Two regularization methods for a class of inverse boundary value problems of elliptic type
© Bouzitouna et al.; licensee Springer 2013
Received: 13 March 2013
Accepted: 18 July 2013
Published: 2 August 2013
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.
1 Formulation of the problem
Throughout this paper, H denotes a complex separable Hilbert space endowed with the inner product and the norm , stands for the Banach algebra of bounded linear operators on H.
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.
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.
where are positive weights with and ε denotes the level noisy.
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
Theorem 2.1 [, Theorem 6, XII.2.5, pp.1196-1198]
, , ,
. In particular, if Φ is a real Borel function, then is self-adjoint.
We denote by , , the -semigroup generated by . Some basic properties of are listed in the following theorem.
Theorem 2.2 (see , Chapter 2, Theorem 6.13, p.74)
the function , , is analytic;
for every real and , the operator ;
for every integer and , ;
for every , , we have .
Theorem 2.3 For , is self-adjoint and one-to-one operator with dense range (, ).
Proof Let , . Then, by virtue of (iv) of Theorem 2.1, we can write .
Let , , then , which implies that , . Using analyticity, we obtain that , . Strong continuity at 0 now gives . This shows that .
we conclude that is dense in H. □
Theorem 2.4 [, Theorem 7.5, p.191]
On the other hand, Theorem 2.4 provides smoothness results with respect to y: whenever , . Under this same hypothesis, we also have smoothness in space: , .
Here we recall a crucial theorem in the analysis of the inverse problems.
Theorem 2.5 [, Generalized Picard theorem, p.502]
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].
for non-expansive operators M.
converge to for any initial data .
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
where ξ is an H-valued function.
Definition 3.1 [, p.250]
Theorem 3.1 Problem (3.1) admits a unique generalized (resp. classical) solution if and only if (resp. ).
If (resp. ), by virtue of Theorem 2.4 and Remark 3.1, we easily check the inclusion (resp. ) and for . □
3.2 Inverse boundary value problem
Injectivity of K (identifiability);
Continuity of K and the existence of its inverse (stability);
The range of K.
Now, to conclude the solvability of problem (3.4) it is enough to apply Theorem 2.5.
From this representation, we see that:
is stable in the interval ();
u is unstable in . This follows from the high-frequency , .
3.3 Regularization by truncation method and error estimates
A natural way to stabilize the problem is to eliminate all the components of large n from the solution and instead consider (3.7) only for .
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.
Since the data g are based on (physical) observations and are not known with complete accuracy, we assume that g and satisfy , where denotes the measured data and δ denotes the level noisy.
where is a given constant.
The main theorem of this method is as follows.
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
Proposition 4.1 The operator is self-adjoint and non-expansive on H. Moreover, it has not 1 as eigenvalue.
Proof The self-adjointness follows from the definition of G (see Theorem 2.1). Since the inequality for , we have , then 1 is not an eigenvalue of G. □
Remark 4.1 [, p.34]
The logarithmic source condition is equivalent to the inclusion .
Lemma 4.1 [, Appendix, Lemma A.1]
Now we are in a position to state the main result of this method.
and by virtue of Lemma 4.1 (estimate (4.7)), we conclude the desired estimate. □
Combining (4.13) and (4.14) and taking the supremum with respect to of , we obtain the desired bound.
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.
where is the unknown source and is the supplementary condition.
is positive, self-adjoint with compact resolvent (A is diagonalizable).
In the following, we consider an example which has an exact expression of solutions .
If , then the function is the exact solution of problem (5.1). Consequently, the data function is .
Kozlov-Maz’ya iteration method
where and .
Truncation method: Relative error
Kozlov-Maz’ya method: Relative error
0.5 × 2.7881 = 1.3941
2/3 × 2.7881 = 1.8587
0.7 × 2.7881 = 1.9517
0.8 × 2.7881 = 2.2305
0.5 × 2.7881 = 1.3941
2/3 × 2.7881 = 1.8587
0.7 × 2.7881 = 1.9517
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 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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