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

This paper discusses the inverse problem of determining an unknown source in a second order differential equation frommeasured 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.

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 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][9][10][11][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][16][17][18][19][20][21].

Mathematical Problems in Engineering
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.

Preliminaries
Let (  ) ≥1 ⊂  be an orthonormal eigenbasis corresponding to the eigenvalues (  ) ≥1 such that We denote by {() =  − } ≥0 the analytic semigroup generated by − on , For  > 0, the space   is given by with the norm We achieve this section by a result concerning nonexpansive operators.
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, selfadjoint positive operator on .Let  ∈  be such that (8) has a solution.If 1 is not eigenvalue of , then the successive approximations converge to a solution to (8) for any initial data  0 ∈ .Moreover,    → 0 for every  ∈ , as  → ∞.

The Direct Problem.
Let  = () ×  with the norm For a given  ∈ , consider the direct problem Making the change of variable   = V, we can write the second order equation in (10) as a first order system in the space  as follows: where  = (  V ),  = ( 0  ), and A = ( 0  − 2 −2 ).The linear operator A is unbounded with the domain (A) = ( 2 ) × () and it is the infinitesimal generator of strongly continuous semigroup {() =  A } ≥0 .Moreover {()} ≥0 is analytic (see [1]) and it admits the following explicit form: where   = ( −2  ) and {  } ≥1 is a complete family of orthogonal projections in  given by   = diag(  ,   ).
Using matrix algebra, we obtain From the semigroup theory (see [23]), the problem (11) admits a unique solution  ∈ ([0, ), ) given by Hence, such that As a consequence, we obtain the following theorem.
Theorem 3. The problem (10) admits a unique solution  ∈ ([0, ), ()) ∩  1 ([0, ), ) 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 so which implies and therefore Note that 1/  → ∞ 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  < 1/‖()‖.
In the next section, we will show that the operator  is nonexpansive and 1 is not eigenvalue of , so it follows from Theorem 2 that (  ) ∈N * converges and ( − ())   → 0, for every  ∈ , as  → ∞.

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  0 ∈  be arbitrary; the initial approximation  0 is the solution to the direct problem Then, if the pair (  ,   ) has been constructed, let where  is such that and Finally, we get  +1 by solving the problem 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 Hence, To study the monotony of  1 , it suffices to determine the sign of ℎ.We have and then ℎ is decreasing; moreover ℎ() ⊂ ] − 2, 0[, ∀ > 0. Hence   1 () < 0, ∀ ≥  1 , which implies that  1 is decreasing and Therefore, sup Proposition 6.For the linear operator  =  − (), one has the following properties: (1)  is positive and self-adjoint, (2)  is nonexpansive, (3) 1 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 ,   () ⊂ ]0, 1[.Then 1 is not eigenvalue of the operator .
Theorem 8. Let  be a solution to the inverse problem (1).Let  0 ∈  be an arbitrary initial data element for the iterative procedure proposed above and let   be the th approximate solution.Then (ii) Moreover, if, for some  = 1 + ,  > 0,  0 −  ∈   , that is, ‖ 0 − ‖   ≤ , then the rate of convergence of the method is given by sup where  is a positive constant independent of .
Proof.(i) From (28), we get and then (ii) By part (i), we have and hence Using the inequality (38), we obtain where Put We compute Setting   () = 0, it follows that  * = /(2 + ) is the critical point of .It is easy to see that the maximum of  is attained at  * .So sup and hence sup Combining ( 58) and ( 62), we obtain sup Since in practice the measured data  is never known exactly but only up to an error of, say,  > 0, it is our aim to solve the equation () =  from the knowledge of a perturbed right-hand side   satisfying where  > 0 denotes a noise level.In the following theorem, we consider the case of inexact data.
Theorem 9. Let  = 1 + , ( > 0),  0 be an arbitrary initial data element for the iterative procedure proposed above such that ( 0 −) ∈   , 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: ) . (65) Using the triangle inequality, we obtain Combining ( 68) and ( 71) and passing to the supremum with respect to  ∈ [0, ], we obtain the estimate (65).
are eigenvalues and orthonormal eigenfunctions, which form a basis for .
The solution of the above problem is given by where   = (,   ) = √ 2 ∫ 1 0 ()sin(),  = 1, 2, . . .Now, to solve the inverse problem, making use of the supplementary condition and defining the operator  :  → , we have is the exact solution of the problem (73).Consequently, () = ((1 − (1 +  2 ) − 2 )/ 4 )sin().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  0 = 0 <  1 < ⋅ ⋅ ⋅ <  +1 = 1, (ℎ = 1/( + 1)), where  is a positive integer.We get the following semidiscrete problem: where  ℎ is the discretisation matrix stemming from the operator  = − 2 / 2 , and is a symmetric, positive definite matrix, with eigenvalues   = 4 ( + 1) 2 sin 2  2 ( + 1) ,  = 1, . . ., , 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  ℎ is a good approximation of the differential operator  whose unboundedness is reflected in a large norm of  ℎ (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 0, variance  2 = 1, and standard deviation  = 1.randn(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 where  ℎ =  −2 ℎ (  − (  +  ℎ ) − ℎ ) and  < 1/‖ ℎ ‖ = ( 2  1 /(1 − (1 +  1 ) − 1 )) .
Figures 1-4 display that as the amount of noise  decreases, the regularized solutions approximate better the exact solution.
Table 1 shows that for  = 4 or  = 5 the relative error decreases with the decease of epsilon which is consistent with our regularization.

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.
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       −       ( 2 (0,1)) 2 ≤ .