Abstract
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 illposed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the illposed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the KozlovMaz’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:
illposed problems; elliptic problems; cutoff spectral regularization; iterative regularization1 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.
Let be a positive, selfadjoint operator with compact resolvent, so that A has an orthonormal basis of eigenvectors with real eigenvalues , i.e.,
In this paper, we are interested in the following inverse boundary value problem: find satisfying
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 secondorder elliptic partial differential equations in a cylindrical domain, for example we mention the following problem.
Example 1.1 An example of (1.1) is the boundary value problem for the Laplace equation in the strip , where the operator A is given by
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 [3], 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 [2], 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
where are positive weights with and ε denotes the level noisy.
The regularizing strategy employed in [2] is essentially based on the Tikhonov regularization and the conditional stability estimate for some a priori constant E.
In practice, the use of Nmeasurements 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.
2.1 Notation
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 wellknown facts in spectral theory.
2.2 Spectral theorem and properties
By the spectral theorem, for each strictly positive selfadjoint operator B,
there is a unique right continuous family of orthogonal projection operators such that with
Theorem 2.1 [[4], Theorem 6, XII.2.5, pp.11961198]
Letbe the spectral resolution of the identity associated toB, and let Φ be a complex Borel function definedEalmost everywhere on the real axis. Thenis a closed operator with dense domain. Moreover,
(iv) . In particular, if Φ is a real Borel function, thenis selfadjoint.
We denote by , , the semigroup generated by . Some basic properties of are listed in the following theorem.
Theorem 2.2 (see [5], Chapter 2, Theorem 6.13, p.74)
For this family of operators, we have:
2. the function, , is analytic;
3. for every realand, the operator;
Theorem 2.3For, is selfadjoint and onetoone 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 .
Thanks to
we conclude that is dense in H. □
Remark 2.1 For , this theorem ensures that is selfadjoint and onetoone operator with dense range . Then we can define its inverse , which is an unbounded selfadjoint strictly positive definite operator in H with dense domain
Let us consider the following problem: for find such that
Theorem 2.4 [[6], Theorem 7.5, p.191]
For any, problem (2.1) has a unique solution, given by
Moreover, for all integer, . If, in addition, , thenand
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 [[7], Generalized Picard theorem, p.502]
Letbe a selfadjoint 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
Moreover,
On the basis , we introduce the Hilbert scale (resp. ) induced by as follows:
2.3 Nonexpansive operators
Definition 2.1 A linear operator is called nonexpansive if
Theorem 2.6 [[8], Theorem 2.2]
Letbe a positive, selfadjoint operator with. Puttingand, we have
i.e.,
For more details concerning the theory of nonexpansive operators, we refer to Krasnosel’skii et al. [[9], p.66].
Let use consider the operator equation
for nonexpansive operators M.
Theorem 2.7LetMbe a linear selfadjoint, positive and nonexpansive 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
converge tofor any initial data.
Proof From the hypothesis and by virtue of Theorem 2.6, we have
By induction with respect to n, it is easily seen that has the explicit form
and (2.4) allows us to conclude that
□
Remark 2.2 In many situations, some boundary value problems for partial differential equations which are illposed can be reduced to Fredholm operator equations of the first kind of the form , where B is compact, positive, and selfadjoint 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 nonexpansive and 1 is not an eigenvalue of L. It follows from Theorem 2.7 that the sequence converges and for every as .
3 Illposedness and stabilization of the inverse boundary value problem
3.1 Cauchy problem with Dirichlet conditions
Consider the following wellposed boundary value problem:
where ξ is an Hvalued function.
Definition 3.1 [[10], p.250]
• A function is called a generalized solution to equation (3.1) if , and for all , and obeys equation (3.1) on the same interval .
• A function is called a classical solution to equation (3.1) if , and for all , and obeys equation (3.1) on the same interval .
Theorem 3.1Problem (3.1) admits a unique generalized (resp. classical) solution if and only if (resp. ).
Proof By using the Fourier expansion and the given Dirichlet boundary conditions
we obtain
This differential equation admits two linearly independent fundamental solutions
Thus, its general solution can be written as
Applying and yields and . Finally, the solution of (3.2) is
Remark 3.1 It is easy to check that the expression (3.3) solves the problem
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
Our inverse problem is to determine from the supplementary condition , then we get
We define
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.
It is easy to see that K is a linear compact selfadjoint operator with the singular values , and by virtue of Remark 2.1, we have
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 other words, the solution f of the inverse problem is obtained from the data g via the unbounded operator defined on functions g in the subspace
Corollary 3.2Problem (1.1)(1.2) admits a unique solutionif and only if
In this case, we have
From this representation, we see that:
• is stable in the interval ();
• u is unstable in . This follows from the highfrequency , .
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 .
Definition 3.2 For , the regularized solution of problem (1.1)(1.2) is given by
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
implies
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.
Let denote the regularized solution of problem (1.1), (1.2) with measured data :
As usual, in order to obtain convergence rate, we assume that there exists an a priori bound for problem (1.2)
Remark 3.4 For given two exact conditions and , let and be the corresponding regularized solutions, respectively. Then
The main theorem of this method is as follows.
Theorem 3.2Letbe the regularized solution given by (3.11), and letfbe the exact solution given by (3.7). If, and if we choose, , then we have the error bound
Proof From direct computations, we have
Using the triangle inequality
we obtain
□
Finally, from (3.4) and (3.15), we deduce the following corollary.
Corollary 3.3Letbe the regularized solution given by (3.12), and letube the exact solution given by (3.8). If, and if we choose, , then we have the error bound
4 Regularization by the KozlovMaz’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 selfadjoint systems. After that, the idea of this method has been successfully used for solving various classes of illposed (elliptic, parabolic and hyperbolic) problems; see, e.g., [1315].
In this section we extend this method to our illposed problem.
4.1 Description of the method
The iterative algorithm for solving the inverse problem (1.1)(1.2) starts by letting be arbitrary. The first approximation is the solution to the direct problem
If the pair has been constructed, let
where ω is such that
Finally, we get by solving the problem
We set . If we iterate backwards in , we obtain
This implies that
Proposition 4.1The operatoris selfadjoint and nonexpansive on H. Moreover, it has not 1 as eigenvalue.
Proof The selfadjointness follows from the definition of G (see Theorem 2.1). Since the inequality for , we have , then 1 is not an eigenvalue of G. □
In general, the exact solution is required to satisfy the socalled source condition; otherwise, the convergence of the regularization method approximating the problem can be arbitrarily slow. Since our problem is exponentially illposed (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 logarithmictype source conditions, i.e.,
where
Remark 4.1 [[16], p.34]
The logarithmic source condition is equivalent to the inclusion .
Proof The proof is based on the following equivalence:
□
Lemma 4.1 [[18], Appendix, Lemma A.1]
Letand, . Then the realvalued functiondefined onsatisfies
Remark 4.2 Let . Then the realvalued function defined on satisfies
Proof Using the mean value theorem, we can write
then
□
Let us consider the following realvalued functions:
Using the change of variables , we obtain
Now we are in a position to state the main result of this method.
Theorem 4.1Letandωsatisfy, letbe an arbitrary element for the iterative procedure suggested above, and letbe thekth approximate solution. Then we have
Moreover, if (), i.e., , , , then the rate of convergence of the method is given by
Proof By virtue of Proposition 4.1 and Theorem 2.7, it follows immediately
We have
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
where
and
By using inequality (4.8), the quantity can be estimated as follows:
Combining (4.13) and (4.14) and taking the supremum with respect to of , we obtain the desired bound.
Remark 4.3 Choosing such that as , we obtain
□
5 Numerical results
In this section we give a twodimensional 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:
where is the unknown source and is the supplementary condition.
It is easy to check that the operator
is positive, selfadjoint with compact resolvent (A is diagonalizable).
In this case, formula (3.7) takes the form
Truncation method
We use trapezoid’s rule to approach the integral and do an approximate truncation for the series by choosing the sum of the front terms. After considering an equidistant grid , , , we get
In the following, we consider an example which has an exact expression of solutions .
Example
If , then the function is the exact solution of problem (5.1). Consequently, the data function is .
Adding a random distributed perturbation (obtained by the Matlab command randn) to each data function, we obtain the vector :
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
Using as a data function, we obtain the computed approximation by (5.5). The relative error is given by
KozlovMaz’ya iteration method
By using the central difference with step length to approximate the first derivative and the second derivative , we can get the following semidiscrete problem (ordinary differential equation):
where is the discretization matrix stemming from the operator :
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 [[19], p.5]). The eigenpairs of are given by
The discrete iterative approximation of (4.12) takes the form
Figures 14, Table 1 show the comparisons between the exact solution and its computed approximations for different values N, M and ε.
Figure 1. TM with (,,).
Figure 2. TM with (,,).
Figure 3. TM with (,,).
Figure 4. TM with (,,).
Table 1. Truncation method: Relative error
Figures 512, Table 2 show the comparisons between the exact solution and its computed approximations for different values N, k, ω and ε.
Table 2. KozlovMaz’ya method: Relative error
Conclusion
The numerical results (Figures 14) 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 512) 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 KozlovMaz’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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have contributed equally. All authors read and approved the final manuscript.
Acknowledgements
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 2011code: 8\92 u23\92 997).
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