Simultaneous identification and reconstruction of the space-dependent reaction coefficient and source term

The inverse problem of simultaneously determining, i.e., identifying and reconstructing, the spacedependent reaction coefficient and source term component from time-integral temperature measurements is investigated. This corresponds to thermal applications in which the heat is generated from a source depending linearly on the temperature, but with unknown space-dependent coefficients. For the resulting nonlinear inverse problem, we first prove the existence of solution based on the Schauder fixed point theorem. Then, under certain additional conditions, the solution is also proved to be unique. For the numerical reconstruction of solution, the problem is reformulated as a least-squares minimisation whose Fréchet gradients with respect to the two unknowns are derived in terms of the solution of an adjoint problem. The conjugate gradient method (CGM) to calculate the numerical solution is developed, and its convergence is proved from the Lipschitz continuity of these gradients. Three numerical examples for oneand two-dimensional inverse problems are illustrated to reveal the accuracy and stability of the solutions applying the CGM regularised by the discrepancy principle when noisy data are inverted.


Introduction
In the mathematical modelling and simulation of complex phenomena, it is often the case that there are more than one physical property of the system that are unknown and they have to be determined (or estimated) from extra measurements of other available physical quantities. Nevertheless, such multiple coefficient inverse problems are more difficult to solve than the single ones due to the extra system-coupling and nonlinearity that arise. In addition, the degree of ill-posedness is expected to increase. When multiple coefficients appearing in the governing parabolic heat transfer equation are unknown, but they depend on the time variable only, their simultaneous identification in terms of local existence and uniqueness was initiated by Professor M. Ivanchov and his co-workers, e.g., [18], followed by numerical implementations realised in [16]. The key starting point in these works is the employment of Green's functions for the leading dominant part ∂ t − a(t)∇ 2 (where a(t) > 0 is the diffusivity) of the governing parabolic heat equation. However, this approach is not extendable to situations when the unknown coefficients depend on the space variables, in which case the non-uniqueness of recovering multiple spacewise dependent properties becomes a main difficulty to overcome, [2,9,14]. In [19,37], the initial temperature and the space-dependent reaction coefficient or the heat source, respectively, were identified and reconstructed from various temperature measurements. Also, very recently, the authors have investigated the simultaneous numerical reconstruction of the spacewise dependent reaction coefficient and heat source, (as well as the initial temperature), from temperature measurements at three distinct time instants [5]. Encouraged by this preliminary numerical study, in this paper, we investigate in more detail the existence and uniqueness of solution, as well as consider the more general (and practical) case of time-integral average temperature measurements.
Thus, we consider the heat transfer process in a bounded domain Ω ⊂ ℝ N , N ≥ 1, with sufficiently smooth boundary ∂Ω, over the time interval from the initial time 0 to T > 0, governed by where u represents the temperature, q is the spatially-dependent reaction coefficient, f , h and g are heat source components, μ is the Dirichlet boundary data and u 0 is the initial temperature. For simplicity, in (1.1), the heat capacity and thermal conductivity were assumed to be constant and taken to be unity. When all the thermal properties and the initial and boundary conditions are specified, the well-posedness of the solution u to the initial-boundary value direct problem (1.1) is classical, see, e.g., [13]. However, in many engineering situations, the thermal coefficients as well as initial and/or boundary data cannot be measured directly. Thus, a wide range of inverse problems have been concerned to determine the physical properties and heat transfer coefficients, boundary and/or initial conditions, [27]. For instance, in the heat transfer in biological tissue, problem (1.1) becomes the well-known Pennes' bio-heat model [29], which is obtained by a balance between the accumulation of energy due to the blood flowing through capillary network and the heat generation due to cell metabolism. The reaction coefficient q, which is also known as the perfusion coefficient, plays an important role in the heat generation in carcinogenic skin and tumours due to the increased nutrition and oxygen demand, [28], and helps to understand the heat transfer through such biological tissues.
The nonlinear inverse problem of determining the reaction coefficient q(x) has been extensively studied, when all the source components f , g and h, the Dirichlet boundary data μ and the initial temperature u 0 are known. Its existence and uniqueness have been established from certain measurements, including the temperature at the terminal time t = T and the time-average integral of temperature, [17,23,26,30,32,33]. Besides, there are several numerical techniques that have been developed to numerically reconstruct q(x) from the above additional measurements, such as the Tikhonov regularization method [8], the Armijo algorithm combined with the finite element method (FEM) [6], the NAG routine E04FCF together with the finite-difference method (FDM) [36], and the CGM [3].
If the source component f(x) is unknown, and the reaction coefficient q and the input data h, g, μ and u 0 are given, the linear inverse problem of identifying it from final time or time-integral temperature measurements has been extensively considered, e.g., [11,20,21,31]. Such inverse source problems arise in various physical and engineering applications, e.g., the identification of sources of water and air pollution in the environment, or the heat sources in thermal processes.
In this work, we assume that the heat source components h(x, t) and g(x, t), the boundary temperature μ(x, t) and the initial temperature u 0 (x) are known, but the reaction coefficient q(x) and the source component f(x) are both unknown and have to be determined from some additional information. Previously, such additional information was taken to be the measurement of u and its time derivative u t at the fixed instant T 0 ∈ (0, T], namely, see [24], or the measurement of u at two different fixed times t = T 1 ∈ (0, T) and t = T, namely, see [25]. In this paper, we generalise the fixed time (instant) observations (1.2), which in practice may be difficult to be realised accurately, to time-average recordings, which smooth out the otherwise possibly large measurement errors in u. More precisely, we investigate the inverse problem of determining the triplet (u(x, t), q(x), f(x)) satisfying problem (1.1) together with the time-integral temperature measurements, where ω 1 (t) and ω 2 (t) are two given weight functions, and ϕ 1 (x) and ϕ 2 (x) are given time-averaged measured data. If one weight function is a close approximation to the Dirac delta function  4) is important to extract useful information for the recovery of the two unknown quantities, see [4] for more details. Efficient schemes of reconstructing two quantities simultaneously have already been established, e.g., the space-dependent reaction coefficient and the initial temperature were simultaneously identified from the final time observation of temperature in [37] and from the time-integral temperature measurement in [4]. Such methods can also be applied to simultaneously determine the reaction coefficient q(x) and the source component f(x) in the inverse problem (1.1), (1.3) and (1.4).
The paper is organised as follows. The approach of dealing with the existence and uniqueness of the nonlinear inverse problem (1.1), (1.3) and (1.4) is to transform it into a nonlinear non-classical direct problem, as described in Section 2. For the numerical solution, the least-squares objective functional is minimised to obtain the quasi-solution of the unknown reaction coefficient q(x) and source component f(x), as described in Section 3. In Section 4, the Fréchet differentiability of this objective functional is established, and the Fréchet gradients are derived and proved to be Lipschitz continuous. The CGM is established based on these gradients and the adjoint problem to reconstruct the unknown coefficients simultaneously. The convergence of the CGM with the Fletcher-Reeves formula [12] is considered based on the arguments in [38]. Since the inverse problem considered in this paper is unstable, the CGM regularised by the discrepancy principle [1] is utilised to obtain a stable numerical solution to the inverse problem. Three numerical examples of one-and two-dimensional inverse problems are presented and discussed in Section 5. Finally, Section 6 highlights the conclusions of the paper.

Unique solvability of the inverse problem
In this section, we shall prove that there exists a unique solution (q(x), f(x), u(x, t)) to the inverse problem (1.1), (1.3) and (1.4), when the source components h(x, t) and g(x, t), the Dirichlet boundary data μ(x, t) and the initial temperature u 0 (x) are specified. In the following mathematical analysis, we will use the standard Sobolev spaces H 1 (Ω), H 1 0 (Ω), H 2 (Ω) and H 2,1 (Q), which have been defined in many textbooks, e.g., [35]. In order to obtain the existence and uniqueness of the solution to the inverse problem (1.1), (1.3) and (1.4), we introduce the following assumptions: (a) the input data h, g ∈ L ∞ (Q), μ ∈ L ∞ (S) and and ω 1 , ω 2 ∈ C 1 [0, T] satisfy the compatibility conditions in Ω, for some positive constants M 1 and M 2 , where We also define the Banach space We now state the well-posedness for the direct initial-boundary value problem (1.1) when all the coefficients and input data are specified, [26], and the Schauder fixed point theorem, which is employed to establish the existence of solution to the inverse problem. Lemma 2.1. Let assumptions (a) and (b) hold, and suppose that 0 < q − ≤ q ≤ q + < ∞ in Ω for some prescribed constants q − and q + , q ∈ L ∞ (Ω) and f ∈ L ∞ (Ω). Then the initial-boundary value direct problem (1.1) satisfying the compatibility condition u 0 (x) = μ(x, 0) for all x ∈ ∂Ω, has a unique solution u ∈ V, which satisfies the following estimate: By the notations in (2.1) and using (2.4), we have where Substituting (2.5) and (2.6) into (1.1), we obtain the nonlinear non-classical parabolic problem given by (2.8) Thus, the inverse problem (1.1), (1.3) and (1.4) of determining the unknown quantities q(x) and f(x) simultaneously is equivalent to obtaining the solution u(x, t) to the nonlinear parabolic problem (2.8).
We utilise the methodology applied in [26], see also [24,25], and consider the following two auxiliary parabolic problems: and and c( ⋅ ) is a Lipschitz continuous function on ℝ defined by and M 0 is some fixed number in the interval (0, M 1 ). Remark that from (2.11) we obtain whose form, in relation to (2.7), suggests to prove that u = U + W solves problem (2.8). Proof. Taking the linear initial-boundary value problem (2.9) into consideration and using Lemma 2.1, there exists a unique solution U(x, t) ∈ V, satisfying the following estimate: (2.14) Consider the following linear initial-boundary value problem: where the functions v 1 and v 2 are defined by the same way as for problem (2.10), namely, and v(x, t) is an arbitrary function belonging to the space V 0 . Multiplying the first equation in (2.15) by W, integrating the result over the space-time domain Q and using the definition of function c( ⋅ ) in (2.12), we have where |Q| = T × diam(Ω) is the size of the solution domain Q, and we have used the inequality αβ ≤ 1 4ϵ α 2 + ϵβ 2 for ϵ = 1 4 (M 1 − M 0 ), twice to estimate the third and fourth-terms on the right-hand side of the first equation in (2.15). Thus, we obtain where C 1 is a positive constant depending on M 0 , M 1 , M 2 , N 1 and |Q|. On dividing inequality (2.17) by min{M 1 − M 0 , 2} > 0 and re-denoting the resulting constant still by C 1 then, we get the estimate From the homogeneous Dirichlet boundary condition and initial condition on W in (2.15) we have Using assumption (e), equation (2.12), and the homogeneous boundary and initial conditions in (2.15), we have and where we have used that for any two positive numbers α and β we have that αβ ≤ 2α 2 + β 2 8 . Then, multiplying the first equation in (2.15) by (W τ − ∇ 2 W) and integrating over Ω × (0, t), along with using (2.18)-(2.21), and that M 0 < M 1 , we get Using inequality (2.3) in the Lemma 2.1 applied to the solution W of problem (2.15), equations (2.14), (2.15) and the assumption (e), we obtain Using (2.2), estimates (2.18), (2.23)-(2.25) imply that the solution W(x, t) of the initial-boundary value problem (2.15) belongs to the space V 0 , and satisfies the following estimate: and C 0 is a positive constant depending on M 0 , M 1 , M 2 , N 1 and |Q|. Thus, problem (2.15) generates an operator F from V 0 to itself and Then X is a bounded and closed convex subset of V 0 . Since the solution W to problem (2.15) satisfies the estimate (2.26), the operator F takes the set X into itself. It remains to show F : X → X is a continuous and compact operator. Let {v n (x, t)} be a sequence of functions from the set X. Then according to (2.16) Applying the same technique used to derive (2.17) for the function W satisfying problem (2.15), to the function w n satisfying problem (2.27), we obtain It then yields that Invoking the approaches utilised in deriving (2.18)-(2.24), we have Multiplying the first equation in (2.27) by ((w n ) τ − ∇ 2 w n ) and integrating over Ω × (0, t), along with using (2.29)-(2.32), we obtain By the definition (2.12) of the function c(ξ), we observe that Then, as n → ∞, it follows that ‖s n ‖ L ∞ (Ω) → 0, as n → ∞. Thus, from (2.28), (2.33) and (2.34), we obtain that, as n → ∞, The above results together with (2.2) imply that ‖w n ‖ V 0 → 0, as n → ∞, and W(x, t) ∈ X due to the convexity and closedness of X. This shows that for any bounded sequence {v n (x, t)} in X, the sequence {W n = F(v n )} has a convergent subsequence in X. Thus, the operator F is a continuous and compact mapping of X into itself. Since X is a convex closed subset of V 0 , the Schauder fixed point Theorem 2.1 implies that the operator F has a fixed point W in V 0 , which is the solution W(x, t) of problem (2.10). Moreover, W satisfies inequality (2.25) rewritten as This conclude the proof of the Existence Theorem 2.2.
Next, we will prove the existence and uniqueness of the solution to the inverse problem (1.1), (1.3) and (1.4). We set Proof. According to Lemma 2.1 and Theorem 2.2, we know that the parabolic problems (2.9) and (2.10) have solutions U(x, t) ∈ V and W(x, t) ∈ V 0 satisfying (2.14) and (2.35), respectively. By the definition of W 1 in (2.11), we have and by the same way, we have Thus, Then these inequalities and the first inequality in (2.38) along with the definition (2.12) of the function c( ⋅ ) Then problem (2.10) becomes it is easy to obtain that u(x, t) belongs to the space V and satisfies problem (2.8). Moreover, u(x, t) satisfies the estimate (2.41) Suppose now that w(x, t) and v(x, t) ∈ V are two solutions to problem (2.8), and set Then z(x, t) satisfies the following problem: .16) and By the first inequality in (2.38) and M 0 ∈ (0, M 1 ), as in (2.39), we have and using that (as in (2.39)), Thus from Lemma 2.1 applied to the solution z of problem (2.42), and inequality (2.41) applied to the function v, we obtain that Since Furthermore, q(x) > 0, a.e. x ∈ Ω, due to (2.5) and the first inequality in (2.38) similar as deriving (2.43). The proof is complete.

Example
Let us now give an example for which the sufficient conditions (a)-(e) and (2.38) of the existence and uniqueness of solution in Theorem 2.3 are satisfied. Consider the one-dimensional (N = 1) case such that Ω = (0, l) with l > 0 and Q = Ω × (0, T) with T > 0. Take the Dirichlet data and the initial temperature Also, take the weight functions Finally, take the source component terms Let us first check assumptions (a)-(e) by hand (or, in some cases, using symbolic computations in MAPLE). The regularity assumptions (a) and (b) are obviously satisfied. Condition (c) is also satisfied since To check that assumption (e) is satisfied, we first calculate the quantities given by expression (2.1), namely, .
is a decreasing function since its derivative is negative. Hence, its minimum value is achieved at x = l, which yields Hence, its maximum is achieved at x = 0, which yields From (2.44), (2.45) and (2.47), we have that and from (2.14) we get Next, letting M 0 = T 4l < M 1 = T 2l in (2.12), from (2.25) we obtain that (note that ‖h‖ L ∞ (Q) = 1) Finally, from (2.36) and (2.37) we obtain that .
Then conditions in (2.38) require that From this, we get that For fixed l and T sufficiently large, this condition is clearly satisfied. For example, if T ≥ 270 and 0 < l ≤ 144 π , then 15πl 8T ≤ 1 and (2.48) is satisfied. We also satisfy that l < T 2 7 which was previously needed to ensure that the function Φ 2 a 1 − Φ 1 a 2 ≤ M 2 is decreasing.
In the next section, we shall numerically recover the unknown coefficients q(x) and f(x), simultaneously, from the time-average temperature measurements ϕ 1 given by (1.3) and ϕ 2 given by (1.4).
In practical cases, the time-average temperature measurements ϕ 1 given by (1.3) and ϕ 2 given by (1.4) may not be smooth functions, but contain some noise. This leads to the main difficulty in the numerical reconstruction of the solution since the inverse problem is ill-posed such that small errors into the input data (1.3) and (1.4) lead to large errors in the output coefficients q(x) and f(x). From the view of numerical implementation, we in fact search for the solution approximated from the noisy measurements (ϕ ϵ where ϵ ≥ 0 represents the noise level. In this noisy case, instead of (1.3) and (1.4) we impose Therefore, the inverse problem is to recover the reaction coefficient q(x), the source term f(x) and the tem- .

(3.4)
In the present study we do not debate on the issue of discretise then regularise or, regularise then discretise, see, e.g., [10]; for simplicity, we adopt the latter approach. Next, we intend to derive the Fréchet derivatives J q (q, f) and J f (q, f) of the objective functional J(q, f), and express these gradients via the solution λ(x, t) of a certain appropriate adjoint problem to problem (1.1) given by (3.5) Lemma 3.1. Suppose that q ∈ A q , ϕ ϵ 1 , ϕ ϵ 2 ∈ L ∞ (Ω) and ω 1 (t), ω 2 (t) ∈ L ∞ [0, T]. Then the adjoint problem (3.5) has a unique solution λ ∈ V 0 ⊂ V, which satisfies

6)
and C is a positive constant depending on T, |Q| and q − .
Proof. The existence and uniqueness of the solution λ ∈ V 0 ⊂ V follows from Lemma 2.1. Multiplying the adjoint problem (3.5) by λ, and integrating the resulting relation over Q, we have for some constant C ≥ 0, this yields (3.6). The lemma is proved Proof. Take ∆q ∈ L ∞ (Ω) such that q + ∆q ∈ A q , and denote by ∆u q := u(q + ∆q, f) − u(q, f) the increment of u(q, f) with respect to q. Based on (1.1), ∆u q satisfies the following problem: (3.9) Multiplying the first equation in (3.9) by ∆u q , and integrating it on Q, we have , which implies that, using also (2.41), where c is a positive constant independent of q and f . Denoting ∆J q := J(q + ∆q, f) − J(q, f), by the adjoint problem (3.5), we have . Via problems (3.5) and (3.9), the first term of the right-hand side of the above equation becomes Thus, . By using (3.10), we have Consequently, we obtain which means that the Fréchet derivative J q (q, f) is given by (3.7). Similarly, take ∆f ∈ L ∞ (Ω) such that f + ∆f ∈ A f , and denote by ∆u f := u(q, f + ∆f) − u(q, f) the increment of u with respect to f . Then ∆u f satisfies the problem (3.11) and . , f), and by the same approach, we have Finally, we obtain that thus the Fréchet gradient J f (q, f) is given by (3.8). The theorem is proved.

Conjugate gradient method
The following iteration process based on the CGM is utilised for the simultaneous reconstruction of the spacedependent reaction coefficient q(x) and the source term f(x) by minimising the objective functional J(q, f) given by (3.4): where k denotes the number of iterations, q 0 (x) and f 0 (x) are the initial guesses for q(x) and f(x), respectively, β k q and β k f are the step sizes, and s k q and s k f are the search directions for q and f given by where J k q := J q (q k , f k ), J k f := J f (q k , f k ), γ k q and γ k f are the conjugate coefficients, and the Fletcher-Reeves formula [12] is applied, i.e., 3) The step sizes β k q and β k f can be determined by the line search method, i.e., J(q k+1 , f k+1 ) = min Then, by using (4.1), we have and similarly ∂J(q k+1 , f k+1 ) This means that the minimisation (4.4) implies that the step sizes β k q and β k f satisfy

Convergence
In this subsection, inspired by the arguments in [7], we discuss the convergence of the above iterative process (4.1)-(4.3).

Theorem 4.1.
Let {q k , f k } k≥0 ∈ A q × A f be the iterations defined by the (4.1), and let β k q and β k f satisfy (4.4). k≥0 defined by (3.4) is a monotone decreasing convergence sequence.
Proof. Since the search step sizes β k q and β k f satisfy the condition (4.4), they also satisfy (4.5). For k = 0, by (4.2) we have For k ≥ 1, we can obtain Therefore, we have that Similarly, we have By (4.4), we have Then, for the case β q ≥ 0 and β f = 0, using the mean value theorem and the Lipschitz continuity of the gradient J q given by Theorem 3.2, we obtain It is obvious that the function of β q on the right-hand side of the above formula attains its maximum at i.e., .
Similarly, taking . Thus, , 4) is nonnegative, it follows that it is a monotonic decreasing convergent sequence.
According to the arguments in [4,7], and the results in Theorems 3.2 and 4.1, we obtain the following result:

Iterative algorithm
According to the above discussion, all the parameters of the iterative process (4.1) are expressed explicitly except for the search step sizes β k q and β k f . By using the line search (4.4), these two parameters can be obtained by minimising Since in this expression the search step sizes β k q and β k f are in implicit form, we linearise it to approximate J(q k+1 , f k+1 ) such that the search step sizes β k q and β k f become explicit in the new formulation. Thus where ∆u q (x, t; q k , f k ) and ∆u f (x, t; q k , f k ) are obtained by solving the sensitivity problems (3.9) and (3.11) with ∆q k = s k q and ∆f k = s k f , respectively. We now define Then The derivatives of J(q k+1 , f k+1 ) with respect to β k q and β k f are given by By setting ∂J(q k+1 , f k+1 ) the search step sizes β k q and β k f are given as follows: The iteration process given by (4.1) does not provide the CGM with the regularisation necessary for the minimisation of the objective functional (3.4) to be classified as well-posed because of the errors inherent in the time-average temperature measurements (1.3) and (1.4). However, the method may become well-posed if the discrepancy principle is applied to stop the iteration procedure. According to the discrepancy principle, the iterative procedure is stopped when the following criterion is satisfied: Then the iteration algorithm based on CGM for the numerical recovery of the reaction coefficient q(x) and the source component f(x) is as follows: Step 1. Set k = 0 and choose initial guesses q 0 (x) and f 0 (x) for the unknown space-dependent coefficients q(x) and f(x), respectively.
Step 2. Solve the direct problem (1.1) numerically by utilising the finite-difference scheme to compute the temperature u(x, t; q k , f k ), and the objective functional J(q k , f k ) given by (3.4).
Step 3. Solve the adjoint problem (3.5) numerically to obtain λ(x, t; q k , f k ), and the Fréchet gradients J q (q k , f k ) by (3.7) and J f (q k , f k ) by (3.8). Calculate the conjugate coefficients γ k q and γ k f in (4.3), and the search directions s k q and s k f in (4.2).
Step 4. Solve the problems ∆u q (x, t; q k , f k ) given by (3.9) and ∆u f (x, t; q k , f k ) given by (3.11) by using ∆q k = s k q and ∆f k = s k f , and compute the search step sizes β k q and β k f given by (4.6).
Step 6. If the stopping condition (4.7) is satisfied, then go to Step 7. Else set k = k + 1 , and go to Step 2.

Numerical results and discussions
In this section, we perform one-and two-dimensional numerical computations to simultaneously reconstruct the space-dependent reaction coefficient q(x) and source component f(x) based on the CGM described in the previous section. The finite-difference method (FDM) based on the Crank-Nicolson scheme is employed in one-dimension (N = 1), whilst the alternating direction implicit (ADI) method is employed in two-dimension (N = 2), see [5], for solving the direct, sensitivity and adjoint problems involved in the CGM. In the twodimensional case we decided to switch to an ADI (of Peaceman-Rachford type) method because, as pointed out in [15], it is more suitable, in the sense of convergence and stability, than the Crank-Nicolson scheme for two-dimensional parabolic problems with self-adjoint elliptic spatial operator, see [34]. The Simpson's rule is utilised for computing the integrals involved in the objective functional J(q, f), its Fréchet derivatives J q and J f , the adjoint problem for λ, and the conjugate coefficients γ q and γ f . We have also tested the accuracy using the trapezoidal rule obtaining no significant differences in the numerical results than when using the Simpson's rule. The accuracy errors E q (k) and E f (k), as functions of the iteration number k, for the reaction coefficient q(x) and the source component f(x), respectively, are defined as where q k and f k are the CGM iterates, and q and f are the analytical expressions of the reaction coefficient and source component, if available. The noisy integral temperature observations ϕ ϵ 1 and ϕ ϵ 2 in (3.2) and (3.3) are simulated by adding to their analytical values, if available, Gaussian additive noise as where σ is the standard deviation given by σ = p 100 × max x∈Ω {|ϕ 1 (x)|, |ϕ 2 (x)|}, p % represents the percentage of noise, and the term random(1) generates random values from a Gaussian distribution with zero mean and standard deviation equal to unity.
In the one-dimensional (N = 1) examples we consider the final time T = 1, the domain Ω = (0, 1), and take the FDM mesh size ∆x = ∆t = 0.01 for solving the direct, sensitivity and adjoint problems in the CGM iterative procedure. In the two-dimensional (N = 2) example, we consider T = 1, Ω = (0, 1) × (0, 1), and the ADI mesh size ∆x 1 = ∆x 2 = ∆t = 0.01. As for the weight functions ω 1 (t) and ω 2 (t) in ( 5) which implies that the time-average temperature measurements ϕ 1 and ϕ 2 can be regarded as the measured temperature at the distinct time instants t 1 and t 2 , see [5]. For all the three numerical examples presented in the following subsections, we take the weight functions as i.e., a = 10 −3 , t 1 = 0.2 and t 2 = T = 1 in (5.4).

Example 1
In the one-dimensional (N = 1) case, we take the input data as and since the weight functions are chosen as in (5.6), the measured temperatures are taken as in (5.5), namely, These approximations already introduce some (numerical) noise into the input data (1.3) and (1.4). An analytical solution of the inverse problem (1.1), (1.3) and (1.4) with the above data is given by Although the conditions in (2.38) of Theorem 2.3 are not satisfied to guarantee the uniqueness of the solution, the numerically obtained results indicate that (5.7) is the closest solution to the chosen initial guess. From (3.5), one can observe that the gradients (3.7) and (3.8) vanish for x on the boundary ∂Ω, hence we need to specify the initial guesses for q 0 and f 0 on ∂Ω be equal to their true values, otherwise, from (4.1) and (4.2) there will be no progress with iterations at the boundary points. Therefore, we take q 0 (x) = 2 and f 0 (x) = 0.5 for x ∈ Ω, which ensures that q 0 (x) = q true (x) and f 0 (x) = f true (x) for x ∈ ∂Ω, and also that these initial guesses are rather far from their true values (5.7) for x ∈ Ω.
In Figures 2 (a) and 2 (b), the numerical solutions for the space-dependent reaction coefficient q(x) and the source term f(x) are plotted at the stopping iteration numbers k * inferred from Figure 1, for the levels of noise p ∈ {0, 1, 2}. Higher levels of noise produced less accurate numerical results and therefore they are not presented. For p = 0 noiseless data, the analytical and numerical solutions overlap and they are graphically undistinguishable. Overall, from Figure 2 it can be seen that the stable and accurate solutions are obtained for both coefficients.

Example 2
In the previous example, we have considered a test with a smooth analytical solution given by (5.7). In this example, we consider a more severe test involving reconstructing a discontinuous reaction coefficient, as follows. We take otherwise, and the measurements (5.5) given by Then an analytical solution of the inverse problem (1.1), (1.3) and (1.4) with this input data is given by 2, x ∈ [ 1 4 , 3 4 ], 1, otherwise, , f(x) = (4x − 1) sin(πx), u(x, t) = 1 + x 2 t.
The initial guesses are taken as q 0 (x) = 1 and f 0 (x) = 0. As in the previous example, the iterative CGM is stopped at the iterations k * ∈ {60, 5, 4} for p ∈ {0, 1, 2}, by using the discrepancy principle (4.7) with  Figure 3 we can observe that the numerical results are stable and reasonably accurate bearing in mind the severe discontinuous reaction coefficient q(x) that had to be retrieved simultaneously with the source term f(x).

Conclusions
In this paper, the simultaneous identification and reconstruction of the space-dependent reaction coefficient and source term from time-integral temperature measurements has been investigated. The existence and uniqueness of the solution have been established. Then the two unknown quantities have been reconstructed simultaneously by minimising the least-squares objective functional, and the CGM has been developed using the newly derived Fréchet gradients, along with the adjoint and sensitivity problems. The convergence of the nonlinear CGM has been considered using the approaches employed in [7]. Three numerical examples in both one-and two-dimensions have been illustrated, and the results indicate the accuracy and stability of the numerical recovery of the two unknown spatially-dependent quantities of interest. Further work will consider the simultaneous numerical reconstruction of the space-dependent convection and reaction coefficients from time-integral temperature measurements, [22].