Computers and Mathematics with Applications 65 (2013) 42–57

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Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

Identification of an unknown time-dependent heat source term fromoverspecified Dirichlet boundary data by conjugate gradient methodAlemdar Hasanov, Burhan Pektaş ∗

Department of Mathematics and Computer Science, Izmir University, Üçkuyular 35350, Izmir, Turkey

a r t i c l e i n f o

Article history:Received 27 March 2012Received in revised form 3 September 2012Accepted 20 October 2012

Keywords:Inverse source problemTime-dependent heat sourceBoundary measured dataGradient formulaConjugate gradient method

a b s t r a c t

The inverse problem of identifying the unknown time-dependent heat source H(t) of thevariable coefficient heat equation ut = (k(x)ux)x + F(x)H(t), with separable sources of theform F(x)H(t), from supplementary temperature measurement h(t) := u(0, t) at the leftend of the rod, is investigated. The Fouriermethod is employed to illustrate the comparisonof spacewise (F(x)) and time-dependent (H(t) heat source identification problems. An ex-plicit formula for the Fréchet gradient of the cost functional J(H) = ∥u(0, ·;H) − h∥2

L2(0,Tf )is derived via the unique solution of the appropriate adjoint problem. The ConjugateGradient Algorithm, based on the gradient formula for the cost functional, is then pro-posed for numerical solution of the inverse source problem. The algorithm is examinedthrough numerical examples related to reconstruction of continuous and discontinuousheat sources H(t), when heat is transferred through non-homogeneous as well as compos-ite structures. Numerical analysis of the algorithm applied to the inverse source problemin typical classes of source functions is presented. Computational results, obtained for ran-dom noisy output data, show how the iteration number of the Conjugate Gradient Algo-rithm can be estimated. Based on these results it is shown that this iteration number playsa role of a regularization parameter. Numerical results illustrate bounds of applicability ofthe proposed algorithm, and also its efficiency and accuracy.

Crown Copyright© 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

We study the following inverse source problem (ISP) of determining the unknown time-dependent heat source termH(t) > 0, in the following heat conduction problemut = (k(x)ux)x + F(x)H(t), (x, t) ∈ ΩT ,

u(x, 0) = u0(x), x ∈ (0, l),ux(0, t) = 0, u(l, t) = 0, t ∈ (0, Tf ],

(1.1)

from the measured temperature

h(t) := u(0, t), t ∈ [0, Tf ], (1.2)

on the left end x = 0 of a rod. Here ΩT := (x, t) ∈ R2: x ∈ (0, l), t ∈ (0, Tf ], Tf > 0 is the parabolic domain. The

temperature distribution h(t), t ∈ [0, Tf ], is defined to be the measured output data. It is assumed that the heat conductioncoefficient satisfies the following conditions: k(x) ∈ L∞[0, l], k∗

≥ k(x) ≥ k∗ > 0, for all x ∈ (0, l). The left flux and the

∗ Corresponding author. Tel.: +90 2322464949.E-mail addresses: burhan.pektas@izmir.edu.tr, burhanps@gmail.com (B. Pektaş).

0898-1221/$ – see front matter Crown Copyright© 2012 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2012.10.009

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 43

temperature at x = l are taken to be zero in (1.1) without loss of generality, since the parabolic problem is linear. Note thatthe measured temperature h(t) > 0 at the left end of a non-homogeneous rod may have noise.

Here and below it is assumed that the functions u0(x) (initial temperature) and h(t) satisfy the following consistencyconditions: u′

0(0) = u0(l) = 0 and u0(0) = h(0).Heat source identification problems are the most commonly encountered inverse problems in heat conduction. These

problems have been studied over several decades due to their significance in a variety of scientific and engineeringapplications (see [1–26]). In many heat conduction and diffusion problems the source terms are unknown and usually arenot easy to detect directly. Hence, one of the following typical measured output data are available and feasible from thepoint of view of experiments:uT (x) := u(x, Tf ), x ∈ (0, l) (measured final data);

h(t) := u(0, t), t ∈ (0, Tf ] (measured temperature at the left end of a rod)f (t) := −k(l)ux(l, t), x ∈ (0, l) (measured flux at the right end of a rod).

These data are defined to be overspecified boundary (measured) data, according to inverse problems terminology.In the inverse source problem considered here, defined as ISPH, the time-dependent source term H(t) needs to be recovered

from the Dirichlet boundary data h(t), defined by (1.2), assuming that the function F(x) is known. For given source terms F(x)and H(t) the problem (1.1) is defined to be the direct problem.

In the case of arbitrary source term F(x, t) difficulties related to identifiability and uniqueness are discussed in well-known studies [1,5,6,10,14,19,27]. Evidently, one can only hope for a well-posed inverse source problem only in the caseif some a priori information derived from the corresponding physical model. When the source is assumed to be dependentonly on the spatial variable (H(t) ≡ 1, ∀t ∈ (0, T ]), the problem of determining the spacewise unknown source F(x) inthe inverse source problem governed by Eq. (1.1) and the measured final data uT (x) := u(x, Tf ) is one of the most studiedproblems (see, for example, [2,5,8,12] and references therein). For the sake of brevity this problem will be defined belowas ISPF. Note that in [8], an inverse source problem with a source dependent only on time, but with interior temperaturemeasurement χ(t) := u(x0, t), x0 ∈ (0, l), which might not be feasible from the viewpoint of measurements, has alsobeen studied. For the temperature dependent source term heat equation ut = uxx + F(u) the inverse source problem basedon the boundary measured output data (1.2) has first been considered in [3]. In the separable sources case of the formF(x, t) = σ(t)f (x), where σ(t) is known, the inverse problem of determining the spacewise unknown source has first beenstudied in [7], then in [4,12,13]. Note that for the general type source term F(x, t) the mathematical analysis of the leastsquare approach for inverse source problems, based on weak solution theory for PDEs, has been proposed in [10,11].

This study presents a systematic analysis of the inverse source problem of determining the time-dependent source termH(t) of the variable thermal conductivity coefficient heat Eq. (1.1), from themeasured Dirichlet data (1.2).We use the quasi-solution approach for the considered inverse problem, introducing the cost functional

J(H) =

Tf

0[u(0, t;H) − h(t)]2dt, (1.3)

andweak solution theory for parabolic PDEs. Based on this approachwederive an explicit gradient formula for the functionalJ(H) via the solution of appropriate adjoint problem, and then implement the Conjugate Gradient Algorithm (CGA) fornumerical solution of the problem ISPH.Note that, different from the numericalmethods given in the above citedworks, herethe thermal conductivity k(x) is assumed to be not a constant, which permits one to analyze non-homogeneous (composite)materials also.

The paper is organized as follows. In Section 2 a brief Fourier analysis of inverse source problems ISPH and ISPF is given.An explicit gradient formula for the cost functional J(H) corresponding to ISPH is derived in Section 3. Then the ConjugateGradient Algorithm (CGA) applied to ISPH is described. Numerical algorithms for the direct and adjoint problems, as wellas numerical examples related to estimation of the computational noise level εu > 0, are given in Section 4. In Section 5,an optimal selection of the iteration parameter of CGA is discussed. In Section 6 CGA is confirmed for ISPH, with noisefree and random noisy output data, by numerical experiments in typical classes of source terms. Some conclusions andrecommendations for use of the algorithm applied to ISPH are presented in the final Section 7.

2. Preliminary Fourier analysis of inverse source problems ISPF and ISPH

Asmentioned above the problem of determining the spacewise unknown source F(x) in ISP governed by Eq. (1.1) and themeasured final data uT (x) := u(x, Tf ), i.e. the inverse source problem defined as ISPF, is one of the most studied problems.For this reason we give a brief comparative Fourier analysis of the inverse source problems ISPH and ISPF.

To employ the Fourier method we assume that k(x) ≡ k = const > 0. Using separation of variables (u(x, t) = X(x)T (t))in the direct problem (1.1) we obtain the following Fourier cosine series:

u(x, t) =

∞n=0

un(t) cosµnx, (x, t) ∈ ΩT , (2.1)

44 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

where cosµnx, n = 0, 1, . . . , are eigenfunctions corresponding to the eigenvalues λn = [(π/2 + πn)/l]2, µn =√

λn,and

un(t) = u0,ne−kλnt + Fn

t

0e−kλn(t−τ)H(τ )dτ . (2.2)

The Fourier coefficients u0,n and Fn of the initial data u0(x) and the spacewise source F(x) in (2.2) are defined as follows:

u0,n =2l

l

0u0(ξ) cosµnξdξ, Fn =

2l

l

0F(ξ) cosµnξdξ .

Consider first ISPF, i.e. the problem of determining the unknown spacewise source F(x) in (1.1) from the measured finaldata uT (x) := u(x, Tf ). Using the condition uT (x) = u(x, Tf ) in (2.2) we obtain the solution of ISPF:

F(x) =

∞n=0

Fn cosµnx, Fn = (uT ,n − u0,ne−kλnTf )

Tf

0e−kλn(Tf −τ)H(τ )dτ

, (2.3)

where

uT ,n =2l

l

0uT (ξ) cosµnξdξ, n = 0, 1, 2, . . . (2.4)

are the Fourier coefficients of the measured output data uT (x).

Lemma 2.1. Let k(x) ≡ k = Const > 0. Assume that the initial/final temperature functions u0(x), uT (x), and the sourcefunctions F(x),H(t) are bounded and satisfy the conditions: u0(x), uT (x) ∈ L2[0, l], F(x) ∈ L2[0, l],H(t) ∈ L2[0, Tf ]. Then theunique solution of ISPF is defined by (2.3)–(2.4).

Lemma 2.1 implies, in particular, that the final overdetermination uT (x) := u(x, T ) uniquely determines the unknownspacewise source F(x) in (1.1). Note thatwhen the source term is a function of both space and time variables, but not additiveor separable, the global uniqueness result for the inverse source problem with final observations has been obtained in [1].

In some applications, the source term of a heat equation might depend on only the space variable, i.e. H(t) ≡ 1. In thiscase the Fourier coefficients, given in (2.3), are defined as follows:

Fn = kλnuT ,n − u0,ne−kλnT

/(1 − e−kλnT ).

Consider now ISPH, i.e. the problem of determining the unknown time dependent source H(t), defined by (1.1)–(1.2).Using condition (1.2) in the series solution (2.1)–(2.2) of the direct problem (1.1) we get:

∞n=0

u0,ne−kλnt +

∞n=0

Fn

t

0e−kλn(t−τ)H(τ )dτ = h(t).

Hence the time-dependent source function H(t) is the solution of the following Volterra integral equation of the first kind t

0K(t − τ)H(τ )dτ = h(t), h(t) = h(t) −

∞n=0

u0,ne−kλnt , (2.5)

with the kernel

K(t − τ) =

∞n=0

Fne−kλn(t−τ). (2.6)

Lemma 2.2. Let the conditions of Lemma 2.1 hold. Assume that h(t) ∈ L2[0, T ]. Then the unique solution of ISPH is defined asthe solution of the Volterra Eq. (2.5) for almost everywhere t ∈ [0, T ].

Proof. The kernel K(η), defined by (2.6), is a continuous function due to the conditions of the lemma.We denote byL[K ⋆H]

the Laplace transform of the convolution on the left hand side of (2.6):

(K ⋆ H)(t) :=

t

0K(t − τ)H(τ )dτ .

Then applying the Convolution Theorem we conclude

L[K ⋆ H] = L[K ]L[H],

where L[K ] and L[H] denote the Laplace transforms of the functions K and H , accordingly.

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 45

Thus the above analysis of the inverse source problems ISPF and ISPH, for the case when k(x) ≡ k = const > 0, showsthat both problems have unique solutions. However, while for ISPF this solution can be found as the exact Fourier seriessolution, for ISPH this solution can only be found as a numerical solution of the Volterra equation of the first kind.

3. Explicit gradient formula for the cost functional J(H), and algorithm of CGM for ISPF and ISPH

LetH ⊂ H0(0, Tf ) be closed convex sets of admissible time-dependent sources.With respect to a given spacewise sourceF(x) and heat conduction coefficient k(x) > 0 we will assume that

F(x) ∈ H0(0, l), k(x) ∈ L∞[0, l], k∗≥ k(x) ≥ k∗ > 0. (3.1)

The weak solution of the direct problem (1.1) will be defined to be the function u ∈ V 1,0(ΩTf ), which satisfies the followingintegral identity

ΩTf

(−uvt + kuxvx)dxdt =

ΩTf

F(x)H(t)v(x, t)dxdt, ∀v ∈ H1,1(ΩTf ),

with v(x, Tf ) = 0. For an existence and uniqueness of this solution in V 1,0(ΩTf ) see [28,29]. Here V 1,0(ΩTf ) := C([0, Tf ];H0(0, l)) ∩ H0((0, Tf );H1(0, l)) is the Banach space of functions with the norm

∥u∥V1,0(ΩTf ) := vrai maxt∈[0,Tf ]

∥u∥H0[0,l] + ∥ux∥H0(ΩTf ),

and V 1,0(ΩTf ) := v ∈ V 1,0(ΩTf ) : v(l, t) = 0, ∀t ∈ (0, Tf ].Here H1,1(ΩTf ) is the Sobolev space of functions with the norm [30]

∥u∥H1,1(ΩTf ) :=

ΩTf

[u2+ u2

x + u2t ]dxdt

1/2

,

and H1,1(ΩTf ) := v ∈ H1,1(ΩTf ) : v(l, t) = 0, ∀t ∈ (0, Tf ). Note that the norms ∥u∥H1(0,l) and ∥ux∥H0(0,l) are equivalentdue to the homogeneous Dirichlet condition u(l, t) = 0 in the direct problem (1.1). Here and below H0(E) ≡ L2(E), and∥ · ∥H0(E) means the H0-norm in E.

Evidently, for each given H ∈ H , from the set of admissible sources H ⊂ H0[0, Tf ], the weak solution u ∈ V 1,0(ΩTf ) of

the parabolic problem (1.1) exists and is unique, if conditions (3.1) hold. We define this solution to be as u(x, t;H) if thesource function F(x) is known and the time-dependent source term H(t) needs to be recovered from the measured output datah(t) > 0, given by (1.2).

A least square solution (or quasi-solution) of ISPH is defined to be as a solution of the following minimization problem:J(H∗) = inf

F∈FJ(H), (3.2)

where J(H) is the cost functional defined by (3.1). Evidently, if J(H∗) = 0, then the quasi-solution H∗ ∈ H is also a strictsolution of ISPH.

Let us assume now that H,H + 1H ∈ H . Consider the first variation 1J(H) := J(H + 1H) − J(H) of the cost functionalJ(H). We have:

1J(H) = 2 Tf

0[u(0, t;H) − h(t)]1u(0, t; 1H)dt +

Tf

0[1u(0, t; 1H)]2dt, (3.3)

where 1u(x, t; 1H) = u(x, t;H + 1H) − u(x, t;H) is the solution of the following parabolic problem1ut = (k(x)1ux)x + F(x) 1H(t), (x, t) ∈ ΩT ,1u(x, 0) = 0, x ∈ (0, l),1ux(0, t) = 0, 1u(l, t) = 0, t ∈ (0, Tf ].

(3.4)

Multiplying both sides of Eq. (3.4) by the arbitrary function ϕ(x, t) we get: Tf

0

l

0[1ut(x, t;H) − (k(x)1ux(x, t; 1H))x]ϕ(x, t)dxdt =

Tf

0

l

0F(x)1H(t)ϕ(x, t)dxdt. (3.5)

Applying the integration by parts formula to the left side of (3.5), we obtain: Tf

0

l

0[1ut(x, t;H) − (k(x)1ux(x, t; 1H))x]ϕ(x, t)dxdt

= −

Tf

0

l

0[ϕt(x, t; F) + (k(x)ϕx(x, t; F))x]1u(x, t; 1H)dxdt

+

l

0[1u(x, t; 1H)ϕ(x, t)]

t=Tft=0 dx −

Tf

0[k(x)1ux(x, t; 1H)ϕ(x, t) − k(x)1u(x, t; 1H)ϕx(x, t)]x=l

x=0 dt. (3.6)

46 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

Now we require that the function ϕ(x, t) is the solution of the following backward problem:ϕt = −(k(x)ϕx)x, (x, t) ∈ ΩT ,ϕ(x, Tf ; F) = 0, x ∈ (0, l),−k(0)ϕx(0, t) = 2[u(0, t;H) − h(t)], ϕ(l, t) = 0, t ∈ (0, Tf ].

(3.7)

The first right hand side integral in (3.6) is equal to zero due to the identityϕt +(k(x)ϕx)x = 0. Further, the second right handside integral is also zero, due to the homogeneous initial (1u(x, 0; 1H) = 0) and final (ϕ(x, Tf ) = 0) conditions. Finally,taking into account the boundary conditions in (3.4) and (3.7) in the last right hand side integral of (3.6) we conclude: Tf

0

l

0[1ut(x, t;H) − (k(x)1ux(x, t; 1H))x]ϕ(x, t)dxdt = 2

Tf

0[u(0, t;H) − h(t)]1u(0, t; 1H)dt.

This, with (3.5), implies:

2 Tf

0[u(0, t;H) − h(t)]1u(0, t; 1H)dt =

Tf

0

l

0ϕ(x, t;H)F(x)1H(t)dxdt, ∀H ∈ H .

By (3.3) we obtain the following formula for the first variation of the cost functional J(H):

1J(H) =

Tf

0

l

0ϕ(x, t; F)F(x)dx

1H(t)dt +

Tf

0[1u(0, t; 1H)]2dt, (3.8)

for all 1H ∈ H . By the definition J(H + 1H) − J(H) = (J ′(H), 1H)H0(0,l) + o∥1H∥

2H0(0,l)

of the Fréchet differential, we

need to prove that the last integral term in (3.8) is of order o∥1H∥

2H0(0,l)

.

Lemma 3.1. Let H ⊂ H0(0, Tf ) and conditions (3.1) hold. Then for the solution 1u := 1u(x, t; 1H) ∈ V 1,0(ΩTf ) of theparabolic problem (3.4), corresponding to a given source term 1H ∈ H , the following estimates hold:

∥1u∥2H0(ΩTf )

≤ε

2σε

∥F∥2H0(0,l)∥1H∥

2H0(0,Tf )

, ∥1ux∥2H0(ΩTf )

≤ε

l2σε

∥F∥2H0(0,l)∥1H∥

2H0(0,Tf )

, (3.9)

where σε = 2k∗/l2 − 1/(2ε), ε > l2/(4k∗).

Proof. Multiplying both sides of the parabolic Eq. (3.4) by 1u, integrating on ΩTf and using the initial and boundaryconditions we obtain the following energy identity

ΩTf

k(x)[1ux]2dxdt +

12

l

0[1u(x, Tf ; 1H)]2dx =

ΩTf

F(x)1H(t)1udxdt. (3.10)

Using here the Cauchy ε-inequality αβ ≤ εα2/2 + β2/(2ε), ∀α, β ∈ R, ∀ε > 0, we conclude:

k∗∥1ux∥2H0(ΩTf )

≤ε

2∥F∥

2H0(0,l)∥1H∥

2H0(0,Tf )

+12ε

∥1u∥2H0(ΩTf )

.

To obtain estimates (3.9) one needs to use the Poincaré inequality ∥1ux∥2H0(ΩTf )

≥ (2/l2)∥1u∥2H0(ΩTf )

on the left, and then

on the right hand sides of the above inequality.

Corollary 3.1. Let the conditions of Lemma 3.1 be satisfied. Then the following estimate holds:

∥1u(0, ·; 1H)∥2H0[0,Tf ]

:=

Tf

0[1u(0, t; 1H)]2dt ≤ γε∥F∥

2H0(0,l)∥1H∥

2H0(0,Tf )

, γε =ε

lσε

> 0. (3.11)

The proof follows from the inequality ∥1u(0, ·;H)∥2H0[0,Tf ]

≤ l∥1ux∥2H0(ΩTf )

and estimate (3.9).

This corollary with (3.8) implies the following result.

Theorem 3.1. Let the conditions of Lemma 3.1 be satisfied. Then the cost functional J(H) defined by (1.3) and corresponding toISPH, is Fréchet-differentiable: J(H) ∈ C1(H). Moreover, the Fréchet derivative at H ∈ H of the cost functional J(H) can bedefined via the solution ϕ(x, t;H) ∈ V 1,0(ΩTf ) of the well-posed adjoint problem (3.7) and the given spacewise source functionF(x) ∈ F as follows:

J ′(H)(t) =

l

0ϕ(x, t;H)F(x)dx, t ∈ (0, Tf ]. (3.12)

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 47

Thus the gradient J ′(H) of the cost functional J(H) can be determined via the solution ϕ(x, t;H) ∈ V 1,0(ΩTf ) of the well-posed adjoint problem (3.7), which contains the measured output data h(t). This result, with the gradient formula (3.12),suggests a use of gradient type iterative methods for the numerical solution of ISPH.

On the other hand, it is well-known that any gradient method for the minimization problem requires an estimation ofthe iteration parameter αn > 0 in the iteration scheme H(n+1)

= H(n)− αnJ ′(H(n)), n = 0, 1, 2, . . ., where H(0)

∈ H is thegiven initial iteration. In the case of Lipschitz continuity of the gradient J ′(H) the parameter αn > 0 can be estimated via theLipschitz constant L1 > 0, i.e. 0 < δ0 ≤ αn ≤ 2/(L1 + 2δ1), where δ0, δ1 > 0 are arbitrary parameters. The following resultshows the continuity of the gradient J ′(H).

Lemma 3.2. Let the conditions of Lemma 3.1 hold. Then the functional J(H) is of Hölder class C1,1(H), and for all H,H + 1H ∈ H ,

∥J ′(H + 1H) − J ′(H)∥H0(0,Tf ) ≤ LF∥1H∥H0(0,Tf ), LF = L1∥F∥2H0[0,l], (3.13)

where LF > 0 is the Lipschitz constant, depending on the H0-norm of the spacewise source function F(x), L1 = (ε1γε/(2σε1))1/2l

> 0, σε1 = k∗ − l/ε1 > 0, the parameter γε > 0 is defined in Corollary 3.1, and the arbitrary parameter ε1 > 0 satisfies thecondition: ε1 > l/k∗.

Proof. By the gradient formula (3.12) we have:

∥J ′(H + 1H) − J ′(H)∥2H0(0,Tf )

=

Tf

0

l

01ϕ(x, t; 1H)F(x)dx

2

dt, (3.14)

where the function 1ϕ(x, t; 1H) := ϕ(x, t;H + 1H) − ϕ(x, t;H) ∈ V 1,0(ΩTf ) is the solution of the following backwardparabolic problem

1ϕt = −(k(x)1ϕx)x, (x, t) ∈ ΩTf ,

1ϕ(x, Tf ) = 0, x ∈ (0, l),−k(0)1ϕx(0, t) = 21u(0, t; 1H), 1ϕ(l, t) = 0, t ∈ (0, Tf ],

(3.15)

due to (3.7). Multiplying both sides of the parabolic Eq. (3.15) by 1ϕ(x, t; 1H), integrating on ΩTf and using then the initialand boundary conditions we obtain the following energy identity:

ΩTf

k(x)[1ϕx(x, t; 1H)]2dxdt +12

l

0[1ϕ(x, 0; 1H)]2dx = 2

Tf

01u(0, t; 1H)1ϕ(0, t; 1H)dt.

Applying to the right hand side integral the Cauchy ε-inequality we conclude:

k∗

ΩTf

[1ϕx(x, t; 1H)]2dxdt ≤ ε1

Tf

0[1u(0, t; 1H)]2dt +

1ε1

Tf

0[1ϕ(0, t; 1H)]2dt.

Using here Corollary 3.1 and the inequality ∥1ϕ(0, ·; 1H)∥2H0[0,Tf ]

≤ l∥1ϕx∥2H0(ΩTf )

we have:

σε1∥1ϕx∥2H0(ΩTf )

≤ ε1γε∥F∥2H0(0,l)∥H∥

2H0(0,Tf )

, σε1 = k∗ − l/ε1 > 0. (3.16)

On the other hand, applying to the right hand side of (3.14) first the Hölder inequality, and then the Poincaré inequality weconclude

∥J ′1(F + 1H) − J ′1(F)∥2H0(0,l) ≤

l2

2

l

0

Tf

0(1ϕx(x, t; 1H))2dxdt

l

0F 2(x)dx

.

This estimate with (3.16) implies the proof.

Formula (3.12) shows that the Fréchet gradient J ′(H)(t) = (ϕ(·, t, ;H), F)H0[0,l] of the cost functional J(H) is theH0-innerproduct of the solution of the corresponding well posed adjoint problem (3.7), and the given spacewise source term F(x).

The scheme of the algorithm of CGM, based on explicit use of the gradient formula (3.12) and defined to be the algorithmCGA, is given in Block Diagram 1.

48 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

Block Diagram 1. The scheme of the CG algorithm for ISPF.

The optimal value εoptJ > 0 of the stopping parameter εJ > 0 in the stopping condition Jh(H(n)) < ε2

J will be defined basedon an analysis of the behavior convergence error e(n; ·; γ ), depending on the iteration number n. Details will be explainedin Section 5.

4. Numerical solution of the direct problem: generation of synthetic noise free and noisy output data

Any numerical method for an inverse problem related to differential problems requires, first of all, construction of anoptimal computational mesh for the corresponding direct problem. This mesh needs to be fine enough in order to obtain anaccurate numerical solution of the direct problem, on the one hand. On the other hand, an implementation of any iterativemethod for inverse problems requires solving the forward problem efficiently in each iteration step. Hence minimummeshsize restrictions need to be taken into account in numerical solution of the direct problem.

For the numerical solution of the direct problem (1.1) as well as the adjoint problem (3.7), the following monotone(conservative) finite-difference schemes are used:

yj+1i − yji

τ=

1h

ki+1/2

yj+1i+1 − yj+1

i

h− ki−1/2

yj+1i − yj+1

i−1

h

+ FiH j, i = 1,Nx − 1, j = 0,Nt − 1;

y0i = u0(xi), i = 0,Nx;

yj+10 − yj0

τ=

2h

k1/2

yj+11 − yj+1

0

h

+ F0H j, j = 0,Nt − 1

yjNx= 0, j = 0,Nt .

(4.1)

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 49

vji − v

j−1i

τ= −

1h

ki+1/2

vj−1i+1 − v

j−1i

h− ki−1/2

vj−1i − v

j−1i−1

h

, i = 1,Nx − 1, j = Nt , 1

vNti = 0, i = 0,Nx;

vj0 − v

j−10

τ= −

2h

k1/2

vj−11 − v

j−10

h+ qj−1

, j = 0,Nt − 1

vjNx

= 0, j = 0,Nt .

(4.2)

Here yji in (4.1), and vji in (4.2) are the approximate values of u(x,tj) and ϕ(xi, tj) at the mesh points (xi, tj), of the solutions

of the direct problem (1.1), and the adjoint problem (3.7), respectively. The left flux qj, at tj, is defined to be q(t) :=

2[u(0, t;H) − h(t)] in (4.2). The order of approximation of this scheme is known to be O(h2+ τ) on a piecewise uniform

meshωhτ := ωh×ωτ = (xi, tj) ∈ ΩT : xi = ih, i = 0,Nx, tj = jτ , j = 0,Nt, with space and timemesh steps h = l/Nx andτ = Tf /Nt [31]. The second-order accuracy with respect to the space mesh step h > 0 is achieved due to the conservativityof the above schemes.

Assuming u(x, t) and uh(xi, tj) ≡ yji the exact and numerical solutions of the direct problem (1.1) for given source termsF(x) and H(t) we define the parameter

εu = ∥u(0, ·) − uh(0, ·)∥L2h[0,Tf ],

to be the computational noise level for the problem ISPH. Here and below ∥ · ∥L2h is the discrete analog of the L2-norm ∥ · ∥L2 .To estimate the computational noise levels for the continuous and discontinuous conductivity cases we consider the

following examples. In the computational experiments here and below l = 1, Tf = 1.

Example 4.1. Estimation of the computational noise level for the continuous conductivity case. The function u(x, t) =

10 exp(−t)x2(1− x)3, x ∈ [0, l], t ∈ [0, Tf ], is the exact solution of the direct problem (1.1), with the thermal conductivityk(x) = 1 + x2, and the initial data u0(x) = 10x2(1 − x)3. The corresponding source functions are F(x) = 31x5 − 63x4 +

59x3 − 43x2 + 18x − 2,H(t) = 10 exp(−t). For continuous thermal conductivity k(x) = 1 + x2 the computationalnoise levels, corresponding to ISPH, obtained by the finite-difference scheme (4.1) and corresponding to various meshparameters, are given in the fourth column of the left Table 1. The minimal mesh size when these levels are of order 10−4, isNx × Nt = 151 × 51. For this reason, for the continuous thermal conductivity this mesh will be defined as an optimal mesh,and will be used in subsequent inversion algorithms.

Example 4.2. Estimation of the computational noise level for the discontinuous conductivity case. The same functionu(x, t) = 10 exp(−t)x2(1 − x)3, x ∈ [0, l], t ∈ [0, Tf ], from Example 4.1, with appropriately chosen source functionsF(x) and H(t) are assumed to be an exact solution of the direct problem (1.1), assuming that the discontinuous thermalconductivity k(x) is given by the piecewise constant function:

k(x) =

k1 x ∈ [0, ξ),k2, x ∈ (ξ , 1],

where k1 = 0.8, k2 = 1.0; ξ = 0.8. Numerical results are given in the fourth column of the right Table 1.

The above results show that for the continuous, as well as discontinuous thermal conductivities, less than 1%,computational noisewill be included in the synthetic output data h(tj) := uh(0, tj), when the optimalmeshwith the parametersNx × Nt = 151 × 51 will be employed for the numerical solution of the discrete direct problem (1.1), as well as the adjointproblem (3.7).

5. Performance analysis of CG algorithm: an optimal choice of the stopping parameter

In this section we will demonstrate a performance comparison of the CG algorithm, described in Block Diagram 1, withother versions of this algorithm, used in [26] for the inverse source problem. Thenwewill estimate an optimal value ε

optJ > 0

of the stopping parameter εJ > 0 in the stopping condition Jh(H(n)) < ε2J . This estimate will be based on an analysis of

the dependence on the iteration number n of the convergence error e(n; ·; γ ) and the accuracy error E(n; ·; γ ), defined asfollows [32]:

e(n;H; γ ) := ∥uh(0, ·;H(n)) − h(·)∥L2h[0,Tf ], E(n;H; γ ) := ∥H − H(n)∥L2h[0,Tf ].

Based on this analysis we will estimate an optimal number of iterations nopt in the CG algorithm, and then show that thisnumber plays a role of a regularization parameter.

50 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

Table 1Computational noise level εu for different mesh parameters.

Nx × Nt h τ εu

Continuous thermal conductivity k(x)

101 × 51 0.01 0.02 0.41 × 10−3

151 × 51 0.0067 0.02 0.65 × 10−4

201 × 51 0.005 0.02 0.16 × 10−3

201 × 101 0.005 0.01 0.39 × 10−4

401 × 101 0.0025 0.01 0.12 × 10−3

Discontinuous thermal conductivity k(x)

101 × 51 0.01 0.02 0.10 × 10−1

151 × 51 0.0067 0.02 0.97 × 10−2

201 × 51 0.005 0.02 0.96 × 10−2

201 × 201 0.005 0.005 0.10 × 10−1

401 × 101 0.0025 0.01 0.99 × 10−2

5.1. Comparison of two versions of the CG algorithm applied to ISPH

First of all, note that different versions of the CG algorithm has been implemented in various studies for an inverseproblem of determining the spacewise heat source F(x) in the heat Eq. (1.1) from the measured final data uT (x) := u(x, Tf ),i.e. for the problem defined as ISPF (see, for example, [12,24,26]). Thus, while in [14] the descent parameter α(n) of the CGalgorithm for ISPF is defined as a solution of the minimum problem

J1[F (n)− α(n)p(n)

] = minα>0

J1[F (n)− αp(n)

], (5.1)

in [26] this parameter is defined as

α(n)=

∥J1[F (n)]∥

2L2(0,l)

∥u(x, T ; p(n))∥2L2(0,l)

. (5.2)

Here J1[F ] := ∥uT (x)−u(x, T ; F)∥2L2(0,l)

is the cost functional corresponding to ISPF.Wewill compare both versions of the CGalgorithm applied to the ISPH considered here. In view of this inverse problem, the first definition of the descent parameterα(n), i.e. (5.1), corresponds to the above described algorithm, given in Block Diagram 1. For convenience, this version will bedefined here as CGA1. The second definition, i.e. formula (5.2) used in [14] for ISPF, corresponds to the formula

α(n)=

∥J[H(n)]∥

2L2(0,T )

∥u(0, t; p(n))∥2L2(0,T )

, (5.3)

for ISPH. This version of the CG algorithm will be defined below as CGA2.

Example 5.1. Reconstruction of a smooth time-dependent source by the algorithms GCA1 and CGA2.For the given smooth convex time-dependent source H(t) = (5/π) exp(−50(t − 0.5)2) (Fig. 1, solid line), the synthetic

noise free output data h(t) := uh(0, t;H) is generated from the numerical solution uh(x, t;H) of the direct problem (1.1).The thermal conductivity and the spacewise source term are assumed to be k(x) = 1+ x2 and F(x) = exp(−x). The randomnoisy output data hγ (t) := h(t) + γ rand(h)∥h∥∞, with the noise levels γ = 1% and γ = 5%, are then generated from thesynthetic noise free output data. The numerical results obtained by CGA1 and CGA2 are plotted in Fig. 1. The accuracy errors,corresponding to these noise levels, are E(n;H; γ ) = 0.14 and E(n;H; γ ) = 0.60 and these errors correspond to n = 20and n = 15 iterations, respectively. These results show that the reconstructions obtained by the algorithms CGA1 and CGA2are almost the same.

However, the algorithm CGA2 depends upon randomness, while the algorithm CGA1 does not depend on the choice of therandomness, as the second series of computational experiments related to the same random noisy output data hγ (t), withγ = 1%, show. The reconstructions obtained by the algorithms CGA1 and CGA2 are plotted in Fig. 2. The reconstructedsmooth time-dependent sources in the left Fig. 2 are obtained for the values εJ = e(n = 100;H; γ ) shows that the algorithmCGA2 may not converge in all randomness.

The right Fig. 2 shows that the algorithm CGA2 may not converge in all randomness, while reconstructions obtained bythe algorithmCGA1 and plotted in the left Figs. 1 and 2 are almost the same in all cases. The behaviors of the convergence andaccuracy errors of these algorithms depending on the iteration number n are shown in Fig. 3. The left Fig. 3, correspondingto the algorithm CGA1, illustrates the typical, for discrete ill-posed problems, L-curve behavior of the convergence errore(n;H; γ ), while the right Fig. 3 shows that the convergence error, corresponding to the algorithm CGA2, increases aftersome iterations.

Similar results are obtained for the random noisy output data hγ (t), with the noise level γ = 5%.

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 51

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

H(t

)

Noise Level 1%ExactCGA2CGA1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

H(t

)

Noise Level 5%

ExactCGA2CGA1

Fig. 1. Comparison of numerical results obtained by CGA1 and CGA2: reconstruction of smooth time-dependent source from random noisy output datahγ (t), with noise levels γ = 1% (left figure) and γ = 5% (right figure).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

H(t

)

ExactAppr. (n=100)

nopt = 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

t

H(t

)

Noise level 1%

ExactCGA2

3107

Fig. 2. An influence of randomness: reconstruction of smooth time-dependent source from random noisy output data hγ (t), with noise levels γ = 1% byCGA1 (left figure) and CGA2 (right figure).

0 50 100 150 200 250

Iteration Number

Err

ors

e()

and

E(

)

Noise level 1% (Alg.−1)

E ( )e( )

0 50 100 150 200 250

Iteration Number

Noise level 1% (Alg.−2)

E( )e( )

100

10-1

10-2

10-3

10-4

108

106

104

102

100

10-2

10-4

Err

ors

e()

and

E(

)

Fig. 3. The convergence error e(n;H; γ ) and accuracy error E(n;H; γ ) corresponding to CGA1 (left figure) and CGA2 (right figure); γ = 1%.

Further computational experiments also show that the algorithm CGA1 does not depend upon randomness. Fig. 4illustrates the reconstructed time-dependent source from the same level but three different randomness noisy data, whenthe noise levels are γ = 1% (left figure) and γ = 5% (right figure). The values of the convergence error and accuracy error areobtained: E(n;H; γ ) = 1.2÷1.4×10−1, e(n;H; γ ) = 0.2÷0.4×10−2, when γ = 1%, and E(n;H; γ ) = 3.5÷0.75×10−1,e(n;H; γ ) = 0.6 ÷ 0.7 × 10−2, when γ = 5%, in n = 20 and n = 15 iterations, respectively.

5.2. An optimal choice of the stopping parameter in the CG algorithm

Before use of the CG algorithm (subsequently, CGA), given by Block Diagram 1, it is important to understand the behaviorof the convergence error e(n;H; γ ) and the accuracy error E(n;H; γ ), depending on the iteration number n of CGA. Note

52 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

H(t

)

0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

tt0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

H(t

)

0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t0 0.5 1

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

NoiseNoise

Fig. 4. The reconstructed time-dependent source from the same level but different randomness noisy output data: γ = 1% (left figure) and γ = 5% (rightfigure).

0 10 20 30 40 50 60 70 80 90 100Iteration Number

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

t

H(t

)

ExactAppr. (n=100)

nopt= 15

100

10-1

10-2

10-3

10-4

Err

ors

e(n;

H;

) an

d E

(n;H

;)

E(n;H; )e(n;H; )

Fig. 5. Reconstructed time-dependent source from random noisy data (γ = 5%) in different values of the iteration number n (left figure); behavior of theconvergence and accuracy errors (right figure).

that the values of the parameter εu > 0 in the last column of Table 1 represent a computational noise. These values showthe lower bounds of the convergence error e(n;H; γ ) for noise free data. The values of the convergence error e(n;H; γ )are of order 10−1

÷ 10−3, depending on noise level, on the smoothness of the thermal conductivity k(x), and also on theclass of unknown source terms. When dealing with inverse source problems, the behavior of convergence error e(n;H; γ ),as a function of the iteration number n, consists of three phases: the initial phase of rapid decrease (convergence) but shortduration, the second phase of slow decrease, and the third phase of almost constant behavior, after some iterations (seethe left Fig. 3 and the right Fig. 5). As computational experiments show, in all types of considered time-dependent sourcefunctions, the minimum value of the accuracy error E(n;H; γ ) is achieved between the first and second phases of the curvee(n;H; γ ) versus n. Any value of the convergence error e(n;H; γ ) between the first and second phases is assumed to be anoptimal value ε

optJ > 0 of the stopping parameter in the CG algorithm. The corresponding number of iterations is defined to

be an optimal one: εoptJ := e(nopt

;H; γ ).In all cases, 10–25 iterations of CGA are sufficient to reach an approximate minimum value of the accuracy error

E(n;H; γ ), depending on the noise level γ > 0. Moreover, the number of iterations to reach the minimum accuracy errordecreases by increasing the parameter γ > 0. Thus, in Example 5.1, the minimum value E(n;H; γ ) ≈ 0.12 × 10−1 of theaccuracy error corresponds to the value e(n;H; γ ) ≈ 0.11 × 10−2 of the convergence error, when γ = 1% (the left Fig. 3),and the minimum value E(n;H; γ ) ≈ 0.49× 10−1 of the accuracy error corresponds to the value e(n;H; γ ) ≈ 0.71× 10−2

of the convergence error, when γ = 5% (the right Fig. 5). These minimum values of errors are achieved at nopt = 18 ÷ 19and nopt = 11 ÷ 12 iterations, respectively.

Note also that, in all classes of time-dependent source functions H(t), the behavior of the convergence error e(n;H; γ ),depending on the iteration number n of CGA, is almost an L-curve (see [33]), although there is no Tikhonov regularizationhere. This, in particular,means that the above defined optimal value nopt of the iteration number plays a role of regularizationparameter in the proposed CG algorithm. A similar situation arises in large-scale problems (see [34,35]) where, as a stoppingcriterion, the first iteration number k for which the condition ∥rk∥L2 ≤ βδ holds is chosen. Recall that for ISPF this momenthas also been observed in [26].

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 53

Table 2Determination of an optimal value ε

optJ > 0 of the stopping parameter in CGA:

reconstruction of a smooth spacewise source term.

Noise levels (%) n e(n;H; γ ) E(n;H; γ ) εoptJ

5 0.14×10−1 0.72×10−1

10 0.29×10−2 0.18×10−1

γ = 1 15 0.16×10−2 0.12×10−1 εoptJ = e(nopt

;H; γ )

25 0.10×10−2 0.30×10−1 nopt≈ 15 ÷ 20

50 0.50×10−3 0.46×10−1

100 0.30×10−3 0.63×10−1

5 0.18×10−1 0.67×10−1

10 0.78×10−2 0.54×10−1

γ = 5 15 0.62×10−2 0.75×10−1 εoptJ = e(nopt

;H; γ )

25 0.41×10−2 0.19 × 100 nopt≈ 11 ÷ 15

50 0.35×10−2 0.32 × 100

100 0.27×10−2 0.64 × 100

Table 2 clearly illustrates how to choose an optimal value εoptJ of the stopping parameter in CGA. As it is seen from the

third column of Table 2, when γ = 1%, the accuracy error increases by one order of magnitude, as the number of iterationsdecreases from n = 5 to n = 15 ÷ 20 (the first phase of the curve e(n;H; γ ) versus n). Then this order remains almostconstant until the number of iterations reaches n = 40÷ 45. This means that any value of the convergence error e(n;H; γ )

between the first and second phases of the curve e(n;H; γ ) versus n can be taken as an optimal value εoptJ > 0 of the

stopping parameter in the CGA algorithm. For the case when γ = 5% an optimal value of the stopping parameter can bedetermined similarly.

To illustrate the role of the optimal value εoptJ of the stopping parameter in CGA consider the following example.

Example 5.2. Reconstruction of a smooth time-dependent source by GCAwith optimal and arbitrary values of the stoppingparameter ε

optJ > 0. The following two values ε

optJ = 0.62 × 10−2 and εJ = 0.27 × 10−2 of the stopping parameter are

used for reconstruction of the smooth time-dependent source, from the noisy data hγ (t) := h(t) + γ rand(h)∥h∥∞, withthe noise level γ = 5%, generated from the synthetic output data, given in Example 5.1. The number of iterations of CGA,corresponding to these values of the stopping parameter, are nopt

= 15 and n = 100, respectively, as Table 2 shows. Thereconstructed time-dependent sources are plotted in the left Fig. 5 which clearly shows that deterioration occurs with anincreasing number of iterations. Thus, the corresponding values of accuracy error are E(nopt

= 15;H; γ ) = 0.54×10−1 andE(n = 100;H; γ ) = 0.64 × 100. However, although these values differ by one order of magnitude, even for the impropervalue εJ = 0.27×10−2 of the stopping parameter, the overall reconstruction is acceptable. Hence the proposed CG algorithmis reliable in the sense that small deviation from an optimal value ε

optJ > 0 of the stopping parameter has negligible effect

on the accuracy error.

6. Performance analysis of the CG algorithm applied to benchmark problems ISPH

The performance testing of the CG algorithm is studied on the following widely used classes of time-dependent sourceterms H(t):

– the class smooth time-dependent source functions of constant and variable oscillatory frequencies;– the class of discontinuous time-dependent sources.

Consider first the inverse problem of reconstructing an unknown time-dependent smooth time-dependent sourcefunction of constant oscillatory frequency.

Example 6.1. Reconstruction of the oscillating time-dependent source function by the CG algorithm. To generate thesynthetic noise free output data h(tj) := uh(0, tj;H) we use the time-dependent oscillating source, given by the function

H(t) = 3.1 +

2i=1

arctan(1 − 0.5((t − ti)/0.05)2), t1 = 0.35, t2 = 0.7, t ∈ [0, 1].

Here the variable thermal conductivity k(x) and spacewise source terms F(x) in the direct problem (1.1) are taken to bek(x) = 1 + x2, F(x) = 5 exp(−3x). The noise free output data h(t) := uh(0, t;H) is generated from the numerical solutionof the direct problem (1.1) (left Fig. 6, solid line). Here and below the function H(0)(t) ≡ 0, 0 ≤ t ≤ 1 is assumed to bean initial iteration. The CG algorithm is applied to ISPH for reconstruction of the time-dependent function H(t). Numericalresults corresponding to random noisy output data hγ (t) := h(t) + γ rand(h)∥h∥∞, with noise levels γ = 1% and γ = 5%are plotted in the left Fig. 6. These results show that the reconstructions are quite accurate. Behavior of the convergenceerror e(n;H; γ ) and accuracy error E(n;H; γ ), depending on the iteration number n, are given in the right Fig. 6. As in the

54 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5

t

H(t

)

Exact H(t)

0 20 40 60 80 100 120

Iteration Number

,nopt = 19,nopt = 11

10 0

10 -1

10 -2

10 -3

10 -4

Err

ors

e(n;

H;

) an

d E

(n;H

;)

E(n;H;γ), γ=1%e(n;H;γ), γ=1%E(n;H;γ), γ=5%e(n;H;γ), γ=5%

Fig. 6. Reconstruction of constant oscillatory frequency source from randomnoisy output data (left figure); behavior of the errors E(n;H; γ ) and e(n;H; γ )

(right figure).

Table 3Performance characteristics of the CG algorithm for oscillat-ing time-dependent source terms.

Noise levels (%) e(nopt;H; γ ) nopt E(nopt

;H; γ )

γ = 1 3.00×10−3 19 4.14 × 10−2

γ = 5 1.40×10−2 11 1.03 × 10−1

Table 4Performance characteristics of the CG algorithm for variableoscillatory frequency source.

Noise levels (%) e(nopt;H; γ ) nopt E(nopt

;H; γ )

γ = 1 1.30×10−3 24 2.68 × 10−2

γ = 5 6.20×10−3 18 1.01 × 10−1

previous cases, minimum values of the accuracy error E(n;H; γ ) are achieved between the first and second phases of thecurve e(n;H; γ ) versus n. Any value of the convergence error e(n;H; γ ) between the first and second phases is assumed tobe an optimal value ε

optJ > 0 of the stopping parameter in the CG algorithm. Specifically, for the noise level γ = 1%, any

value of the convergence error e(n;H; γ ) from the interval 1.00÷5.00×10−3 can be assumed to be an optimal value of thestopping parameter ε

optJ > 0, since the reconstructed source functions corresponding to these values are almost the same.

For the noise levels γ = 1% and γ = 5%, the reconstructed source functions in the left Fig. 6 are obtained in the optimalvalues ε

optJ = 3.00 × 10−3 and ε

optJ = 1.40 × 10−2 of the stopping parameter, correspondingly.

Performance characteristics of the CG algorithm applied to ISPH in the class of oscillating time-dependent source termsare given in Table 3. The results presented here show that for random noisy output data, with noise levels γ = 1% andγ = 5%, reconstructions are quite accurate (the last column of Table 3). The second and third columns of this table showoptimal values of the stopping parameter εJ > 0 and the corresponding iteration numbers.

Example 6.2. Reconstruction of the smooth time-dependent variable oscillatory frequency source. In this example thesynthetic noise free output data h(tj) := uh(0, tj;H) is generated by using the variable oscillatory frequency oscillating time-dependent sourceH(t) = 5t sin(7π t) (left Fig. 7, solid line). Other inputs are taken to be as k(x) = 1+x2, F(x) = 5 exp(−3x).The reconstructed sources from noisy output data hγ

:= h(t) + γ rand(h)∥h∥∞, with relative noise levels γ = 1% andγ = 5% and different randomness, are plotted in the left Fig. 7. These reconstructions correspond to the optimal valuesεoptJ = 1.30×10−3 and ε

optJ = 6.20×10−3 of the stopping parameter. The right Fig. 7 shows the behavior of the convergence

error e(n;H; γ ) and accuracy error E(n;H; γ ), depending on the iteration number n. Optimal values of these errors, as wellas the iteration number nopt, are given in Table 4.

Example 6.3. Reconstruction of the high frequency oscillatory smooth time-dependent source. The synthetic noise freeoutput data h(tj) := uh(0, tj;H) is generated by using the variable oscillatory frequency oscillating time-dependent sourceH(t) = t sin(10π t) − 5t2 sin(15π t) (left Fig. 8, solid line), with the same k(x) = 1 + x2 and F(x) = 5 exp(−3x) as in theprevious example. The random noisy output data hγ

:= h(t) + γ rand(h)∥h∥∞ are used with noise levels γ = 1% andγ = 5%. The reconstructed sources are plotted in the left Fig. 8.

The reconstructed sources from the noisy output data, with relative noise levels γ = 1% and γ = 5% and differentrandomness, are plotted in the left Fig. 8. These reconstructions correspond to the optimal values ε

optJ = 0.70 × 10−3 and

εoptJ = 2.78 × 10−3 of the stopping parameter. The right Fig. 8 shows the behavior of the convergence error e(n;H; γ ) and

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−3

−2

−1

0

1

2

3

4

5

t

H(t

)

ExactNoisy 1 (1%)Noisy 2 (5%)

0 10 20 30 40 50 60 70 80 90 100

Iteration Number

Err

ors

e(n;

H;

) an

d E

(n;H

;)

101

100

10-1

10-2

10-3

10-4

E(n;H;γ), γ=1%e(n;H;γ), γ=1%E(n;H;γ), γ=5%e(n;H;γ), γ=5%

Fig. 7. Reconstruction of variable frequency oscillatory source from randomnoisy output data (left figure); behavior of the errors E(n;H; γ ) and e(n;H; γ )

(right figure).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−4

−2

0

2

4

6

t

H(t

)

ExactNoisy 1 (1%)Noisy 2 (5%)

0 10 20 30 40 50 60 70 80 90 100Iteration Number

Err

ors

e(n;

H;

) an

d E

(n;H

;)

101

100

10-1

10-2

10-3

10-4

E(n;H;γ), γ=1%e(n;H;γ), γ=1%E(n;H;γ), γ=5%e(n;H;γ), γ=5%

Fig. 8. Reconstruction of high frequency oscillatory source from random noisy output data (left figure); behavior of the errors E(n;H; γ ) and e(n;H; γ )

(right figure).

Table 5Performance characteristics of the CG algorithm for highoscillatory frequency source.

Noise levels (%) e(nopt;H; γ ) nopt E(nopt

;H; γ )

γ = 1 0.70 × 10−3 38 1.82 × 10−2

γ = 5 2.78 × 10−3 28 7.45 × 10−2

accuracy error E(n;H; γ ), depending on the iteration number n. Optimal values of the convergence and accuracy errors, aswell as the corresponding iteration number nopt, are given in Table 5.

Finally consider the important case when an unknown time-dependent source is a discontinuous function. This case canbe further divided into two subcases:

– the thermal conductivity k(x) is a continuous spacewise function;– the thermal conductivity k(x) is a discontinuous spacewise function.

The first case corresponds to a non-homogeneous heated bar. The second case, which is the most difficult one, arises incomposite materials.

Example 6.4. Reconstruction of a discontinuous time-dependent source (homogeneous rod). Let F(x) = 5 exp(−3x), u0(x)= 0 in the direct problem (1.1). Assume that the thermal diffusivity is given by the following continuous function:k(x) = 1+ x2. The synthetic output data h(tj) := uh(0, tj;H) is generated from the numerical solution of the direct problem(1.1), assuming

H(t) =

2, 0 ≤ t < 0.25;4, 0.25 ≤ t < 0.5;2 + 2 sin(5.5π t), 0.5 ≤ t ≤ 1,

(6.1)

the discontinuous time-dependent source in the heat conduction problem (1.1) (left Fig. 9, solid line). Then the CG algorithmis applied for reconstruction of the discontinuous source function H(t) from the random noisy output data hγ (t), with noise

56 A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t

H(t

)

Exact H(t)

0 10 20 30 40 50 60 70 80 90 100 110Iteration Number

101

100

10-1

10-2

10-3

10-4

Err

ors

e(n;

H;

) an

d E

(n;H

;)

nopt =25

nopt =12

E(n;H;γ), γ=1%e(n;H;γ), γ=1%E(n;H;γ), γ=5%e(n;H;γ), γ=5%

Fig. 9. Reconstruction of discontinuous time-dependent from random noisy output data (left figure); behavior of the errors E(n;H; γ ) and e(n;H; γ )

(right figure): homogeneous rod.

Exact H(t)nopt =25

nopt =8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

H(t

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1t

101

100

10-1

10-2

10-3

Err

ors

e(n;

H;

) an

d E

(n;H

;)

0 20 40 60 80 100 120

Iteration Number

E(n;H;γ), γ=1%e(n;H ;γ), γ=1%E(n;H;γ), γ=5%e(n;H ;γ), γ=5%

Fig. 10. Reconstruction of discontinuous time-dependent from random noisy output data (left figure); behavior of the errors E(n;H; γ ) and e(n;H; γ )

(right figure): non-homogeneous rod.

Table 6Performance characteristics of the CG algorithm for discontinuous source.

k(x) Noise levels (%) e(nopt;H; γ ) nopt E(nopt

;H; γ )

Continuous γ = 1 4.92 × 10−3 25 1.07 × 10−1

γ = 5 2.50 × 10−2 12 2.61 × 10−1

Discontinuous γ = 1 4.72 × 10−3 25 1.23 × 10−2

γ = 5 4.15 × 10−2 8 2.88 × 10−1

levels γ = 1% and γ = 5%. The reconstructed sources are plotted in the left Fig. 9. It is seen that the reconstruction qualityis high enough in both cases, taking into account discontinuity of the reconstructed function. The right Fig. 9 shows thebehavior of the convergence error e(n;H; γ ) and accuracy error E(n;H; γ ), depending on the iteration number n.

Example 6.5. Reconstruction of a discontinuous time-dependent source for a heated composite bar with discontinuousthermal conductivity k(x). In this computational experiment the thermal conductivity of a heated bar is assumed to begoverned by the discontinuous (piecewise constant) function

k(x) =

k1 x ∈ [0, ξ),k2, x ∈ (ξ , 1],

with k1 = 0.8, k2 = 1.0; ξ = 0.8. The discontinuous time-dependent source H(t) is assumed to be given by (6.1), andall other data are taken from Example 6.4. The synthetic noise free h(tj) := uh(0, tj;H) and the random noisy outputdata hγ (t) := h(t) + γ rand(h)∥h∥∞ are generated from the numerical solution of problem (1.1) with these data. Thereconstructed sources from the random noisy output data hγ (t), with noise levels γ = 1% and γ = 5%, are presented inthe left Fig. 10. The right Fig. 10 shows the behavior of the convergence error and accuracy error, depending on the iterationnumber n.

Performance characteristics of the CG algorithm applied to ISPH in the class of discontinuous time-dependent sourceterms are given in Table 6. These results show that even in the case of discontinuous thermal conductivity reconstructionquality is high enough.

A. Hasanov, B. Pektaş / Computers and Mathematics with Applications 65 (2013) 42–57 57

7. Conclusions

This paper presents a systematic study of the inverse problems of identifying the time-dependent heat sourceH(t) of thevariable coefficient heat conduction equation ut = (k(x)ux)x + F(x)H(t), from supplementary temperature measurementh(t) := u(0, t), t ∈ (0, T ], at one end of a non-homogeneous rod. The explicit gradient formula for the correspondingcost functional is derived. Then the Conjugate Gradient Algorithm based on these explicit gradient formulas is proposed forthe time-dependent source identification problem. The results presented for most used classes of time-dependent sourcesshow that the proposed CG algorithm, based on the explicit gradient formula, is a very fast and effective reconstructionalgorithm for ISPH, if the optimal value ε

optJ > 0 of the stopping parameter is properly chosen. Numerical experiments also

show that the CG algorithm can be used in the variable as well as discontinuous thermal conductivity k(x) > 0 cases. Allreconstructions are high enough and typically reached in 10–30 iterations, when the random noise levels are of 1% ÷ 5%.

Acknowledgments

The research has been supported by the Izmir University Research Fund, and also by the Scientific and TechnologicalResearch Council of Turkey (TUBITAK). The authors gratefully thank the anonymous referees for valuable suggestions whichimproved the manuscript.

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