Simultaneous identification of Robin coefficient and ...

International Journal of Computer Mathematics

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Simultaneous identification of Robin coefficient and heat flux in an elliptic system Daijun Jiang & T. A. Talaat To cite this article: Daijun Jiang & T. A. Talaat (2017) Simultaneous identification of Robin coefficient and heat flux in an elliptic system, International Journal of Computer Mathematics, 94:1, 185-196, DOI: 10.1080/00207160.2015.1099634 To link to this article: http://dx.doi.org/10.1080/00207160.2015.1099634

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Date: 16 July 2017, At: 19:57

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, 2017 VOL. 94, NO. 1, 185–196 http://dx.doi.org/10.1080/00207160.2015.1099634

Simultaneous identification of Robin coefficient and heat flux in an elliptic system Daijun Jianga and T. A. Talaata,b a School of Mathematics and Statistics, Central China Normal University, Wuhan, China; b Physics and Mathematical Engineering Department, Faculty of Electronic Engineering, Menoufiya University, Egypt

ABSTRACT

ARTICLE HISTORY

In this paper, we shall derive and propose an efficient algorithm for simultaneously reconstructing the Robin coefficient and heat flux in an elliptic system from part of the boundary measurements. The uniqueness of the simultaneous identification is demonstrated. The ill-posed inverse problem is formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization, while the regularizing effects of the regularized system are justified. The Levenberg–Marquardt method is applied to change the non-convex minimization into convex minimization, which will be solved by surrogate functional method so as to get the explicit expression of the minimizer. Numerical experiments are provided to show the accuracy and efficiency of the algorithm.

Received 1 March 2015 Revised 14 August 2015 Accepted 12 September 2015 KEYWORDS

Simultaneous identification; Tikhonov regularization; Levenberg–Marquardt method; surrogate functional method 2010 AMS SUBJECT CLASSIFICATIONS

35J57; 65M32

1. Introduction The goal of this work is to investigate the simultaneous reconstruction of Robin coefficient and heat flux in the following elliptic partial differential equation: −∇ · (a(x)∇u) + c(x)u = f (x) a(x)

in ,

∂u + γ (x)u = g(x) on 1 , ∂n ∂u a(x) = h(x) on 2 , ∂n ∂u a(x) = 0 on 3 , ∂n

(1)

where ⊂ Rd (d = 2, 3) is an open bounded and connected domain, and denote the boundary by ∂, which consists of three disjointed parts ∂ = 1 ∪ 2 ∪ 3 , with 1 , 2 and 3 being the union of some (d − 1)-dimensional polyhedral domains. The coefficients a(x) and c(x) are heat conductivity and radiation, which assumed to satisfy the conditions that 0 < a1 ≤ a(x) ≤ a2 and 0 < c1 ≤ c(x) ≤ c2 in . The functions f (x) and g(x) are source strength and ambient temperature respectively, while γ (x) and h(x) are the Robin coefficient and heat flux respectively to be identified and satisfy γ (x) ∈ K = {γ (x) ∈ L2 (1 ) : 0 < γ1 ≤ γ (x) ≤ γ2 , a.e. on 1 }, h(x) ∈ L2 (2 ). In order to simultaneously recover the Robin coefficient and heat flux, we need to have some extra measurement data from the forward solution u of system (1). We shall assume the measurement data of u is available on 3 . So the inverse problem to be considered can be formulated as follows: CONTACT Daijun Jiang

[email protected]

© 2015 Informa UK Limited, trading as Taylor & Francis Group

186

D. JIANG AND T.A. TALAAT

Inverse Problems I. Let u solve system (1) and a(x), c(x), f (x) and g(x) be known. Given the partial boundary measurable data z of u on 3 , we will simultaneously reconstruct the Robin coefficient γ (x) on 1 and heat flux h(x) on 2 . Inverse problems of parameter identifications have attracted a great attention in the recent three decades due to their practically important applications in engineering and scientific computing; see, for example, [1,5] and the references therein. The proposed inverse problem I is highly nonlinear and ill-posed and arises in several applications of practical importance. The Robin coefficient may characterize the thermal properties of conductive materials on the interface or certain physical processes near the boundary, for example, it represents the corrosion damage profile in corrosion detection [7,9], and indicates the thermal property in quenching processes [17], while the heat flux may represent the rate of heat energy transfer through a given surface [18]. Several numerical methods have been proposed for identifying the Robin coefficient or heat flux. In [7], the Gauss–Newton method was used to solve Robin inverse problem, but with no consideration of regularizations. In [11,12], mathematical and numerical justification of the effectiveness of regularization approaches for Robin inverse problem was given and the conjugate gradient method was applied to the least squares formulation. In [18], a conjugate gradient method was formulated for reconstructing the heat flux. However, very little has been done for the theoretical and numerical analysis of simultaneous identification of the Robin coefficient and heat flux, which motivates the central topic of this current investigation. The rest of the paper is arranged as follows. In Section 2, we demonstrate the uniqueness result of the inverse problem. In Section 3, the Tikhonov regularization system of the nonlinear inverse problem is formulated and its regularizing effects are justified and analysed. In Section 4, the combination of Levenberg–Marquardt method and surrogate functional method is applied to solve the nonlinear minimization system. In Sections 5, two numerical experiments are presented to illustrate the efficiency and accuracy of the proposed algorithm. Some concluding remarks are given in Section 6. Throughout the paper, C is often used for a generic constant. We shall use the symbol ·, · for the general inner product, and write the norms of the spaces H m (), L2 (), H 1/2 () and L2 () (for some ⊂ ∂) respectively as · m, , · , · 1/2, and · .

2. Uniqueness of the simultaneous identification In this section, we shall demonstrate the uniqueness of the proposed inverse problem I. Firstly, we give a preliminary lemma for stating the well-posedness of forward solution u of system (1). Lemma 2.1 (see Lions and Magenes [15]): Let be an open bounded and connected domain with C1 boundary ∂, a(x) ∈ H 1 (), c(x) ∈ L∞ () and γ (x) ∈ L∞ (1 ) with positive lower and upper bounds a1 , a2 , c1 , c2 and γ1 , γ2 respectively, f (x) ∈ L2 (), g(x) ∈ H 1/2 (1 ) and h(x) ∈ L2 (2 ), then there exists a unique solution u ∈ H 1 () of system (1) and it satisfies u 1, ≤ C( f + g 1/2,1 + h 2 ). Now we will establish the following uniqueness result for Inverse Problems I. For simplicity, we shall write the solution of system (1) as u(γ , h) to emphasize its dependence on the Robin coefficient γ and heat flux h. Theorem 2.1: Let pairs (γ1 , h1 ) and (γ2 , h2 ) be two solutions to Inverse Problems I, and assume that ∂ is C2 continuous, meas({x ∈ 1 : u(γ1 , h1 ) = 0}) = 0 and u(γ1 , h1 ) = u(γ2 , h2 ) on 3 , then γ1 = γ2 a.e. on 1 and h1 = h2 a.e. on 2 .

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS

187

Proof: It is straightforward to verify that ω =: u(γ1 , h1 ) − u(γ2 , h2 ) satisfies the Cauchy problem −∇ · (a(x)∇ω) + c(x)ω = 0 a(x)

∂ω =0 ∂n ω=0

in , on 3 ,

(2)

on 3 ,

and on the boundary 1 a(x)

∂ω + γ1 u(γ1 , h1 ) − γ2 u(γ2 , h2 ) = 0; ∂n

(3)

on the boundary 2 ∂ω (4) = h1 − h2 . ∂n The unique continuation principle (see [10]) implies that ω = 0 in . Hence, by trace theorem, it yields that a(x)

ω ∂ ≤ ω 1/2,∂ ≤ C ω 1, = 0, which implies that ω = 0 a.e. on ∂, that is, u(γ1 , h1 ) = u(γ2 , h2 ) a.e. on ∂. Furthermore, from the variational form of system (2) ∂ω a(x)∇ω · ∇ϕdx + c(x)ωϕ dx = 0, ∀ϕ ∈ H 1 (), ϕ ds = 1 ∪2 ∂n we obtain ∂ω/∂n = 0 a.e. on 1 ∪ 2 . Therefore, from Equations (3) and (4), it immediately yields that u(γ1 , h1 )(γ1 − γ2 ) = 0

a.e. on 1

and h1 = h2

a.e. on 2 .

Now the assumption meas ({x ∈ 1 : u(γ1 , h1 ) = 0}) = 0 deduces that γ1 = γ2 a.e. on 1 and h1 = h2 a.e. on 2 . Remark 2.1: For the stability of Robin inverse problems, Choulli [2] proved a local Lipschitz stability estimate for an arbitrary smooth domain and a log-log stability estimate in the case of a rectangular domain in an elliptic equation with a third kind boundary value and Choulli and Yamamoto [3] established a stability estimate from the final overdetermination in a mixed boundary value problem for a parabolic equation. However, very little has been done for the stability of Inverse Problems I, which is difficult due to the simultaneous identification.

3. Tikhonov regularization and its regularizing effects We assume the noise level in the observation data z of the true solution u to the elliptic system (1) is of order δ, namely ˜ − zδ ≤ δ, u(γ˜ , h) 3

(5)

where γ˜ and h˜ are the exact Robin coefficient and heat flux. As the inverse problem is a well-known mathematically ill-posed problem [1], we formulate it into a mathematically stabilized minimization system with Tikhonov regularization: min

(γ ,h)∈K×L2 (2 )

J(γ , h) = u(γ , h) − zδ 23 + β γ 21 + η h 22 ,

where β, η are the regularization parameters.

(6)

188


In the following, we shall justify the regularizing effects of the nonlinear optimization system (6) that it always has solutions and its solutions are stable with respect to the noise error in the observation data zδ . The first theorem establishes the existence of solutions. Theorem 3.1: The minimization system (6) has at least one minimizer. Proof: As J(γ , h) ≥ 0, inf J(γ , h) is finite over K × L2 (2 ). Thus there exists a minimizing sequence {(γ n , hn )} in K × L2 (2 ) such that lim J(γ n , hn ) =

n→∞

inf

(γ ,h)∈K×L2 (2 )

J(γ , h).

(7)

Then by the definition of J(γ , h), we know γ n 1 and hn 2 are bounded. Therefore, there exists a subsequence, still denoted as {(γ n , hn )}, and some (γ ∗ , h∗ ) in K × L2 (2 ) such that (γ n , hn ) (γ ∗ , h∗ ) in L2 (1 ) × L2 (2 ), where means weak convergence. Next, we will prove that (γ ∗ , h∗ ) is a minimizer of Equation (6). Let un ≡ u(γ n , hn ), then it satisfies the following variational form associated with the system (1): for any ϕ ∈ H 1 (), a∇un · ∇ϕ dx + cun ϕ dx + γ n un ϕ ds = f ϕ dx + gϕ ds + hn ϕ ds. (8)

1

1

2

Taking ϕ = un in Equation (8) and using the trace theorem un ∂ ≤ C un 1, and lower bounds of the functions a(x), c(x) and γ n , we have min{a1 , c1 } un 21, + γ1 un 21 ≤ f un + g 1 un 1 + hn 2 un 2 ≤ C( f + g 1 + hn 2 ) un 1, ≤ C un 1, , from which we get un 1, ≤ C. This implies the existence of a subsequence of {un }, still denoted as {un }, and some u∗ ∈ H 1 () such that un u∗ in H 1 (). The Sobolev trace theorem ensures the continuous embedding from H 1 () into H 1/2 (∂) and the compact embedding of H 1/2 (∂) into Lq (∂) for q < q∗ , where the exponent q∗ satisfies the Sobolev embedding theorem [12]: 1 1 1/2 = − , ∗ q 2 d−1

i.e. q∗ =

2(d − 1) . d−2

Therefore, q∗ > 2 for d = 2, 3, and we obtain un → u∗ in L2 (∂), where → means strong convergence. Hence, it yields that lim un − zδ 23 = u∗ − zδ 23 .

n→∞

Now, we shall show that u∗ = u(γ ∗ , h∗ ). To do this, we first prove for any ϕ ∈ H 1 (), n n lim γ u ϕ ds = γ ∗ u∗ ϕ ds. n→∞ 1

1

(9)

(10)

Indeed, 1

(γ n un − γ ∗ u∗ )ϕ ds =

1

γ n (un − u∗ )ϕ ds +

1

(γ n − γ ∗ )u∗ ϕ ds ≡ I1 + I2 .

For I1 , using the fact that γ n ≤ γ2 and un → u∗ in L2 (∂), we have |I1 | ≤ γ2 un − u∗ 1 ϕ 1 → 0 as n → ∞. To estimate I2 , as u∗ , ϕ ∈ H 1 (), then by the trace theorem and embedding theorem for


189

d = 2, 3, we have u∗ , ϕ ∈ H 1/2 (∂) ⊂ L4 (∂), which implies that u∗ ϕ ∈ L2 (∂). Therefore, by the weak convergence γ n γ ∗ in L2 (1 ), it immediately yields that I2 → 0 as n → ∞. Hence, letting n → ∞ in Equation (8) and using the weak convergence of un , γ n , hn and relation (10), we obtain ∗ ∗ ∗ ∗ a∇u · ∇ϕ dx + cu ϕ dx + γ u ϕ ds = f ϕ dx + gϕ ds + h∗ ϕ ds ∀ϕ ∈ H 1 (),

1

1

2

from which and the definition of u(γ ∗ , h∗ ), we have u∗ = u(γ ∗ , h∗ ). Finally, using Equations (9), (7) and the lower semi-continuity of a norm, we have J(γ ∗ , h∗ ) = u(γ ∗ , h∗ ) − zδ 23 + β γ ∗ 21 + η h∗ 22 = lim u(γ n , hn ) − zδ 23 + β γ ∗ 21 + η h∗ 22 n→∞

≤ lim u(γ n , hn ) − zδ 23 + β lim inf γ n 21 + η lim inf hn 22 n→∞

n→∞

≤ lim inf J(γ n , hn ) = n→∞

inf

(γ ,h)∈K×L2 (2 )

n→∞

J(γ , h),

which implies that (γ ∗ , h∗ ) is a minimizer of Equation (6).

The next theorem demonstrates that the minimization system (6) is indeed a stabilization with respect to the changes of the observation errors. Theorem 3.2: Let {znδ } be a sequence such that znδ → zδ in L2 (3 ) as n → ∞ and {(γ (n) , h(n) )} be the sequence of the minimizers of problem (6) with zδ replaced by znδ . Then there is a subsequence of {(γ (n) , h(n) )} that converges weakly in L2 (1 ) × L2 (2 ) to a minimizer of Equation (6). Proof: By the definition of (γ (n) , h(n) ), we know that for any (γ , h) ∈ K × L2 (2 ) u(γ (n) , h(n) ) − znδ 23 + β γ (n) 21 + η h(n) 22 ≤ u(γ , h) − znδ 23 + β γ 21 + η h 22 , which implies the boundedness of γ (n) 21 and h(n) 22 . Therefore, there exists a subsequence, still ¯ in K × L2 (2 ) such that (γ (n) , h(n) ) (γ¯ , h) ¯ in L2 (1 ) × denoted as {(γ (n) , h(n) )}, and some (γ¯ , h) ¯ is a global minimizer of Equation (6). L2 (2 ). Next, we will prove that (γ¯ , h) ¯ in H 1 () and From the proof of Theorem 3.1, we know that u(γ (n) , h(n) ) u(γ¯ , h) (n) (n) 2 ¯ in L (∂) for d = 2, 3. Hence u(γ , h ) → u(γ¯ , h) ¯ − zδ 2 | | u(γ (n) , h(n) ) − znδ 23 − u(γ¯ , h) 3 (n) (n) ¯ ¯ − 2zδ ) ds = (u(γ (n) , h(n) ) − u(γ¯ , h))(u(γ , h ) + u(γ¯ , h) n 3

+

3

(z

δ

¯ − zδ − znδ )(2u(γ¯ , h)

− znδ ) ds

¯ u(γ (n) , h(n) ) + u(γ¯ , h) ¯ − 2zδ ≤ u(γ (n) , h(n) ) − u(γ¯ , h) 3 n 3 ¯ − zδ − zδ + zδ − znδ 3 2u(γ¯ , h) n 3 ¯ + zδ − zδ ), ≤ C( u(γ (n) , h(n) ) − u(γ¯ , h) 3 n 3

(11)

¯ in L2 (∂) and zδ → zδ in L2 (3 ). which converges to 0 as n → ∞, since u(γ (n) , h(n) ) → u(γ¯ , h) n

190


¯ is a minimizer of Equation (6). To see this, for any (γ , h) ∈ K × Now we can show that (γ¯ , h)

L2 (2 ), we derive from Equation (11) and the lower semi-continuity of a norm that ¯ 2 ¯ = u(γ¯ , h) ¯ − zδ 2 + β γ¯ 2 + η h J(γ¯ , h) 3 1 2 ¯ 2 = lim u(γ (n) , h(n) ) − znδ 23 + β γ¯ 21 + η h 2 n→∞

≤ lim u(γ (n) , h(n) ) − znδ 23 + β lim inf γ (n) 21 + η lim inf h(n) 22 n→∞

n→∞

≤ lim inf { u(γ

(n)

n→∞

,h

(n)

) − znδ 23

n→∞

+ β γ (n) 21

+ η h(n) 22 }

≤ lim inf { u(γ , h) − znδ 23 + β γ 21 + η h 22 } n→∞

= u(γ , h) − zδ 23 + β γ 21 + η h 22 = J(γ , h), ¯ is a minimizer of Equation (6). which verifies that (γ¯ , h)

4. The combination of Levenberg–Marquardt method and surrogate functional method for the minimization In this section, we shall mainly propose an efficient algorithm for the nonlinear minimization (6). As the forward solution u(γ , h) is nonlinear with respect to γ , so the minimization (6) is also nonconvex, which is more difficult to solve than convex minimization. Therefore, we will first change the non-convex minimization (6) into convex minimization by using the Levenberg–Marquardt iterative method [13,16]. Then the surrogate functional method [4] will be applied to the convex minimization so as to get the explicit expression of the minimizer. For our later use, we derive the partial Gâteaux derivatives of the forward solution and their adjoint operators. We denote by uγ (γ , h)d and uh (γ , h)p the partial Gâteaux derivatives of the forward solution u(γ , h) with respect to the Robin coefficient γ and heat flux h for any directions d ∈ L2 (1 ) and p ∈ L2 (2 ) respectively. It is standard to get that uγ (γ , h)d and uh (γ , h)p satisfy the following systems −∇ · (a(x)∇(uγ (γ , h)d)) + c(x)(uγ (γ , h)d) = 0 a(x)

∂(uγ (γ , h)d) ∂n

in ,

+ γ (uγ (γ , h)d) = −du(γ , h)

a(x)

∂(uγ (γ , h)d) ∂n

=0

on 1 ,

(12)

on 2 ∪ 3

and −∇ · (a(x)∇(uh (γ , h)p)) + c(x)(uh (γ , h)p) = 0 in , a(x)

∂(uh (γ , h)p) + γ (uh (γ , h)p) = 0 on 1 , ∂n ∂(uh (γ , h)p) = p on 2 , a(x) ∂n ∂(uh (γ , h)p) = 0 on 3 , a(x) ∂n

which are linear with respect to d and p, respectively.

(13)


191

Next, we introduce their adjoint operators. For any ω, v ∈ L2 (3 ), we define uγ (γ , h)∗ ω and by solving the following systems

uh (γ , h)∗ v

−∇ · (a(x)∇(uγ (γ , h)∗ ω)) + c(x)(uγ (γ , h)∗ ω) = 0 a(x)

∂(uγ (γ , h)∗ ω) ∂n

+ γ (uγ (γ , h)∗ ω) = 0

a(x) a(x)

∂(uγ (γ , h)∗ ω)

=0

∂n ∂(uγ (γ , h)∗ ω)

in , on 1 ,

= −ωu(γ , h)

∂n

(14)

on 2 , on 3

and −∇ · (a(x)∇(uh (γ , h)∗ v)) + c(x)(uh (γ , h)∗ v) = 0 in , a(x)

∂(uh (γ , h)∗ v) + γ (uh (γ , h)∗ v) = 0 on 1 , ∂n ∂(uh (γ , h)∗ v) a(x) = 0 on 2 , ∂n ∂(uh (γ , h)∗ v) = v on 3 , a(x) ∂n

(15)

then we propose the following lemma, which gives important adjoint relations. Lemma 4.1: For any directions d ∈ L2 (1 ), p ∈ L2 (2 ) and ω, v ∈ L2 (3 ), we have u(γ , h)(uγ (γ , h)d), ω3 = d, u(γ , h)(uγ (γ , h)∗ ω)1 ,

(16)

uh (γ , h)p, v3 = p, uh (γ , h)∗ v2 .

(17)

Proof: We first prove Equation (16). For any ϕ ∈ H 1 (), the variational forms of system (12) and (14) are

a(x)∇(uγ (γ , h)d) · ∇ϕ dx +

c(x)(uγ (γ , h)d)ϕ dx =

1

(−du(γ , h) − γ (uγ (γ , h)d))ϕ ds,

a(x)∇(uγ (γ , h)∗ ω) · ∇ϕ dx + c(x)(uγ (γ , h)∗ ω)ϕ dx =− γ (uγ (γ , h)∗ ω)ϕ ds − ωu(γ , h)ϕ ds.

(18)

1

3

(19)

Taking ϕ = uγ (γ , h)∗ ω in Equation (18) and ϕ = uγ (γ , h)d in Equation (19), it immediately shows that u(γ , h)(uγ (γ , h)d), ω3 = d, u(γ , h)(uγ (γ , h)∗ ω)1 .

192


Now we start to show Equation (17). For any ϕ ∈ H 1 (), the variational forms of system (13) and (15) are

a(x)∇(uh (γ , h)p) · ∇ϕ dx +

c(x)(uh (γ , h)p)ϕ dx = −

1

γ (uh (γ , h)p)ϕ ds +

a(x)∇(uh (γ , h)∗ v) · ∇ϕ dx + c(x)(uh (γ , h)∗ v)ϕ dx γ (uh (γ , h)∗ v)ϕ ds + vϕ ds. =−

2

pϕ ds, (20)

1

3

(21)

Taking ϕ = uh (γ , h)∗ v in Equation (20) and ϕ = uh (γ , h)p in Equation (21), it immediately implies that uh (γ , h)p, v3 = p, uh (γ , h)∗ v2 .

Next, we shall linearize the forward solution, then apply the Levenberg–Marquardt method to solve Equation (6), which changes the non-convex minimization (6) into convex minimization. So ¯ ∈ K × L2 (2 ), we use the linearization for a given (γ¯ , h) ¯ + u (γ¯ , h)(γ ¯ ¯ ¯ u(γ , h) ≈ u(γ¯ , h) − γ¯ ) + uh (γ¯ , h)(h − h), γ then we may solve the minimization system (6) by the following Levenberg–Marquardt iterative minimization [6,8,19]: j(γ k+1 , hk+1 ) =

min

(γ ,h)∈K×L2 (2 )

j(γ , h)

≡ uγ (γ k , hk )(γ − γ k ) + uh (γ k , hk )(h − hk ) − (zδ − u(γ k , hk )) 23 + β γ 21 + η h 22 .

(22)

As Equation (22) is a simple convex quadratic minimization, we can write its necessary condition, that is, (∂j/∂γ )(γ k+1 , hk+1 )d = 0 for any d ∈ L2 (1 ) and (∂j/∂h)(γ k+1 , hk+1 )p = 0 for any p ∈ L2 (2 ). By direct computation, we have ∂j (γ , h)d = 2uγ (γ k , hk )(γ − γ k ) + uh (γ k , hk )(h − hk ) − (zδ − u(γ k , hk )), ∂γ

uγ (γ k , hk )d3

+ 2βγ , d1 , ∂j (γ , h)p = 2uγ (γ k , hk )(γ − γ k ) + uh (γ k , hk )(h − hk ) − (zδ − u(γ k , hk )), ∂h + 2ηh, p2 .

uh (γ k , hk )p3

One may see that we need to use some iterative method to get the minimizer (γ k+1 , hk+1 ) from the above systems, which is still difficult. Therefore, we would like to apply the surrogate functional technique to further simplify the numerical solutions of minimization (6).


193

Now we introduce an surrogate functional J s (γ , h) of j(γ , h) [4]: J s (γ , h) = j(γ , h) + A γ − γ k 21 + B h − hk 22 − uγ (γ k , hk )(γ − γ k ) + uh (γ k , hk )(h − hk ) 23 ,

(23)

where A and B are two positive constants such that A d 21 + B p 22 − uγ (γ k , hk )d + uh (γ k , hk )p 23 ≥ 0 for any d ∈ L2 (1 ) and p ∈ L2 (2 ) to guarantee the positivity of J s (γ , h). Indeed, we only need to choose A ≥ 2 uγ (γ k , hk ) 2 (L2 ( ),L2 ( )) and B ≥ 2 uh (γ k , hk ) 2 (L2 ( ),L2 ( )) . 1 3 2 3 Here · (U,V) denotes the norm of a function from space U to V , and the boundedness of uγ (γ k , hk ) (L2 (1 ),L2 (3 )) and uh (γ k , hk ) (L2 (2 ),L2 (3 )) can be readily seen from Lemma 2.1. Then using the adjoint relations (16) and (17), we can rewrite J s (γ , h) as follows: J s (γ , h) = −2uγ (γ k , hk )(γ − γ k ) + uh (γ k , hk )(h − hk ), zδ − u(γ k , hk )3 + β γ 21 + η h 22 + A γ − γ k 21 + B h − hk 22 + zδ − u(γ k , hk ) 23 δ − u(γ k , hk ) z = −2 γ − γ k , u(γ k , hk ) uγ (γ k , hk )∗ u(γ k , hk ) 1

k

− 2h − h

, uh (γ k , hk )∗ (zδ

k

− u(γ , h

k

))2 + β γ 21

+ η h 22

+ A γ − γ k 21 + B h − hk 22 + zδ − u(γ k , hk ) 23

2

δ k k 1

k k k k k ∗ z − u(γ , h ) = A γ − γ − u(γ , h ) uγ (γ , h )

+ β γ 21

A u(γ k , hk ) 1

2

1 k k ∗ δ k k k

2 h − h + B

− (γ , h ) (z − u(γ , h )) u h

+ η h 2 B 2 ⎛

2

1 δ − u(γ k , hk ) z

+ ⎝ zδ − u(γ k , hk ) 23 + A u(γ k , hk ) uγ (γ k , hk )∗

k k

A u(γ , h ) 1

2

1 k k ∗ δ

k k

+B

. (24)

B uh (γ , h ) (z − u(γ , h ))

2 As this is a quadratic minimization with respect to γ and h and the last part is not dependent on γ and h, so we use the necessary condition ∂J s /∂γ = 0 and ∂J s /∂h = 0 to find its exact minimizer (γ ∗ , h∗ ): δ k k 1 ∗ k k k k k ∗ z − u(γ , h ) γ = Aγ + u(γ , h ) uγ (γ , h ) , (25) A+β u(γ k , hk ) h∗ =

1 (Bhk + uh (γ k , hk )∗ (zδ − u(γ k , hk ))). B+η

(26)

Now we are ready to propose an efficient algorithm for the simultaneous reconstruction of the Robin coefficient and heat flux. Algorithm 4.1: Choose two tolerance parameters 1 , 2 > 0 and initial values (γ 0 , h0 ), set k = 0. (1) Compute (γ k+1 , hk+1 ) such that (γ k+1 , hk+1 ) = argmin(γ ,h)∈K×L2 (2 ) J s (γ , h).

(27)

194


(2) If γ k+1 − γ k 1 / γ k 1 ≤ 1 and hk+1 − hk 2 / hk 2 ≤ 2 , stop the iteration; otherwise set k := k + 1, go to Step 1. Remark 4.1: The same as for Equations (25) and (26), we have explicit expressions for the minimizers (γ k+1 , hk+1 ) in Equation (27). Hence, we will only need to solve the forward systems (1), (14) and (15) once for u(γ k , hk ), uγ (γ k , hk )∗ ((zδ − u(γ k , hk ))/u(γ k , hk )) and uh (γ k , hk )∗ (zδ − u(γ k , hk )), respectively, at (k + 1)th iteration. Therefore, it is very convenient and inexpensive to implement Algorithm 4.1.

5. Numerical experiments In this section, we shall apply Algorithm 4.1 that was proposed in the previous Sections 4 to simultaneously identify the Robin coefficient and heat flux in the elliptic system (1). We choose the domain = (0, 1) × (0, 2), 1 = {(x, y); x = 1, 0 ≤ y ≤ 2}, 2 = {(x, y); y = 0, 0 ≤ x ≤ 1} and 3 = ∂\(1 ∪ 2 ). Then we triangulate domain into N × M small squares of equal size and further divide each square through its diagonal into two triangles, which leads to a consistent finite element triangulation. All the forward elliptic problems involved in Algorithm 4.1 are solved by the continuous linear finite element method. The general settings involved in Algorithms 4.1 are chosen as follows. The initial guesses are set to be identically equal to some constants, which as we see are rather poor initial guesses for all the test problems. The noisy data zδ is obtained by adding some uniform random noise to the exact data, that is, zδ = u + δRu on 3 , where R is a uniform random function varying in the range [−1, 1]. We take N = 14, M = 28, the noise level δ = 1%, two parameters A = 1, B = 2, two regularization parameters β = η = 10−4 and two tolerance parameters 1 = 2 = 2 × 10−3 . Then we present two numerical tests for simultaneous reconstruction of the Robin coefficient and heat flux in the system (1), where we take a(x) = c(x) = 1 and source function f (x) = (π 2 + x) cos(π y) in . Example 5.1: We take the exact heat flux h(x) = 12 (x − 1)2 + 32 on 2 , Robin coefficient γ (x) = −(y − 1)2 + 2 on {(x, y) ∈ 1 ; 0 ≤ y ≤ 1} and γ (x) = (y − 1)2 + 2 on {(x, y) ∈ 1 ; 1 ≤ y ≤ 2}, ambient temperature g(x) = 2 + (cos(πy) + 1)γ (x) on 1 and the initial guess (γ 0 , h0 ) = (2, 1.3) at all mesh points. Figure 1 shows the exact parameters and the numerical simultaneously recovered parameters by Algorithms 4.1.

2.2

3 γ 2.8

h

(k)

γ

h(k)

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2.6 2 2.4 1.9

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2 1.8

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0

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1

1.5

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0

0.2

0.4

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Figure 1. Exact and reconstructed Robin coefficient (left) and heat flux (right) by Algorithms 4.1 for Examples 5.1: iteration number k = 21, the relative errors γ k − γ 1 / γ 1 = 0.0422 and hk − h 2 / h 2 = 0.0347.

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 3.4

3.2 γ

h h(k)

(k)

γ

3

3.2

2.8

3

2.6

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0

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1.5

2

2

0

0.2

0.4

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Figure 2. Exact and reconstructed Robin coefficient (left) and heat flux (right) by Algorithms 4.1 for Examples 5.2: iteration number k = 15, the relative errors γ k − γ 1 / γ 1 = 0.0331 and hk − h 2 / h 2 = 0.0314.

Example 5.2: We take the exact heat flux h(x) = 3 − x on 2 , Robin coefficient γ (x) = sin((π/2)y) + 2, ambient temperature g(x) = (cos(πy) + 1)γ (x) on 1 and the initial guess (γ 0 , h0 ) = (3, 3) at all mesh points. Figure 2 shows the exact parameters and the numerical simultaneously recovered parameters by Algorithms 4.1. Remark 5.1: We can see from Figures 1 and 2 that even with very bad initial guesses (initial guesses γ 0 and h0 are taken to be some constants), the numerical simultaneously reconstructed Robin coefficients and heat fluxes, with a 1% noise in the data, appear to be quite satisfactory, in view of the highly ill-posedness of the inverse problem. As the norms uγ (γ k , hk ) (L2 (1 ),L2 (3 )) and uh (γ k , hk ) (L2 (2 ),L2 (3 )) are not easy to estimate, so in order to guarantee the positivity of J s (γ , h), one can choose the two positive constants A and B in the numerical experiments as follows: firstly we implement Algorithm 4.1 by choosing any two positive constants A0 and B0 . If Algorithm 4.1 does not converge to some element, then enlarge the two constants A and B until Algorithm 4.1 converges; else reduce A and B so that Algorithm 4.1 converges faster.

6. Concluding remarks In this work, we have demonstrated the uniqueness of the proposed inverse problem for simultaneously identifying the Robin coefficient and heat flux. The non-convex and nonlinear minimization has been formulated by adopting the classical Tikhonov regularization. The Levenberg–Marquardt iterative method has been applied to change the non-convex minimization into convex minimization, which has been solved by the surrogate functional method and derived the explicit expression of the minimizer. Two numerical experiments have been presented to show the accuracy and efficiency of the algorithm. Our future work includes the extension of the combination of the L–M iterative method and surrogate functional method to some time-dependent nonlinear inverse problems, such as the constructions of the time-dependent Robin coefficient, diffusivity coefficient and radiative coefficient in a parabolic system.

Acknowledgements The author would like to thank Professor Masahiro Yamamoto (University of Tokyo) and Professor Jun Zou (Chinese University of Hong Kong) for their valuable discussions and comments, two anonymous referees for their many constructive comments and suggestions, which have helped us improve the organization and the quality of the paper.

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Disclosure statement No potential conflict of interest was reported by the authors.

Funding The work of Daijun Jiang has been financially supported by self-determined research funds of CCNU from the colleges’ basic research and operation of MOE [No. CCNU14A05039], National Natural Science Foundation of China [Nos. 11326233, 11401241 and 11571265] and China Postdoctoral Science Foundation [Grant no. 2012M521444].

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