On an inverse problem in the parabolic equation arising from groundwater pollution problem

In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose both methods, a priori and a posteriori parameter choice rules. In addition, the proposed methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the a posteriori parameter choice rule method gives a better the convergence speed in comparison with the a priori parameter choice rule method in some specific applications.


Introduction
Groundwater is crucial to human being, environment and economy, because a large portion of drinking water comes from groundwater, and it is extracted for commercial, industrial and irrigation uses. Groundwater also sustains stream flow during dry periods, and is critical to the function of streams, wetlands and other aquatic environments. Therefore protecting the safety and security of groundwater is essential for communities, and for the environment. In recent years, mathematical models have used to analyze ground water system. There are two notable approaches in dealing with groundwater modeling, the forward and backward approaches. The former is going to predict unknown variables by solving appropriate governing equations, while the latter is going to determine unknown physical parameters. Most of groundwater models are distributed parameter models, where the parameters used in the modeling equations are not directly obtained from physical observations, but from trial-and-error and graphical fitting techniques. If large errors are included in mathematical model structure, model parameters, sink/source terms and boundary conditions, the model cannot produce accurate results. To deal with this issue, the inverse problem of parameter identification has been applied. In groundwater applications such as in finding a previous pollution source intensity from observation data of the pollutant concentrations at a later time, or in designing the final state of melting and freezing processes, it is necessary to construct a heat source at any given time from the final outcome state data. The groundwater inverse problem has been studied since the middle of 1970s by McLaughin (1975), Yeh (1986), Kuiper (1986), Carrera (1987), Ginn and Cushman (1990) and Sun (1994), etc. Some remarkable results on this research area should be mentioned by McLaughlin and Townley (1996) and Poeter and Hill (1997). Under consideration of a solute diffusion, the flow and self-purifying function of watershed system, the concentration of pollution u(x, t) at any time in a watershed is described by following one-dimensional linear parabolic equation: x ∈ Ω, t > 0, (1.1) where Ω ∈ R is the spatial studied domain, η is the diffusion coefficient, ν is mean velocity of water in the watershed, and γ is the self-purifying function of the watershed, P (x, t) is the source term causing the pollution function u (x, t). By setting . This equation is a well-known parabolic heat equation with time-dependent coefficients. This equation has been investigated for the heat source with either temporal [6,6,6] or spatial depending [6,6,6,6,6] only. There are few studies on identification of the source term depending on both time and space in term of a separable form of F(x, t), as F(x, t) = ϕ(t) f (x); where ϕ (t) is a given function; for instance Hassanov [6] identified the heat source in the form of F(x, t) = F(x)H(t) for the variable coefficient heat conduction equation; u t = (k(x)u x ) x + F(x)H(t) under the variational method. However, in the case with the time-dependent coefficient of ∂ 2 w ∂x 2 , there are still limited results. In this study, we consider the equation for groundwater pollution as follows.
with initial and final conditions and boundary condition Here, a (t) > 0, g (x) and ϕ (t) are given functions. In this paper, we will determine the source term f (x) from the inexact observed data of ϕ (t) and g (x). Let . and ., . be the norm and the inner product in L 2 (0, π), respectively. Now, we take an orthonormal basis in L 2 (0, π) satisfying the boundary condition (1.4), particularly the basic function 2 π sin (nx) for n ∈ N is satisfied this condition. Then, by an elementary calculation the prob- By setting A (t) = t 0 a (s) ds, we can solve the ordinary differential equation (1.5) with the contitions (1.6). We thus obtain which leads to Note that e n 2 A(T ) increases rather quickly once n becomes large. Thus, the exact data function g (x) must satisfy that g n decays at least as the same speed of e n 2 A(t) . However, in application the input data g (x) from observations will never be exact due to the measurements. We assume the data functions g (x) ∈ L 2 (0, π), and ϕ (t) , ϕ (t) ∈ L 2 (0, T ) satisfy where > 0 represents a noise from observations. The main objective of this paper is to determine a conditional stability, and provide the revised generalized Tikhonov regularization method. In addition, the stability estimate between the regularization solution and the exact solution is obtained. For explanation of this method, we impose an a priori bound on the data where M ≥ 0 is a constant, and . H k (0,π) denotes the norm in the Sobolev space H k (0, π) of order k can be naturally defined in terms of Fourier series whose coefficients decay rapidly; namely, equipped with the norm where f n defined by f n = f, X n , X n = 2 π sin (nx) is the Fourier coefficient of f .
As a regularization method, the Tikhonov method has been used to solve ill-posed problems in a number of publications. However, most of previous works focus on an a priori choice of the regularization parameter. There is usually a defect in any a priori method; i.e. the a priori choice of the regularization parameter depends obviously on the a priori bound M of the unknown solution. In fact, the a priori bound M cannot be known exactly in practice, and working with a wrong constant M may lead to a bad regularization solution. In this paper, we mainly consider the a posteriori choice of a regularization parameter for the mollification method. Using the discrepancy principle we provide a new posteriori parameter choice rule.
The outline of this paper is as follows. In Section 2, a conditional stability is introduced. A Tikhonov regularization and its convergence under an a priori parameter choice rule is presented in Section 3. Similarly to Section 3, another Tikhonov regularization and its convergence under a posteriori parameter choice rule is shown in Section 4. In Section 5, we introduce two numerical examples, which are implemented from proposal regularization methods, the numerical results are compared with exact solutions.

A conditional stability
Let a, ϕ, ϕ : [0, T ] → R be continuous functions. We suppose that there exist constants (2.12) Hereafter, let us set where h is ϕ, and ϕ by implication. Then, we can obtain the following conditional stability.
for n ∈ N.
Proof. The proof is simple by elementary calculation. (2.15) Proof. Using Holder's inequality, we first have (2.16) From (1.7), the inequality will be obtained continuously. (2.17) We pay attention to the integral on the right-hand side by direct estimate and computation. From (2.12), we thus get (2.18) Theorem 2 has been proved.
T 0 e n 2 B(t) dtX n (x) X n (ξ). Due to k (x, ξ) = k (ξ, x), K is self-adjoint. Next, we prove its compactness. We define finite rank operators K m by Then, from (3.19) and (3.20), we have Therefore, K m − K → 0 in the sense of operator norm in L L 2 (0, π) ; L 2 (0, π) , as m → ∞. K is also a compact operator. Next, the singular values for the linear self-adjoint compact operator are and corresponding eigenvectors is X n which is known as an orthonormal basis in L 2 (0, π). From (3.19), the inverse source problem introduced above can be formulated as an operator equation.
In general, it is an ill-posed problem. From the point of view, we intend to solve it by using Tikhonov regularization method tconvergenceo minimize the quantity in L 2 (0, π) As shown in [20] by Theorem 2.12, its minimizer f µ satisfies Due to singular value decomposition for compact self-adjoint operator, we have If the given data is noised, we can establish From (2.13), (3.26) and (3.27), we get (3.29) Following we will deduce an error estimate for f − f µ and show convergence rate under a suitable choice of regularization parameters. It is clear that the entire error can be decomposed into the bias and noise propagation as follows: We first give the error bound for the noise term.
Lemma 3.1. If the noise assumption holds and assume that g − g ≤ and ϕ − ϕ L 2 [0,T ] ≤ , then the solution depends continuously on the given data. Moreover, we have the following estimate.
Proof. We notice that We consider two following estimates by diving into two steps.
In order to obtain the boundedness of bias, we usually need some a priori conditions. By Tikhonov's theorem, the K −1 restricted to the continuous image of a compact set M. Thus, we assume f is in a compact subset of L 2 (0, π). Hereafter, we assume that f H 2k (0,π) ≤ M for k > 0. (3.37) Proof. From (1.8) and (3.26), we deduce that where P (n) = µ 4 µ 2 + (Φ (n, ϕ)) 2 2 1 + n 2 −2k . (3.39) Next, we estimate P (n). Without loss of generality, we assume that µ − 1 4 is not integer. Therefore, (3.38) can be divided into the sum of A 1 and A 2 as follows: where n 0 ≤ µ − 1 4 ≤ n 0 + 1. In A 1 , we have For 0 < k ≤ 2, we deduce that In addition, we observe in A 2 that From (3.42)-(3.44), we thus obtain Hence, by using the assumption, we conclude that , then , then

48)
where Q is constant and depends on T, B 1 , C 1 , D 1 , D 2 , M.

Tikhonov regularization under a posteriori parameter choice rule
In this section, we consider an a posteriori regularization parameter choice in Morozov's discrepancy principle (see in [6 and 6]). First, we introduce following lemma: Lemma 4.1. Set ρ (µ) = K f µ − g and assume that 0 < < g , then the following results hold: d. ρ (µ) is a strictly increasing function.
Therefore, combining (4.57) and (4.58), we conclude that which gives the desired result.
Theorem 4.1. Assume the a priori condition and the noise assumptions hold, and there exists τ > 1 such that 0 < τ < g . Then, we choose a unique regularization parameter µ > 0 such that where constants P and Q depend on the constants T, µ, k, τ, B 1 , B 2 , C 1 , C 2 , D 1 , D 2 and M.
Proof. For 0 < k ≤ 1, we have (4.61) It follows that (4.62) We set K 1 and K 2 as follows.

Numerical Examples
In this section, we implemented numerically above proposed regularization methods. Two different numerical examples corresponding to k = T = 1 are shown as follows. The first example is to consider an example with a in Eq. (2) is a constant, and the function f obtained from exact data function. The second example is to consider an example with a is a non-constant function, and f obtained from observation data of g and ϕ.
The couple of (g , ϕ ), which are determined below, play as measured data with a random noise as follows: where rand() ∈ (−1, 1) is a random number. We can easily verify the validity of the inequality: In addition, we can take the regularization parameter for the a priori parameter choice rule µ = where M plays a role as a priori condition computed by f H 2 (0,π) . The absolute and relative errors between regularized and exact solutions are estimated. The regularized solution are computed by: where N is the truncation number; whereby N = 1000 is chosen in the following examples. In general, the whole numerical procedure is shown in the following steps: Step 1. Choosing L and K to generate temporal and spatial discretizations as follows: Of course, the higher value of L and K will provide more stable and accurate numerical calculation, however in the following examples L = K = 100 are satisfied.
Step 2. Setting f µ x j = f µ, j and f x j = f j , constructing two vectors contained all discrete values of f µ and f denoted by Λ µ and Ψ, respectively.
Step 3. Error estimate between the exact solutions and regularized solutions; Absolute error estimation: Relative error estimation: (5.86)

Example 1
As mentioned above, in this example we consider a is constant, and f is an exact data function. Specifically, we consider a type of the problem (1.2)-(1.4) as follows This implies that a (t) = 1, ϕ (t) = e t − 1, g (x) = 10 −1 (e − 1) sin 2x and f (x) = 2 −1 sin 2x. It is easy to see that u (x, t) = 10 −1 e t − 1 sin 2x is the unique solution of the problem (87) Next, we establish the regularized solution according to composite Simpson's rule.

Example 2
Similar to the first example, however in this example we consider a (t) = 2t + 1, then Thus, the exact solution is obtained by: Unlike the first example, from the analytical solution, we can have f H 2 (0,π) < 5500 which implies that µ 1 = 5500 1 3 for the a priori parameter choice rule. Afterwards, based on (4.59) with τ = 1.1 again, we can compute the regularization parameter for the a posteriori parameter choice rule, . Therefore, the regularized solution can be computed by n 4 µ 2 + (1 + · rand (.)) 1 − e −2n 2 2 sin (nx) .  Table 1; when a is constant, and f is an exact data function; it shows the convergence speed of both parameter choice rule methods are quite similar and slow as tends to 0. Whereas, in the second example shown in Table 2; when a is not a constant, and f is obtained from measured data; it shows the convergence speed of the a posteriori parameter choice rule is better than (by second order) the a priori parameter choice rule as tends to 0. In addition, Figures 1 and 2 show a comparison between the exact solution and its regularized solution for the a priori parameter and the a posteriori parameter choice rules in the first example, respectively. It again shows that for the both parameter choice rule methods, the regularized solution was strong oscillated around the exact solution when around 0.1; nevertheless it converges to the exact solution as tends to 0. In the second example, Figures 3 and 4 show the same tendency as in the first example for both methods.

Conclusion
In this study, we solved the problem (1.2)-(1.4) to recover temperature function of the unknown sources in the parabolic equation with the time-dependent coefficient (i.e. inhomogeneous source) by suggesting two methods, the a priori and a posteriori parameter choice rules.
In the theoretical results, we obtained the error estimates of both methods based on a priori condition. From the numerical results, it shows that the regularized solutions are converged to the exact solutions. Furthermore, it also shows that the a posteriori parameter choice rule method is better than the a priori parameter choice rule method in term of the convergence speed.  Table 1: Error estimate between the exact solution and its regularized solution for the a priori parameter and the a posteriori parameter choice rules in Example 1.