Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.


Introduction
The Cauchy problem for the Laplace equation in particular, and for other elliptic equations in general, occurs in the study of many practical problems in areas such as plasma physics [1], electrocardiology [2,3], bioelectric field problems [4], nondestructive testing [5], magnetic recording [6], and the Cauchy problem for elliptic equations [7,8].These problems are known to be severely ill-posed [9], in the sense that the solution, if it exists, does not depend continuously on the Cauchy data in some natural norm (see, e.g., [5] and references therein).This is because the Cauchy problem is an initial value problem which represents a transient phenomenon in a time-like variable while elliptic equation describes steady-state processes in physical fields.A small perturbation in the Cauchy data, therefore, can affect the solution largely.
In [9], the authors constructed a regularization method based on the Meyer wavelet, but the convergence rate of the method was not obtained.In [10,11], a modification method and a Fourier method for this problem were given, respectively, and some error estimates with satisfactory convergence rates were also proved.However, some very important and difficult problems in the theoretical study, that is, the optimal error bound, are not discussed.The major object of this paper is to give the optimal error bounds in theory for problem (1) by employing a regularized theory based on spectral decomposition.Meanwhile we provide two optimal regularized methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition method, which realize the optimal error bound.
The motivation of this paper is inspired by Tautenhahn, in [12], where he discussed a Cauchy problem for the elliptic equation in a bounded domain and used the eigenvalues of the elliptic operator to express the exact solution of the problem.However, this method does not suit problem (1) in an unbounded strip region.Instead of using eigenvalues we employ the technique of Fourier transform.
Let () and   () denote the exact and measured data, respectively, which satisfy where ‖ ⋅ ‖ denotes the  2 -norm and the noise level  > 0 is determined by the accuracy of the instruments.We assume denote the Fourier transform of function ().We now analyze problem (1) in the frequency space.Taking Fourier transform for problem (1) with respect to the variable , we get The unique solution of ( 4) is [9][10][11] û (, ) = ĝ () cosh () .
Due to Parseval formula, û(, ⋅) ∈  2 (R), and therefore (5) implies that () must decay rapidly as || → ∞.However, as the measurement data ĝ (), we can not expect ĝ () has the same decay in high frequency components; that is, small errors in high frequency components can blow up and completely destroy the solution for 0 <  ≤ 1, noting that the factor cosh() in (5) increases exponentially as || → ∞, so the problem (1) is severely ill-posed.
In order to obtain explicit stability estimate for problem (1), some "source condition" is needed.For this we introduce the Sobolev space   | ∈R + according to  0 =  2 (R),   = {V() : ‖V‖  < ∞}, where is the norm in   .We require the a priori smoothness condition for problem (1) concerning the unknown solution (, ) according to This paper is organized as follows: In Section 2 we briefly recount some preliminary results, which are the basis of the discussion for other sections.In Section 3 we give the optimal error bounds between the exact solution and its regularized approximation, which depend on the noise level  either in a Hölder continuous way or in a logarithmic continuous way.In Section 4 we discuss two concrete regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, where both regularization methods realize the optimal error bounds.

Preliminary Result
We consider arbitrary ill-posed inverse problem [12][13][14][15][16][17] where  ∈ L(, ) is a linear injective bounded operator between infinite dimensional Hilbert spaces  and  with nonclosed range () of .We assume that   ∈  are available noisy data with ‖ −   ‖ ≤ .Any operator  :  →  can be considered as a special method for solving (8), and the approximate solution of ( 8) is given by   .However, the convergent rate of   to  can be arbitrarily slow without assuming additional quantitative a prior restrictions on the unknown solution , which is typical for ill-posed problem.
Assume we want to solve (8), we have the a priori information that the exact solution satisfies a source condition; that is,  belongs to the source set where the operator function ( * ) is well defined via spectral representation [13,14]: where is the spectral decomposition of  * , {  } denotes the spectral family of the operator  * , and  is a constant such that ‖ * ‖ ≤ .In the case when  :  2 (R) →  2 (R) is a multiplication operator, () = ()(), the operator function ( * ) attains the form Let us assume that  :  →  is an arbitrary mapping to approximately recover  from   .Then the worst case error for  under the a priori information  ∈  , is [16,17] Δ  (,  , , ) This worst case error characterizes the maximal error of the method  if the solution  of problem (8) varies in the set  , .The best possible worst case error (or the optimal bound) is defined as where the minimum is taken over all methods  :  → .It can be shown (cf.[13,18]) that the minimum in ( 14) is actually obtained and with the modulus of continuity defined by In order to derive explicitly the optimal error bounds for the worst case error Δ  (,  , , ) defined in ( 13) and obtain optimality results for special regularization methods, we assume that the function  in (9) satisfies the following assumption.
Theorem 2 (see [13,14]).Let  , be given by ( 14), let Assumption 1 be satisfied, and let  2 / 2 ∈ ( * ( * )), where ( * ) denotes the spectrum of operator  * ; then In the following we consider two special methods: the method of generalized Tikhonov regularization and the method of generalized singular value decomposition.
For the method of generalized Tikhonov regularization, a regularized approximation    is determined by solving the minimization problem [13,14,16,17] min or equivalently, by solving the Euler equation and the following statement holds.
The regularized approximation    based on the method of generalized singular value decomposition is given by For this method the following result holds [13,14].
It is easy to know that the operator Â() is self-adjoint, so Â * () Â() :  2 (R) →  2 (R) is given by Advances in Mathematical Physics Due to Parseval formula, (, ) ∈  , is equivalent to û(, ) ∈ M, , where Note that the source condition (9) for problem (1) can be written as and then its equivalent form in Fourier frequency space is given by Due to the equivalence of conditions ( 30) and ( 7), we know the conditions (28) and (31) are equivalent and we have the following result.Proposition 5.For operator equation ( 25), the set  , given by ( 7) is equivalent to the general source set  , given by (30) provided  = () is given (in parameter representation) by Proof.From (5) we have Together with (27),  is given by (32) in its parameter representation.The proof is complete.
Indeed, if we denote it is easy to prove that lim So, the representation (40) holds.
(vi) We know that the function () is strong convex if and only if   () > 0. Denoting () = ()() with () = cosh 2 (), from (38) we have where It is easy to know that   () is even function about variable ; therefore we only need to consider the case  ≥ 0.