The Tikhonov Regularization Method in Hilbert Scales for Determining the Unknown Source for the Modified Helmholtz Equation

In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a priori bound of exact solution are not needed for the a posteriori choice rule. Numerical results are presented to show the stability and effectiveness of the method.


Introduction
A variety of important problems in science and engineering involve the solution to the modified Helmholtz equation, e.g., in implicit marching schemes for the heat equation, in Debye-Huckel theory, and in the linearization of the Poisson-Boltzmann equation [1]- [5].In this paper, we consider the following problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation [6]: where ( ) f x is the unknown source and ( ) ( ) is the supplementary condition and the constant 0 k > is the wave number.Our purpose is to identify the source term ( ) f x from the input data ( ) problem is called the inverse source problem.In practice, the data at ( ) g x are often obtained on the basis of reading of physical instrument.So only a perturbed data ( ) g x δ can be obtained.We assume that the exact and measured data satisfy where 0 δ > denotes the noise level, ⋅ denotes the 2 L -norm.
Inverse source problems arise in many branches of science and engineering, e.g., heat conduction, crack identification electromagnetic theory, geophysical prospecting and pollutant detection.The main difficulty of these problems is that they are ill-posed (the solution, if it exists, does not depend continuously on the data).Thus, the numerical simulation is very difficult and some special regularization is required.Many papers have presented the mathematical analysis and efficient algorithms of these problems.The uniqueness and conditional stability results for these problems can be found in [7]- [12].Some numerical reconstruction schemes can be found in [13]- [23].
Up to now, only a few papers for identifying the unknown source on the modified Helmholtz equation have been reported.In [1], an integral equation method has been proposed and a simplified Tikhonov regularization has been presented in [6].In this paper, we will use the Tikhonov regularization method to solve the problem (1).Unlike the one in [6], a different Tikhonov functional will be used and we show that the regularization parameter can be chosen by a discrepancy principle in Hilbert scales which is proposed by Neubauer [24] and better convergence rates have been obtained.Moreover, the smoothness parameter of the exact solution is not needed for the new method.
This paper is organized as follows.In Section 2, we will give the method to construct approximate solution.The choices of the regularization parameter and corresponding convergence results will be found in Section 3. Some numerical results are given in Section 4 to show the effectiveness of the new method.

The Tikhonov Regularization Method
Let ( ) ( ) where , l k δ is the Kronecher symbol.So for any ( ) , where It is easy to derive a solution of problem (1) by the method of separation of variables [6] ( ) ( ) . 1 e Note that the exact data ˆl g must decay faster than the rate 2 l − .As for the measured data function g δ is only in , we cannot expect that it possess such a decay property.So some special regularization methods are required.In the following, we apply the Tikhonov regularization method in Hilbert scales to reconstruct a new function h δ from the perturbed data g δ and Th δ will be used as an approximation of f.It is well known that for any ill-posed problems an a priori bound assumption for the exact solution is needed and necessary.In this paper, we assume the following a priori bound holds: , 0, where 0 E > is a constant and p ⋅ denotes a slightly different norm from the one in [25] which is defined by: .
We let , h h δ α δ = be the minimizer of the Tikhonov functional ( ) where 0 α > is a regularization parameter and q is a positive real number.The real number 2 q l ρ − − is going to occur throughout this paper and will be denoted by l γ .
If we let ( ) ( ) , then we can derive that ˆl h δ satisfy ( ) So we have Which means that Then the approximate solution can be given as ( ) Lemma 1.For any l ∈  , we have Lemma 2. [26] For 0 1 b ≤ ≤ , we have ( ) Lemma 3.
( ) ( ) where h α is the unique minimizer of (10) with g instead of g δ .Proof.

The Choices of Regularization Parameter α and Convergence Results
In this section, we consider the choices of the regularization parameter.An a priori strategy and an a posteriori choice rule will be given.Under each choice of the regularization parameter, the convergence estimate can be obtained.

The a Priori Choice Rule
Take ( ) we can obtain the following theorem.Theorem 5.If (2) holds and (7) holds with f α δ is defined by ( 14) and (23), then Proof.With Lemma 3, Lemma 4 and (23) we obtain Moreover, by using Hölder inequality, we have ( ) The assertion of the Lemma follows from ( 25)-(27).

The a Posteriori Choice Rule
For any ( ) .
It is apparent that the function for some 1 C > .Then there is a unique In the following, we denote the unique α determined in (31) by 2 α .In the next lemma, we consider the behavior of 2 α .Lemma 7. Let g, g δ and 0 δ > satisfy (2) and (30) for where ( ) ) ( ) then from Lemma 1 The rest follows from a).Theorem 8. Suppose that the conditions (2) and (30) hold, the condition (7) hold with f α δ is defined by ( 14) and (31), then 2 , 2 .
Proof.By using the triangle inequality we know So, in terms of Equations ( 17), ( 19) and (33), we have From ( 26), The assertion of the theorem follows from (27).

So we can get the following lemma Lemma 6 .
Let g, g δ and 0 δ > satisfy (2) and 2