This paper deals with the problem of determining an unknown boundary condition in the boundary value problem , , , with the aid of an extra measurement at an internal point. It is well known that such a problem is severely ill-posed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the ill-posed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the Kozlov-Maz’ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.
MSC: 35R25, 65J20, 35J25.
Keywords:ill-posed problems; elliptic problems; cut-off spectral regularization; iterative regularization
1 Formulation of the problem
where f is the unknown boundary condition to be determined from the interior data
This problem is an abstract version of an inverse boundary value problem, which generalizes inverse problems for second-order elliptic partial differential equations in a cylindrical domain, for example we mention the following problem.
which takes the form
To our knowledge, there are few papers devoted to this class of problems in the abstract setting, except for [1,2]. In , the author studied a similar problem posed on a bounded interval. In this study, the algebraic invertibility of the inverse problem was established. However, the regularization aspect was not investigated.
We note here that this inverse problem was studied by Levine and Vessella , where the authors considered the problem of recovering from the experimental data associated to the internal measurements , in which the temperature is measured at various depths as approximate functions such that
The regularizing strategy employed in  is essentially based on the Tikhonov regularization and the conditional stability estimate for some a priori constant E.
In practice, the use of N-measurements or the average of a series of measurements is an expensive operation, and sometimes unrealizable. Moreover, the numerical implementation of the stabilized solutions by the Tikhonov regularization method for this class of problems will be a very complex task.
For these reasons, we propose in our study a practical regularizing strategy. We show that we can recover from the internal measurement under the conditional stability estimate for some a priori constant E. Moreover, our investigation is supplemented by numerical simulations justifying the feasibility of our approach.
2 Preliminaries and basic results
In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis.
We denote by the set of all closed linear operators densely defined in H. The domain, range and kernel of a linear operator are denoted as , and ; the symbols , and are used for the resolvent set, spectrum and point spectrum of B, respectively. If V is a closed subspace of H, we denote by the orthogonal projection from H to V.
For the ease of reading, we summarize some well-known facts in spectral theory.
2.2 Spectral theorem and properties
By the spectral theorem, for each strictly positive self-adjoint operator B,
Theorem 2.1 [, Theorem 6, XII.2.5, pp.1196-1198]
Theorem 2.2 (see , Chapter 2, Theorem 6.13, p.74)
For this family of operators, we have:
Remark 2.1 For , this theorem ensures that is self-adjoint and one-to-one operator with dense range . Then we can define its inverse , which is an unbounded self-adjoint strictly positive definite operator in H with dense domain
Theorem 2.4 [, Theorem 7.5, p.191]
Here we recall a crucial theorem in the analysis of the inverse problems.
Theorem 2.5 [, Generalized Picard theorem, p.502]
Letbe a self-adjoint operator and the Hilbert spaceH, and letbe its spectral resolution of unity. Letand. We suppose that the seteither is empty or contains isolated point only. Then the vectorial equation
is solvable if and only if
2.3 Non-expansive operators
Theorem 2.6 [, Theorem 2.2]
For more details concerning the theory of non-expansive operators, we refer to Krasnosel’skii et al. [, p.66].
Let use consider the operator equation
for non-expansive operators M.
Theorem 2.7LetMbe a linear self-adjoint, positive and non-expansive operator onH. Letbe such that equation (2.3) has a solution. If 1 is not an eigenvalue ofM, i.e., is injective (), then the successive approximations
Proof From the hypothesis and by virtue of Theorem 2.6, we have
and (2.4) allows us to conclude that
Remark 2.2 In many situations, some boundary value problems for partial differential equations which are ill-posed can be reduced to Fredholm operator equations of the first kind of the form , where B is compact, positive, and self-adjoint operator in a Hilbert space H. This equation can be rewritten in the following way:
where , and ω is a positive parameter satisfying . It is easily seen that the operator L is non-expansive and 1 is not an eigenvalue of L. It follows from Theorem 2.7 that the sequence converges and for every as .
3 Ill-posedness and stabilization of the inverse boundary value problem
3.1 Cauchy problem with Dirichlet conditions
Consider the following well-posed boundary value problem:
where ξ is an H-valued function.
Definition 3.1 [, p.250]
Proof By using the Fourier expansion and the given Dirichlet boundary conditions
This differential equation admits two linearly independent fundamental solutions
Thus, its general solution can be written as
Remark 3.1 It is easy to check that the expression (3.3) solves the problem
3.2 Inverse boundary value problem
The operator equation (3.5) is the main instrument in investigating problem (3.4). More precisely, we want to study the following properties:
1. Injectivity of K (identifiability);
2. Continuity of K and the existence of its inverse (stability);
3. The range of K.
Now, to conclude the solvability of problem (3.4) it is enough to apply Theorem 2.5.
Corollary 3.1The inverse problem (3.4) is uniquely solvable if and only if
In this case, we have
In this case, we have
From this representation, we see that:
3.3 Regularization by truncation method and error estimates
Remark 3.2 If the parameter N is large, is close to the exact solution f. On the other hand, if the parameter N is fixed, is bounded. So, the positive integer N plays the role of regularization parameter.
Remark 3.3 In view of
As usual, in order to obtain convergence rate, we assume that there exists an a priori bound for problem (1.2)
The main theorem of this method is as follows.
Proof From direct computations, we have
Using the triangle inequality
Finally, from (3.4) and (3.15), we deduce the following corollary.
4 Regularization by the Kozlov-Maz’ya iteration method and error estimates
In [11,12] Kozlov and Maz’ya proposed an alternating iterative method to solve boundary value problems for general strongly elliptic and formally self-adjoint systems. After that, the idea of this method has been successfully used for solving various classes of ill-posed (elliptic, parabolic and hyperbolic) problems; see, e.g., [13-15].
In this section we extend this method to our ill-posed problem.
4.1 Description of the method
where ω is such that
This implies that
In general, the exact solution is required to satisfy the so-called source condition; otherwise, the convergence of the regularization method approximating the problem can be arbitrarily slow. Since our problem is exponentially ill-posed (the eigenvalues of K converge exponentially to 0), it is well known in this case [16,17] that the best choice to accelerate the convergence of the regularization method is to use logarithmic-type source conditions, i.e.,
Remark 4.1 [, p.34]
Proof The proof is based on the following equivalence:
Lemma 4.1 [, Appendix, Lemma A.1]
Proof Using the mean value theorem, we can write
Let us consider the following real-valued functions:
Now we are in a position to state the main result of this method.
Proof By virtue of Proposition 4.1 and Theorem 2.7, it follows immediately
and by virtue of Lemma 4.1 (estimate (4.7)), we conclude the desired estimate. □
Theorem 4.2Letandωsatisfy, letbe an arbitrary element for the iterative procedure suggested above, and let (resp. ) be thekth approximate solution for the exact datag (resp. for the inexact data) such that. Then, under condition (4.6), the following inequality holds:
Proof Using (4.4) and the triangle inequality, we can write
5 Numerical results
In this section we give a two-dimensional numerical test to show the feasibility and efficiency of the proposed methods. Numerical experiments were carried out using MATLAB.
We consider the following inverse problem:
It is easy to check that the operator
is positive, self-adjoint with compact resolvent (A is diagonalizable).
In this case, formula (3.7) takes the form
where ε indicates the noise level of the measurement data and the function ‘’ generates arrays of random numbers whose elements are normally distributed with mean 0, variance , and standard deviation . ‘’ returns an array of random entries that is the same size as g. The bound on the measurement error δ can be measured in the sense of Root Mean Square Error (RMSE) according to
Kozlov-Maz’ya iteration method
is a symmetric, positive definite matrix. We assume that it is fine enough so that the discretization errors are small compared to the uncertainty δ of the data; this means that is a good approximation of the differential operator , whose unboundedness is reflected in a large norm of (see [, p.5]). The eigenpairs of are given by
The discrete iterative approximation of (4.12) takes the form
Figure 1. TM with (,,).
Figure 2. TM with (,,).
Figure 3. TM with (,,).
Figure 4. TM with (,,).
Table 1. Truncation method: Relative error
Table 2. Kozlov-Maz’ya method: Relative error
The numerical results (Figures 1-4) are quite satisfactory. Even with the noise level , the numerical solutions are still in good agreement with the exact solution. In addition, the numerical results (Figures 5-12) are better for (, ) and (, ) and the other values are also acceptable.
In this study, a convergent and stable reconstruction of an unknown boundary condition has been obtained using two regularizing methods: truncation method and Kozlov-Maz’ya iteration method. Both theoretical and numerical studies have been provided.
Future work will involve the error effect arising in computing eigenfunctions and eigenvalues of the operator A on the truncation method. The question is how to obtain some optimal balance between the accuracy of eigensystem and the noise level of input data.
The authors declare that they have no competing interests.
All authors have contributed equally. All authors read and approved the final manuscript.
The authors would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of our paper. This work is supported by the DGRST of Algeria (PNR Project 2011-code: 8\92 u23\92 997).
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