Research Article | Open Access
Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem
We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.
Throughout this paper denotes a complex Hilbert space endowed with the inner product , and the norm , stands for the Banach algebra of bounded linear operators on .
Let be a linear unbounded operator with dense domain . Assume that is self-adjoint, positive definite in , which has a continuous spectrum , .
We consider the elliptic Cauchy problem (ECP) of finding such that where is some prescribed data in the Hilbert space .
This problem is an abstract version of Cauchy problem, which generalizes Cauchy problem for second-order elliptic partial differential equations in a cylindrical domain; for example, we mention the following problem.
Example 1. An example of (1) is the Cauchy problem for the modified Helmholtz equation in the infinite strip : where the operator is given by It is well known that this operator is self-adjoint with continuous spectrum We note here that the discrete eigenfunctions expansion method cannot be used, but we can use the Fourier diagonalization method to deal with this kind of problems.
Such problem arises in many practical situations, nondestructive testing techniques , geophysics , cardiology , and other applications. There are many various monographs about the historical development of this topic, for more details, we refer the reader to [5, 6]. Recently there has been an excellent topic review  of this problem.
Because problem (1) is severely ill-posed; that is, a small perturbation in the given Cauchy data may result in a very large error on the solution. In order to overcome this instability character, the regularization methods are required.
Some regularization methods for the Cauchy problem for elliptic equations have been proposed by many authors. For instance: Tikhonov regularization method , the quasi-reversibility method , the quasi-boundary-value method [10–13], Kozlov-Maz’ya iteration method , and the mollification method .
This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when is positive and induces the elliptic case, that is, has the following properties: for any , the resolvent exists and satisfies the estimates
In the case when is a linear positive self-adjoint operator with compact inverse, problem (1) has been treated by a different method and there is a large literature in this direction. However, in the case where has a continuous spectrum the literatures are quite scarce.
In the present paper we shall use two spectral regularization methods to construct a stable solution to our original ill-posed problem.
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. Spectral Theorem and Properties
By the spectral theorem, for each positive self-adjoint operator , there is a unique right continuous family of orthogonal projection operators such that with
In our case, we have because , .
Theorem 2 (see [16, Theorem 6, XII.2.5, pages 1196–1198]). Let be the spectral resolution of the identity associate to and let be a complex Borel function defined -almost everywhere on the real axis. Then is a closed operator with dense domain. Moreover(i),(ii), , ,(iii), ,(iv). In particular, if is real Borel function, then is self-adjoint, (v)the operator is bounded if and only if is bounded on . In this case, .
We denote by , , the -semigroup generated by . Some basic properties of are listed in the following theorem.
Theorem 3 (see [17, chapter 2, Theorem 6.13, page 74]). For this family of operators one has:(1), ; (2)the function , , is analytic; (3)for every real and , the operator ; (4)for every integer and , ; (5)for every , , one has .
Theorem 4. For , is self-adjoint and one to one operator with dense range (, ).
Proof. Let , . Then by virtue of (iv) of Theorem 2, we can write .
Let , , then , which implies that , . Using analyticity, we obtain that , . Strong continuity at now gives . This shows that .
Thanks to we conclude that is dense in .
Remark 5. For , , this Theorem ensures us 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 with dense domain
Definition 6 (Hilbert scales). Let be an unbounded self-adjoint strictly positive definite operator in . Following the definition in , one introduces the Hilbert scale according to definition
Here is a Hilbert space with inner product and norm .
In our setting we take . In this case, one has
Following , one defines the following.
Definition 7 (mollification operator). For and , one introduces the Yosida approximation of identity
Remark 8. The idea of the mollification method is very simple and natural: if the data are given inexactly, then we try to find a sequence of mollification operators which map the improper data into well-posed classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed.
Theorem 9. One has(1) and , (2)for all , ,(3)for all , .
Proof. (1) .
(3) Assume that . Then and thus, .
Now for , there exists such that , (since is dense in (see Remark 5)). We have Thus, , , and so .
Using the change of variables
Cauchy’s problem (1) reduces to the two Cauchy problems Thus, the solution of the original problem (1) can be written in the form It is well known that the operator generates a strongly continuous analytic semigroup . In addition, the spectral radius of the semigroup for any . Hence, it follows that the Cauchy problem (15) is well-posed and its solution may be written in the form As opposed to problem (15), Cauchy problem (15) (backward parabolic equation) is not correctly posed, and its (unique) formal solution is given by
Remark 10 (see [19, page 375]). The uniqueness solvability of problem (16) results from the logarithmic convexity of the function :
A useful characterization of the admissible set for which problem (16) has a solution is as follows.
Lemma 11. Problem (16) has a solution if and only if , and its unique solution is represented by
Proof. Suppose that problem (16) has a solution . Then The function if and only if Observing that the function is increasing, then This last inequality is exactly equivalent to .
As a consequence, we have the following corollary.
Corollary 12. Problem (1) has a solution if and only if , and its unique solution is represented by
3. Regularization and Error Estimates
3.1. The Truncation Method
From (25) we can see that the term is the cause of unstability. In order to overcome the ill-posedness of problem (1), we modify the solution by filtering the high frequencies using a suitable method and instead consider (25) only for , where is some constant which satisfies .
According to spectral theory of self-adjoint operators , for any bounded Borel set , we can define the orthogonal projection
To solve (1) in a stable way we approximate by its projection , and instead of considering (1) with we take its projected version where is the characteristic function of the interval for . The quantity is referred to as a cut-off frequency.
Let (resp., ) be the exact (resp., the measured data) at , such that .
The approximated solution corresponding to the measured data is denoted by For simplicity, we denote the solution of problem (1) by , and the regularized solution associated to the data by .
Our first main theorem is the following theorem.
Theorem 13. The solution defined in (27) depends continuously in on the data ; that is, if and are two regularized solutions corresponding to and , respectively, then one has
This inequality implies that the solution of the regularized problem (27) depends continuously on the data .
Now we compute the difference between the original solution and the approximate solution .
Theorem 14. Let be a solution of problem (1) with the exact data ; then the following estimate holds:
Remark 15. If we choose , where , then we have the error bound
From (36) we see that (28) is an approximation of the exact solution . The approximation error depends continuously on the measurement error for fixed . However, as , the accuracy of the regularized solution becomes progressively lower. Consequently, we have not any information about the continuous dependence of the solution if is close to .
In the theory of ill-posed Cauchy problems, we can often obtain continuous dependence on the data for the closed interval by assuming additional smoothness and using a stronger norm.
Now we show two error estimates under the following conditions:(H1),(H2), .
Remark 16. In practice, we know that it is very difficult to verify the conditions (H1) and (H2), so we give different assumptions on the given data as follows:
Theorem 17. If (resp., ), , , then one has the following estimates:
Proof. From the expansions
Then Using Theorem 17 and the triangle inequality, we can write By choosing , we obtain the desired inequality.
Using the same techniques we have hence Using (29) and the triangle inequality, we obtain By choosing , we obtain
3.2. The Mollification Method
Now, we approximate the original problem (1) by the sequence of problems
Theorem 18. If the approximate Cauchy problem (47) admits a unique solution , which depends continuously upon the data with respect to uniform topology of .
Proof. From the representation
(i)If , we obtain
(ii)If , the function with achieves its maximum at , , from which we deduce
From this bound, we derive
From the linear property of our problem, stability estimate of problem (47) may be written precisely in the following corollary.
Corollary 19. If (resp., ) is the approximate solution corresponding to (resp., ), then
Remark 20. We have
It is easy to show that
This remark shows that for all .
Proof. The inclusion is equivalent to . We have where .
Theorem 21. If , then
Proof. We compute
where and .
This implies that and by virtue of of Theorem 9, we conclude the desired convergence.
The following technical lemmas play the key role in our analysis and calculations.
Lemma 22. Let where , , , , , and . Then one has where , .
Proof. Differentiating the expression and setting the derivative equal to zero, we find The function admits a unique solution Therefore We have By using the inequality (, ), then for , we obtain and we can write which implies that . Hence, we obtain where , .
Lemma 23. Let
where , , , , and . Then one has the following.
If , then
If , , , then
Proof. By a simple differential calculus, we show that the function achieves its maximum at . Consequently
Now we assume the following a priori bounds hold:
Using the change of variables , we obtain the new expressions
By virtue of Lemma 22 (inequality (60) and Lemma 23 (inequalities (68) and (69)), we can write where , . Consider
We have Using (78) and (79), we derive
Corollary 25. Let (resp., ) be the solution of problem (1) (resp., (47)) with the exact data (resp., the inexact data ) such that . If (71) (resp., (72)) is satisfied, then one has the following error estimates: where
If we choose with , then we have
3.3. Example: Cauchy Problem for the Modified Helmholtz Equation
In this paragraph, we give a concrete example to see how to apply the theoretical results developed in this study.
Let us consider the Cauchy problem (modified Helmholtz equation) in the infinite strip : where is a real positive constant.
Let be the Fourier transform of : With the help of the Fourier transformation, problem (1) can be transformed to an equivalent problem in the frequency domain: It is easy to check that the formal solution of problem (92) has the form or equivalently, the formal solution of problem (90) is given by
Putting . Then as . From this remark, it is easy to see that a small perturbation in the data may cause a dramatically large error in the solution . In addition, the magnifying factor is , hence, the problem is severely ill-posed.
Since the data are based on (physical) observations and are not known with complete accuracy, we assume that and satisfy where and belong to , denotes the measured data, and denotes the noise level.
For this problem, we define the regularized solutions with noisy data : where is the characteristic function of the interval . Consider where . The quantities and are the parameters which were defined in Sections 3.1 and 3.2.
4. The Nonlocal Boundary Value Problem Method and Some Extensions
In this section we give the connection between the mollification method and the nonlocal boundary value problem method; also we give some extensions to our investigation.
4.1. The Nonlocal Boundary Value Problem Method
Consider the following problem: (i)The case coincides with the method treated by Hào et al. in . (ii)In the case , the solution of is coincides with the solution resulting from the mollification method. (iii)The error estimates obtained in our analysis by using the mollification method are similar to those obtained in [8, 10].
This shows that our study framework is more general and includes many results obtained in this direction.
Let us consider where is a self-adjoint, linear unbounded operator in and changes the sign with ( exists and ).
We assume , . The spectral theory of self-adjoint operators enables us to write
that is, the Hilbert space decomposes into the direct sum , and
This decomposition gives us two problems: one is elliptic (ill-posed) and the other is hyperbolic (well-posed). Consider
The formal solution of problem (99) is