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Two regularization methods for a class of inverse boundary value problems of elliptic type
Boundary Value Problems volume 2013, Article number: 178 (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.
Let be a positive, self-adjoint 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 second-order 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 , 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
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
By the spectral theorem, for each strictly positive self-adjoint operator B,
there is a unique right continuous family of orthogonal projection operators such that with
Theorem 2.1 [, Theorem 6, XII.2.5, pp.1196-1198]
Let be the spectral resolution of the identity associated to B, and let Φ be a complex Borel function defined E-almost everywhere on the real axis. Then is a closed operator with dense domain. Moreover,
, , ,
. 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)
For this family of operators, we have:
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. □
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
Let us consider the following problem: for find such that
Theorem 2.4 [, Theorem 7.5, p.191]
For any , problem (2.1) has a unique solution, given by
Moreover, for all integer , . If, in addition, , then and
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]
Let be a self-adjoint operator and the Hilbert space H, and let be its spectral resolution of unity. Let and . We suppose that the set either is empty or contains isolated point only. Then the vectorial equation
is solvable if and only if
On the basis , we introduce the Hilbert scale (resp. ) induced by as follows:
2.3 Non-expansive operators
Definition 2.1 A linear operator is called non-expansive if
Theorem 2.6 [, Theorem 2.2]
Let be a positive, self-adjoint operator with . Putting and , we have
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 2.7 Let M be a linear self-adjoint, positive and non-expansive operator on H. Let be such that equation (2.3) has a solution . If 1 is not an eigenvalue of M, i.e., is injective (), then the successive approximations
converge to for 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 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]
Theorem 3.1 Problem (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
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
The operator equation (3.5) is the main instrument in investigating problem (3.4). More precisely, we want to study the following properties:
Injectivity of K (identifiability);
Continuity of K and the existence of its inverse (stability);
The range of K.
It is easy to see that K is a linear compact self-adjoint 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 3.1 The 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 3.2 Problem (1.1)-(1.2) admits a unique solution if 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 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 .
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
and if , i.e., , then
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)
where is a given constant.
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 3.2 Let be the regularized solution given by (3.11), and let f be 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
By choosing , , we obtain
Finally, from (3.4) and (3.15), we deduce the following corollary.
Corollary 3.3 Let be the regularized solution given by (3.12), and let u be the exact solution given by (3.8). If , and if we choose , , then we have the error bound
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
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 4.1 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. □
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]
The logarithmic source condition is equivalent to the inclusion .
Proof The proof is based on the following equivalence:
Lemma 4.1 [, Appendix, Lemma A.1]
Let and , . Then the real-valued function defined on satisfies
Remark 4.2 Let . Then the real-valued function defined on satisfies
Proof Using the mean value theorem, we can write
Let us consider the following real-valued functions:
Using the change of variables , we obtain
Now we are in a position to state the main result of this method.
Theorem 4.1 Let and ω satisfy , let be an arbitrary element for the iterative procedure suggested above, and let be the kth 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
and by virtue of Lemma 4.1 (estimate (4.7)), we conclude the desired estimate. □
Theorem 4.2 Let and ω satisfy , let be an arbitrary element for the iterative procedure suggested above, and let (resp. ) be the kth approximate solution for the exact data g (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
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 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:
where is the unknown source and is the supplementary condition.
It is easy to check that the operator
is positive, self-adjoint with compact resolvent (A is diagonalizable).
The eigenpairs of A are
In this case, formula (3.7) takes the form
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 .
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
Kozlov-Maz’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 semi-discrete 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 [, p.5]). The eigenpairs of are given by
The discrete iterative approximation of (4.12) takes the form
where and .
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.
Cosner C, Rundell W: Extension of solutions of second order partial differential equations by the method of quasireversibility. Houst. J. Math. 1984, 10(3):357-370.
Levine HA, Vessella S: Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type. Rend. Circ. Mat. Palermo 1985, 34: 141-160. 10.1007/BF02844893
Ivanov DY: Inverse boundary value problem for an abstract elliptic equation. Differ. Equ. 2000, 36(4):579-586. 10.1007/BF02754252
Dunford N, Schwartz J: Linear Operators, Part II. Wiley, New York; 1967.
Pazy A: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer, New York; 1983.
Brezis H: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York; 2011.
Prilepko AI, Orlovsky DG, Vasin IA Monographs and Textbooks in Pure and Applied Mathematics 222. In Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker, New York; 2000.
Shlapunov A: On iterations of non-negative operators and their applications to elliptic systems. Math. Nachr. 2000, 218: 165-174. 10.1002/1522-2616(200010)218:1<165::AID-MANA165>3.0.CO;2-W
Krasnosel’skii MA, Vainikko GM, Zabreiko PP, Rutitskii YB: Approximate Solutions of Operator Equations. Wolters-Noordhoff, Groningen; 1972.
Krein SG: Linear Differential Equations in Banach Space. Am. Math. Soc., Providence; 1971.
Kozlov VA, Maz’ya VG: On iterative procedures for solving ill-posed boundary value problems that preserve differential equations. Leningr. Math. J. 1990, 1: 1207-1228.
Kozlov VA, Maz’ya VG, Fomin AV: An iterative method for solving the Cauchy problem for elliptic equations. U.S.S.R. Comput. Math. Math. Phys. 1991, 31(1):45-52.
Bastay G Linköping Studies in Science and Technology. In Iterative Methods for Ill-Posed Boundary Value Problems. Linköping University, Linköping; 1995. Dissertations No. 392
Baumeister J, Leitao A: On iterative methods for solving ill-posed problems modeled by partial differential equations. J. Inverse Ill-Posed Probl. 2001, 9(1):13-29.
Maxwell, D: Kozlov-Maz’ya iteration as a form of Landweber iteration (2011). arXiv:1107.2194v1 [math.AP] 12 Jul.
Bakushinsky AB, Kokurin MY: Iterative Methods for Approximate Solution of Inverse Problems. Springer, Dordrecht; 2004.
Hohage T: Regularization of exponentially ill-posed problems. Numer. Funct. Anal. Optim. 2000, 21: 439-464. 10.1080/01630560008816965
Deuflhardy P, Engl HW, Scherzer O: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Probl. 1998, 14: 1081-1106. 10.1088/0266-5611/14/5/002
Eldén L, Simoncini V: A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces. Inverse Probl. 2009., 25: Article ID 065002 (22pp)
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).
The authors declare that they have no competing interests.
All authors have contributed equally. All authors read and approved the final manuscript.
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