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 datawhere 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 systemwhich 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 equationwe 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 problemMaking 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, sowhich 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, letwhere is such that and .
Finally, we get by solving the problem Let us iterate backwards in (25) to obtainNow, 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 thatwhich 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 getHence, 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 thenwhich implies that Hence,From Lemma 5 and (39) we haveCombining (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 thatPut 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 henceCombining (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 satisfyingwhere 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. LetUsing the triangle inequality, we obtain From Theorem 8, we haveOn the other hand, Since it follows thatCombining (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 systemDenoteare 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 satisfyingIt 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 , andis 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.
Table 1 shows that for or the relative error decreases with the decease of epsilon which is consistent with our regularization.
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.