2  k 4 7 k V which implies 97 k 9V 8 8 21 9m9V   9 k 9V 6 Therefore 3 42 97 k 92V 8 8 21 9m9V   9 k 9V 8 2 8 22 9m92V   2 2 9 k 92V 6

(34)

16 M. S. GOCKENBACH and A. A. KHAN By choosing  sufficiently small, and combining the above inequality with (34), we obtain 9 k 9 8 C6 Therefore, we have shown that 1ak 4 7 k 4 k 2 B 6 V 6 V is a bounded sequence.    Consequently, we obtain a subsequence of 1ak 4 7 k 4 k 24 still denoted by 1ak 4 7 k 4 k 24 such that for some 1a  4 7  4  2 L 6 V 6 V4 we have ak 7 a 

in

7 k 7 7 

weakly in

V

(35b)

k 7 

weakly in

V6

(35c)

L

(35a)

The above convergence, in view of Lemma 4.1 further implies that e1ak 4 7 k 2 7 e1a  4 7  24

weakly in

V6

Since e1ak 4 7 k 2 5 04 we get that e1a  4 7  2 5 06 Consequently

1r 1a  4 7   2 8 1r 1a  4 7    2 for all V6

(36)

Notice that, in view of (28), we have J 1a  4 7  2 5

1 T 1a  4 7  2 z4 7  2 z2   R1a  2 2

1 T 1ak 4 7 k 2 z4 7 k 2 z2   lim inf R 1k2 1ak 2 k70 2   1 T 1ak 4 7 k 2 z4 7 k 2 z2   R 1k2 1ak 2 8 lim inf k70 2 8 lim inf

k70

8 lim inf J 1k2 1ak 4 7 k 26 k70

(37)

In the above we have used that 1 1 T 1ak 4 7 k 2 z4 7 k 2 z2 5 T 1a  4 7  2 z4 7  2 z26 k7 2 2 lim

Let a A and 7 V be arbitrary. By the assumption there exists ak  such that ak 7 a in 9 4 9 L 6 We set 1ak 4 7 k 2 5 1ak 4 Pk 72 Ak 6 Vk in (31) to obtain: r J 1k2 1ak 4 7 k 2 8 J 1k2 1ak 4 Pk 72  ek 1ak 4 Pk 724 k   9e1ak 4 Pk 7292V 6 2 Passing the above inequality to k 7 04 and taking into account (37), we obtain that r J 1a  4 7  2 8 J 1a4 72  e1a4 724    9e1a4 7292V 6 2

(38)

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 17

Consequently, L r 1a  4 7    2 5 J 1a  4 7  2 8 L r 1a4 7  2 for all 1a4 72 A 6 V6

1

This along with (31) completes the proof. We shall use the following Uzawa type algorithm to solve (30).

Algorithm 1: Let 0 Vk be known. For n  0 and a known n 4 we compute 1a n 4 7 n 2

Ak 6 Vk by solving L r1k2 1a n 4 7 n 4 n 2 8 L r1k2 1ak 4 7 k 4 n 2 for all 1ak 4 7 k 2 Ak 6 Vk

(39)

and update n by

n1 5 n   n e1a n 4 7 n 24

 n 06

(40)

The following result proves the convergence properties of the above algorithm. Theorem 4.3 For n 34 assume that 0   8  n 8 !  r6 Then for each subsequence computed by Algorithm 1, there exists a subsequence, still denoted by

9  of a sequence 1ak 4 7 k 4 k 2 k 3 4 with ak 7 a  in L4 7 k 7 7  weakly in V and k 7  weakly in V4 so that 1a  4 7  4  2 is a saddle point of L r1k2 . Proof. Let 1ak 4 7 k 4 k 2 be a saddle-point of L 1k2 0 4 that is 1k2  1k2      L 1k2 for all 1ak 4 7 k 4 k 2 Ak 6 Vk 6 Vk 6 0 1ak 4 7 k 4 k 2 8 L 0 1ak 4 7 k 4 k 2 8 L 0 1ak 4 7 k 4 k 2

In view of the inequality 1k2      L 1k2 0 1ak 4 7 k 4 k 2 8 L 0 1ak 4 7 k 4 k 2

we infer that ek 1ak 4 7 k 2 5 06 This, in view of the second inequality 1k2     L 1k2 0 1ak 4 7 k 4 k 2 8 L 0 1ak 4 7 k 4 k 2

implies that J 1k2 1ak 4 7 k 2 8 J 1k2 1ak 4 7 k 2  ek 1ak 4 7 k 24 k  for all 1ak 4 7 k 2 Ak 6 Vk 6 By choosing 1ak 4 7 k 2 5 1ak 4 7 k 2 in (39) we obtain

(41)

18 M. S. GOCKENBACH and A. A. KHAN r J 1k2 1a n 4 7 n 2  ek 1a n 4 7 n 24 n   9ek 1a n 4 7 n 292 2

L r1k2 1a n 4 7 n 4 n 2 5 8

L r1k2 1ak 4 7 k 4 n 2

5

J 1k2 1ak 4 7 k 2

8

J 1k2 1a n 4 7 n 2  ek 1a n 4 7 n 24 k 

where the last inequality comes from using (41). Consequently r ek 1a n 4 7 n 24 n 2 k   9ek 1a n 4 7 n 292 8 06 2

(42)

We set 9 n 5 n 2 k which results in the relation 9 n1 5 n1 2 k 5  n   n e1a n 4 7 n 2 2 k 5 9 n   n e1a n 4 7 n 26 The above identity further yields e1a n 4 7 n 24 9 n  5 5

1 9 n1 2 9n4 9n 9 n

1 9 n1 2 99 9 2 99 n 92 2  2n 9e1a n 4 7 n 292 2 n

The above estimate, in view of (42), implies that

1 1 9 n1 2 99 9 2 99 n 92 8 1 n 2 r 29e1a n 4 7 n 292 6 2 n 2

(43)

If we choose 0   n  r then

1 9 n1 2 99 9 2 99 n 92 8 0 2 n which ensures that 99 n1 92 8 99 n 92 6 Therefore, the sequence 99 n 92  is monotonically nonincreasing. This, in view of (43), ensures that 9e1a n 4 7 n 292 7 06

(44)

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 19 By choosing 1ak 4 7 k 2 5 1ak 4 7 k 2 in (39), we obtain J 1k2 1a n 4 7 n 2 8 J 1k2 1ak 4 7 k 2 2 ek 1a n 4 7 n 24 n  8 C where C is some constant which is independent of n6 In view of the definition of J 1k2 1a n 4 7 n 2 and the inequality J 1k2 1a n 4 7 n 2 8 C4 we obtain 9a n 9 B  97 n 9V 8 C6 Therefore, we have shown that 1a n 4 7 n 4 n 2 is a bounded sequence in Ak 6 Vk 6 Vk 6 We have a subsequence 1a n 4 7 n 4 n 2 such that an

7 a

in

7n

7 7

weakly in

V

(45b)

n

7 

weakly in

V

(45c)

L

(45a)

for some 1a  4 7  4  2 Ak 6 Vk 6 Vk 6 From Lemma 4.1 and (44), we obtain that ek 1a  4 7  2 5 06 Finally passing the limit n 7 in (39), we deduce that L r1k2 1a  4 7  4  2 8 L r1k2 1ak 4 7 k 2 1ak 4 7 k 2 Bk 6 Vk 6 This, however, implies that 1a  4 7  4  2 Ak 6 Vk 6 Vk is a saddle point of L r1k2 .

1

5. NUMERICAL EXAMPLES In the following we show the numerical feasibility of the augmented Lagrangian approach by presenting two examples, one for smooth Lamé moduli and one for nonsmooth. 5.1. Example 1

We consider an isotropic elastic membrane occupying the unit square 3 5 104 12 6 104 126 The exact Lamé moduli are 5 1 and 5 1   S 4 where S 5 1x4 y2 S : y  065 and  S is the intex function of S6 That is, is the discontinuous function whose value is 2 on S and 1 on the rest of 3. We perform one “experiment” of stretching the membrane by a boundary traction h and measuring the resulting displacement u. The boundary traction chosen is   1 1 1 h5 n4 10 1 1 where n is the outward point unit normal to 53. This traction is applied to the bottom, left, and right edges of the membrane, while the top edge (y 5 1) is fixed by a Dirichlet condition.

20 M. S. GOCKENBACH and A. A. KHAN

Figure 1. The exact coefficients (top left) and the estimated (bottom left) on a mesh with 1156 elements. The plots on the bottom left show the element-wise error and the bottom right show the nodal error.

In all finite element computations for this example, we use a sequence of uniform triangulations on 36 The coefficients were identified in a finite-dimensional space of dimension 613 on a mesh with 1156 triangles. On the other hand the dimension of u was 23466 The discrete L 2 error between the exact and the computed was 06076 The discrete L 2 error between the exact and the computed was 0600826 These results seem quite satisfactory. We remark that the discontinuous coefficient was identified quite accurately, except along the line of discontinuity. 5.2. Example 2

In this example the Lamé parameters are smooth functions. As previously we consider an isotropic elastic membrane occupying the unit square 3 5 104 12 6 104 12 and the exact Lamé moduli are 1x4 y2 5 1  x  y

1x4 y2 5 x6

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 21

Figure 2. The exact coefficients (top left) and the estimated (bottom left) on a mesh with 1024 elements. The plots on the bottom left show the element-wise error and the bottom right show the nodal error.

We perform another experiment of stretching the membrane by a boundary traction h given by h1x4 y2 5 060216  6x4 26 2 10x2T

on the bottom edge

h1x4 y2 5 060218  2y4 212 2 6y2T

on the left edge

h1x4 y2 5 0602122 2 2y4 6  6y2T

on the right edge

The membrane is fixed on the top edge by a Dirichlet boundary condition. Then the exact displacement u 5 1u 1 4 u 2 2 is given by u 1 1x4 y2 5 06021x 2 6y2 u 2 1x4 y2 5 0606y6

22 M. S. GOCKENBACH and A. A. KHAN

The coefficients were identified in a finite-dimensional space of dimension of 545 on a mesh with 1024 triangles. On the other hand the dimension of u was 20806 The discrete L 2 error between the exact and the computed was 0600076 The discrete L 2 error between the exact and the computed was 0600096 The H 1 seminorm regularization was used and the regularization parameter was 1027 6

6. CONCLUDING REMARKS The analysis presented in this paper shows that the augmented Lagrangian method, which has already been shown to be effective for problems with smooth coefficients, can be used to estimate discontinuous coefficients in elliptic boundary value problems. Moreover, the results are valid in a setting that encompasses any elliptic inverse problem based on interior measurements, including the system of linear isotropic elasticity and more complicated linearly elastic systems. NOTE 1. A set is termed as solid if it has a nonempty topological interior.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Banks, H. T. and Kunisch, K. Estimation Techniques for Distributed Parameter Systems, Systems & Control: Foundations & Applications, 1. Birkhauser, Boston, 1989. Engl, H. W., Hanke, M. and Neubauer, A. Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer, Dordrecht, 1996. Chen, Z. and Zou, J. An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM Journal on Control and Optimization, 37 (1999), 892–910. Luce, R. and Perez, S. Parameter identification for an elliptic partial differential equation with distributed noisy data. Inverse Problems, 15 (1999), 291–307. Gockenbach, M. S. and A.A. Khan, An abstract framework for elliptic inverse problems. Part 1: an output leastsquares approach. Mathematics and Mechanics of Solids, 12, 259–276 (2007). Gockenbach, M. S. Understanding and Implementing the Finite Element Method, The Society for Industrial and Applied Mathematics, Philadelphia, 2007. Ito, K. and Kunisch, K. The augmented Lagrangian method for parameter estimation in elliptic systems. SIAM Journal on Control and Optimization, 28, 113–136 (1990). Gao, D. Y. Duality Principles in Nonconvex Systems, Theory, Methods, and Applications, Kluwer, Boston, 1999. Gao, D. Y. Complementary finite element method for finite deformation nonsmooth mechanics. Journal of Engineering Mathematics, 30, 339–353 (1996). Falk, R. S. Error estimates for the numerical identification of a variable coefficient. Mathematics of Computation, 40, 537–546 (1983). Knowles, I. Parameter identification for elliptic problems. Journal of Computational and Applied Mathematics, 131, 175–194 (2001). Acar, R. Identification of the coefficient in elliptic equations. SIAM Journal on Control and Optimization, 31, 1221–1244 (1993). Kravaris, C. and Seinfeld, J. H. Identification of parameters in distributed parameter systems by regularization. SIAM Journal on Control and Optimization, 23, 217–241 (1985). Kohn, R. V. and Lowe, B. D. A variational method for parameter identification. RAIRO Modélisation Mathématique et Analyse Numérique, 22, 119–158 (1988).

AN ABSTRACT FRAMEWORK FOR ELLIPTIC INVERSE PROBLEMS 2 23

[15] Zou, J. Numerical methods for elliptic inverse problems. International Journal of Computer Mathematics, 70, 211–232 (1998). [16] Chen, J. and Gockenbach, M. S. A variational method for recovering planar Lame moduli. Mathematics and Mechanics of Solids, 7, 191–202 (2002). [17] Cox, S. J. and Gockenbach, M. S. Recovering planar Lame moduli from a single-traction experiment. Mathematics and Mechanics of Solids, 2, 297–306 (1997). [18] Gockenbach, M. S. and Khan, A. A. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial and Management Optimization, 1, 487–497 (2005). [19] Gockenbach, M. S. Jadamba, B. and Khan, A. A. Numerical estimation of discontinuous coefficients by the method of equation error. International Journal of Mathematics and Computer Science, 1, 343–359 (2006). [20] Ji, L. and McLaughlin, J. Recovery of Lamé parameter in biological tissues. Inverse Problems, 20, 1–24 (2004). [21] Kim, H. and Seo, J. K. Identification problems in linear elasticity. Journal of Mathematical Analysis and Applications, 215, 514–531 (1997). [22] McLaughlin, J. and Yoon, J. R. Unique identifiability of elastic parameters from time-dependent interior displacement measurement. Inverse Problems, 20, 25–45 (2004). [23] Oberai, A. A., Gokhale, N. H. and Feijóo, G. R. Solution of inverse problems in elasticity imaging using the adjoint method. Inverse Problems, 19, 297–313 (2003). [24] Raghavan, K. R. and Yagl, A. E. Forward and inverse problems in elasticity imaging of soft tissues. IEEE Transactions on Nuclear Science, 41, 1639–1648 (1994). [25] Giusti, E. Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhauser, Basel, 1984. [26] Jahn, J. Vector Optimization. Theory, Applications, and Extensions, Springer, Berlin, 2004.