Modified Tikhonov Method for Cauchy Problem of Elliptic Equation with Variable Coefficients

A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.


Introduction
In this paper, we consider the following Cauchy problem for the elliptic equation with variable coefficients in a strip region where , , a b c are given functions such that for given positive constants 1 2 Λ ≤ Λ , Without loss of generality, in the following section we suppose that 1 1 Λ ≥ .Let ( ) 2 0,1 Ω = × ⊂   , as in [1], we assume that the unique solution of problem (1) exists in ( )

2
H Ω for the exact Cauchy data ( ) ( ) This problem is severely ill-posed and the regularization methods are required to stabilize numerical computations [2] [3].
In 2007, Hào et al. [4] regularized problem (1) by adopting Poisson kernel to mollify the Cauchy data, and prove some condition stability estimates of H o  lder and logarithm types for the solution and its derivatives.In 2008, Qian [5] used a wavelet regularization method to treat it.In 2010, [6] investigated the high dimension case for this problem, and constructed a stable regularization solution by using Gauss kernel to mollify Cauchy data.[7] treated this problem by a modified quasi-boundary value method in 2011.Following the above works, recently the reference [8] also solved problem (1) by using two iterative regularization methods, and obtained the convergence estimates of optimal order.In this article, we continue to consider the problem (1).We adopt a modified Tikhonov regularization method to solve it.Under the a-priori bound assumptions for the exact solution, we give and proof the convergence estimates for this method.It can be seen that the convergence result is order optimal [9]- [11] as ( ) 1 a y = for 0 < < 1 y .In addition, for the Cauchy problem with non-homogeneous Dirichlet and Nuemann datum, it can be transformed into the above problem (1) by an auxiliary well-posed boundary problem.Hence, as in [1] [8], here we only need to consider problem (1).This paper is constructed as follows.In Section 2, we give some auxiliary results for this paper.In Section 3, we make the description for modified Tikhonov regularization method, and Section 4 is devoted to the convergence estimates for this method.Numerical results and some conclusions are shown in Sections 5-6, respectively.

Some Auxiliary Results
For a function ( ) 2 f L ∈  , we define its Fourier transform as follow ( ) ( ) Firstly, we consider the following Cauchy problem in the frequency domain Lemma 2.1 [4] There exists a unique solution of (5) such that here ( ) , the definition of entire function can be found in [12].
Secondly, Take the Fourier transform of problem (1) with respect to x , then It can be shown that, for ξ ∈  , the solution of problem (1) in the frequency domain is then, the exact solution of problem (1) can be expressed by Note that ( ) Further, we suppose that there exists a constant 0 E > , such that the following a-priori bounds exists ( ) denotes the Sobolev space p H -norm defined by

Modified Tikhonov Regularization Method
We firstly give the description for this method.Note that, from (9), we have According to (15), for 0 1 y < ≤ , we define the operator ( ) ( ) ( ) 1) can be expressed as the following operator equation and Let the exact and noisy datum where ⋅ denotes the 2 L -norm, the constant 0 δ > denotes a noise level.
Denote I be the identical operator in By Theorem 2.11 of Chapter 2 in [3], the functional ( ) has a unique minimizer f which is the unique solution of the following Euler equation According to Parseval equality, we get ( ) ( ) and from (20), we have Combing with ( 22), ( 23), (24), we can obtain that ( ) ( ) , using the inverse Fourier transform, we get the following Tikhonov regularization solution for problem (1) Note that, the above Tikhonov regularization solution (27) can be interpreted as using the regularized kernel

Convergence Estimates
Now, we choose the regularization parameter by the a-priori rule and give the convergence estimates for this method.
Theorem 4.1 Suppose that u given by (10) is the exact solution of problem (1) with the exact data ϕ and where, .

( )
f s has a unique maximum value point * s , such that and note that, thus, we get From (34), we can derive that combing with (36), (37), we have Now we estimate 2 I .Note that, adopting the similar proof procedure, we have : , 1 From the selection of regularization parameter E α δ = , (30), (39), (43), for the fixed 0 1 y < < , we can derive that

A A y A y y A A A y A y A y A y y
Theorem 4.1 shows that, for the fixed 0 1 y < < , the regularization solution u δ α defined by ( 28) is a stable approximation to the exact solution u and the convergence result is the order optimal (Hölder type), but the estimate (29) gives no information about the error estimate at 1 y = as the constraint ( 12) is too weak.For this purpose, as common, we can suppose that the stronger a-priori assumption (13) is satisfied.Theorem 4.2 Suppose that u given by (10) is the exact solution of problem (1) with the exact data ϕ and Proof.From ( 10), (28), ( 18), ( 13) and ( 14), we have . sup sup By Lemma 2.1, we can know using the similar derivation processes with 1 I , 2 I in Theorem 4.1, we have Case 1: for the large values with  , the first term is asymptotically negligible compared to this term. ϕ the adjoint operator for ( ) K y .In the ordinary Tikhonov regularization, we need solve the following minimum value problem δ α u is the regularization solution defined by (28) with the measured data δ ϕ which satisfy (18), and the a priori bound (12) is satisfied.If we choose the regularization parameter E α δ = , then for fixed 0 1 y < < , we have the following convergence estimate

u δα
is the regularization solution defined by (28) with the measured data δ ϕ which satisfy (18), and the a priori bound (13) is satisfied.If we choose the regularization parameter E
From the convergence estimate (44), we can see that the logarithmic term with respect to δ is the dominating term.Asymptotically this yields a convergence rate of order