Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 713403, 9 pages
http://dx.doi.org/10.1155/2015/713403

Research Article

An Iterative Regularization Method for Identifying the Source Term in a Second Order Differential Equation

Fairouz Zouyed and Sebti Djemoui

Applied Math Lab, University Badji Mokhtar Annaba, P.O. Box. 12, 23000 Annaba, Algeria

Received 28 May 2015; Accepted 21 September 2015

Academic Editor: Peter Dabnichki

Copyright © 2015 Fairouz Zouyed and Sebti Djemoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper discusses the inverse problem of determining an unknown source in a second order differential equation from measured final data. This problem is ill-posed; that is, the solution (if it exists) does not depend continuously on the data. In order to solve the considered problem, an iterative method is proposed. Using this method a regularized solution is constructed and an a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

1. Introduction

Let be a separable Hilbert space with the inner product and the norm . Consider the problem of finding the source term in the following system:

with the additional data

where is a positive self-adjoint linear operator with a compact resolvent; we denote by the spectrum of the operator .

The problem (1) is an abstract version of the system

which arises in the mathematical study of structural damped vibrations of string or a beam [1–3]. Also this problem can be considered as a biparabolic problem in the abstract setting. For physical motivation we cite the biparabolic model proposed in [4] for more adequate mathematical description of heat and diffusion processes than the classical heat equation. For other models we refer the reader to [5–7].

For most classical partial differential equations, the reconstruction of source functions from the final data or a partial boundary data is an inverse problem with many applications in several branches of sciences and engineering, such as geophysical prospecting and pollutant detection [8–12].

The main difficulty of inverse source identification problems is that they are ill-posed, that is, even if a solution exists, it does not depend continuously on the data; in other words, small error in the data measurement can induce enormous error to the solution. Thus, special regularization methods that restore the stability with respect to measurements errors are needed. In the present work, we focus on an iterative method proposed by Kozlov and Maz’ya [13, 14] for solving the problem; it is based on solving a sequence of well-posed boundary value problems such that the sequence of solutions converges to the solution for the original problem. It has been successfully used for solving various classes of ill-posed elliptic, parabolic, and hyperbolic problems [5, 15–21].

We note that although the interest in inverse problem has rapidly increased during this decade, the literature devoted to the class of problems (1) is quite scarce.

The paper is organized as follows. Section 2 gives some tools which are useful for this study; in Section 3 we introduce some basic results and we show the ill-posedness of the inverse problem; Section 4 gives a regularization solution and error estimation between the approximate solution and the exact one; the numerical implementation is described in Section 5 to illustrate the accuracy and efficiency of this method.

2. Preliminaries

Let be an orthonormal eigenbasis corresponding to the eigenvalues such that

We denote by the analytic semigroup generated by on ,

For , the space is given by

with the norm

We achieve this section by a result concerning nonexpansive operators.

Definition 1. A linear bounded operator is called nonexpansive if .

Let be an nonexpansive operator; to solve the equation

we state a convergence theorem for a successive approximation method.

Theorem 2 (see [22], p. 66). Let be a nonexpansive, self-adjoint positive operator on . Let be such that (8) has a solution. If is not eigenvalue of , then the successive approximations

converge to a solution to (8) for any initial data . Moreover, for every , as .

3. Basic Results

3.1. The Direct Problem

Let with the norm , .

For a given , consider the direct problem

Making the change of variable , we can write the second order equation in (10) as a first order system in the space as follows:

where , , and .

The linear operator is unbounded with the domain and it is the infinitesimal generator of strongly continuous semigroup . Moreover is analytic (see [1]) and it admits the following explicit form:

where and is a complete family of orthogonal projections in given by .

Using matrix algebra, we obtain

From the semigroup theory (see [23]), the problem (11) admits a unique solution given by

Hence,

such that

As a consequence, we obtain the following theorem.

Theorem 3. The problem (10) admits a unique solution given by

3.2. Ill-Posedness of the Inverse Problem

Now, we wish to solve the inverse problem, that is, find the source term in the system (1). Making use of the supplementary condition (2) and defining the operator , we have

where .

It is easy to see that is a self-adjoint compact linear operator. On the other hand,

so

which implies

and therefore

Note that as , so the inverse problem is ill-posed; that is, the solution does not depend continuously on the given data. Hence this problem cannot be solved by using classical numerical methods.

Remark 4. As many boundary inverse value problems for partial differential equations which are ill-posed, the study of the problem (1) is reduced to the study of the equation , where is a compact self-adjoint operator in the Hilbert space . This equation can be rewritten in the following way:

where is a positive number satisfying .
In the next section, we will show that the operator is nonexpansive and is not eigenvalue of , so it follows from Theorem 2 that converges and , for every , as .

4. Iterative Procedure and Convergence Results

The alternating iterative method is based on reducing the ill-posed problem (1) to a sequence of well-posed boundary value problems and consists of the following steps.

First, we start by letting be arbitrary; the initial approximation is the solution to the direct problem

Then, if the pair has been constructed, let

where is such that

and .

Finally, we get by solving the problem

Let us iterate backwards in (25) to obtain

Now, we introduce some properties and tools which are useful for our main theorems.

Lemma 5. The norm of the operator is given by

Proof. We aim to find the supremum of the function , , and for this purpose, fix , let , and define the function

We compute

Put

Hence,

To study the monotony of , it suffices to determine the sign of . We have

and then is decreasing; moreover , . Hence , , which implies that is decreasing and

Therefore,

Proposition 6. For the linear operator , one has the following properties:

(1)is positive and self-adjoint,(2) is nonexpansive,(3) is not an eigenvalue of .

Proof. Form properties of operator and the definition of it follows that is self-adjoint and nonexpansive positive operator and from the inequality

it follows that the point spectrum of . Then is not eigenvalue of the operator .

Lemma 7. If , one has the estimates

Proof. To establish (38), let us first prove that

which is equivalent to prove that

We have

Then, is nondecreasing and it follows that . So, .
Choosing in (40), we obtain

So,

From (44), we deduce (38).
Now, we prove the estimate (39). It is easy to verify that

Then, if we choose , we get

Hence, from (46), (39) follows.

Theorem 8. Let be a solution to the inverse problem (1). Let be an arbitrary initial data element for the iterative procedure proposed above and let be the th approximate solution. Then

(i)The method converges; that is,

(ii)Moreover, if, for some , , , that is, , then the rate of convergence of the method is given by

where is a positive constant independent of .

Proof. (i) From (28), we get

and then

which implies that

Hence,

From Lemma 5 and (39) we have

Combining (52) and (53) and passing to the supremum with respect to , we obtain

(ii) By part (i), we have

and hence

Using the inequality (38), we obtain

where
So, it follows that

Put

We compute

Setting , it follows that is the critical point of . It is easy to see that the maximum of is attained at . So

and hence

Combining (58) and (62), we obtain

Since in practice the measured data is never known exactly but only up to an error of, say, , it is our aim to solve the equation from the knowledge of a perturbed right-hand side satisfying

where denotes a noise level. In the following theorem, we consider the case of inexact data.

Theorem 9. Let , , be an arbitrary initial data element for the iterative procedure proposed above such that , let be the th approximations solution for the exact data , and let be the th approximations solution corresponding to the perturbed data such that (64) holds. Then one has the following estimate:

Proof. Let

Using the triangle inequality, we obtain

From Theorem 8, we have

On the other hand,

Since

it follows that

Combining (68) and (71) and passing to the supremum with respect to , we obtain the estimate (65).

Remark 10. If we choose the number of the iterations so that as , we obtain

5. Numerical Implementation

In this section, an example is devised for verifying the effectiveness of the proposed method. Consider the problem of finding a pair of functions , in the system

Denote

are eigenvalues and orthonormal eigenfunctions, which form a basis for .

The solution of the above problem is given by

where ,

Now, to solve the inverse problem, making use of the supplementary condition and defining the operator , we have

Example 11. In the following, we first selected the exact solution and obtained the exact data function through solving the forward problem. Then we added a normally distributed perturbation to each data function and obtained vectors . Finally we obtained the regularization solutions through solving the inverse problem with noisy data satisfying

It is easy to see that if , then

is the exact solution of the problem (73). Consequently,
Now, we propose to approximate the first and second space derivatives by using central difference and we consider an equidistant grid points to a spatial step size , (), where is a positive integer. We get the following semidiscrete problem:

where is the discretisation matrix stemming from the operator , and

is a symmetric, positive definite matrix, with eigenvalues

and orthonormal eigenvalues

We assume that it is fine enough so that the discretization errors are small compared to the uncertainty of the data; this means that is a good approximation of the differential operator whose unboundedness is reflected in a large norm of (see [24]).
Adding a random distributed perturbation to each data function, we obtain

where indicates the noise level of the measurements data and the function randn() generates arrays of random numbers whose elements are normally distributed with mean , variance , and standard deviation . (size()) returns an array of random entries that is of the same size as . The noise level can be measured in the sense of root mean square error (RMSE) according to

The relative error is given as follows:

The discrete iterative approximation of (66) is given by

where and .

Figures 1–4 display that as the amount of noise decreases, the regularized solutions approximate better the exact solution.

Figure 1: The comparison between the exact solution and its computed approximations for    and noisy level .

Figure 2: The comparison between the exact solution and its computed approximations for    and noisy level .

Figure 3: The comparison between the exact solution and its computed approximations for   and noisy level .

Figure 4: The comparison between the exact solution and its computed approximations for    and noisy level

Table 1 shows that for or the relative error decreases with the decease of epsilon which is consistent with our regularization.

Table 1: Relative error RE().

RE()

60
4
10⁻³
0.2039

60
4
10⁻⁴
0.0945

60
5
10⁻³
0.3032

60
5
10⁻⁴
0.0305

6. Conclusion

In this paper, we have extended the iterative method to identify the unknown source term in a second order differential equation, convergence results were established, and error estimates have been obtained under an a priori bound of the exact solution. Some numerical tests have been given to verify the validity of the method.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors read and approved the paper.

Acknowledgments

The authors would like to thank the anonymous referees for their suggestions.

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