Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

and Applied Analysis 3 the definition in [18], one introduces theHilbert scale {Gθ}θ≥0 according to definition G0 = H, Gθ = D(B θ ) . (9) Here Gθ is a Hilbert space with inner product (ξ, ζ)θ = (B θ ξ, B θ ζ) and norm ‖ ξ‖ θ = (B θ ξ, B θ ξ). In our setting we takeB = S(L) = e√. In this case, one has Gθ = {h ∈ H : ‖h‖ 2 θ = 󵄩󵄩󵄩󵄩󵄩 e θL√A h 󵄩󵄩󵄩󵄩󵄩 2


Introduction
Throughout this paper  denotes a complex Hilbert space endowed with the inner product (⋅, ⋅), and the norm ‖⋅‖, L() stands for the Banach algebra of bounded linear operators on .
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 R × (0, 1) [1]:   (, ) +   (, ) −  (, ) = 0,  ∈ R,  ∈ (0, ) ,  (, 0) =  () ,   (, 0) = 0,  ∈ R, (2) where the operator   is given by It is well known that this operator is self-adjoint with continuous spectrum  (  ) =  ( 0 ) +  = [0, +∞[ +  = [, +∞[ .(4) 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 [2], geophysics [3], cardiology [4], 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 [7] 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 [8], the quasireversibility method [9], the quasi-boundary-value method [10][11][12][13], Kozlov-Maz'ya iteration method [14], and the mollification method [15].
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  ∈ (−∞, 0], the resolvent (; ) = ( − ) −1 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 illposed problem.

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.

Spectral Theorem and Properties
. By the spectral theorem, for each positive self-adjoint operator , there is a unique right continuous family In our case, we have  = ∫ ∞    because  ≥ ,  > 0.
Let {  ,  ≥  > 0} 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 We denote by () =  −√ = ∫ ∞   − √    ∈ L(),  ≥ 0, the  0 -semigroup generated by − √ .Some basic properties of () are listed in the following theorem.
Thanks to we conclude that (( 0 )) is dense in .
Definition 7 (mollification operator).For  > 0 and  ≥ 1, 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.
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 − 1/2 generates a strongly continuous analytic semigroup In addition, the spectral radius of the semigroup ‖ −1/2 ()‖ < 1 for any  > 0. 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) This last inequality is exactly equivalent to  ∈ G 1 .
As a consequence, we have the following corollary.

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 lim  → 0 () = +∞.
According to spectral theory of self-adjoint operators [20], for any bounded Borel set Δ  = { ≤  ≤ } ⊆ () = [, +∞[, 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 1 [,] is the characteristic function of the interval [, ] for  < .The quantity  is referred to as a cut-off frequency.
The approximated solution V   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 V   ().
Our first main theorem is the following theorem.
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 V   = V   (;   ).
Theorem 14.Let  ∈ ([0, ]; ) be a solution of problem (1) with the exact data  ∈ ; then the following estimate holds: Proof.From relations (25) and (27) we have then Using the inequality From (36) we see that ( 28) is an approximation of the exact solution ().The approximation error depends continuously on the measurement error for fixed 0 <  < .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 [0, ] by assuming additional smoothness and using a stronger norm.Now we show two error estimates under the following conditions: (H1) () ∈ (  ), (H2) () ∈ G  ,  > 0.
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: 2 ),  > 0,  > 0, then one has the following estimates: Proof.From the expansions we have Then Proof.From the representation we have Remark 20.We have It is easy to show that This remark shows that   () ∈ ( /2 ) for all  ∈ [0, ].
Since the data (⋅) are based on (physical) observations and are not known with complete accuracy, we assume that  and   satisfy      −       ≤ , where  and   belong to  2 (R),   denotes the measured data, and  denotes the noise level.
For this problem, we define the regularized solutions with noisy data   : ) f () , where  ≥ 1.The quantities  = () and  = () are the parameters which were defined in Sections 3.1 and 3.2.

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.
has a solution is as follows.