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.

• Ill-Posed Problem
• Unknown Source
• Regularization Method
• Discrepancy Principle in Hilbert Scales

1. 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 is the unknown source and is the supplementary condition and the constant is the wave number. Our purpose is to identify the source term from the input data. This

problem is called the inverse source problem. In practice, the data at are often obtained on the basis of reading of physical instrument. So only a perturbed data can be obtained. We assume that the exact and measured data satisfy

where denotes the noise level, denotes the―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.

2. The Tikhonov Regularization Method

Let, it is well known that is an orthonormal basis in, i.e.,

where is the Kronecher symbol. So for any, we can write, where

It is easy to derive a solution of problem (1) by the method of separation of variables [6]

where

Note that the exact data must decay faster than the rate. As for the measured data function 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 from the perturbed data and 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:

where is a constant and denotes a slightly different norm from the one in [25] which is defined by:

where

We let be the minimizer of the Tikhonov functional

where is a regularization parameter and q is a positive real number. The real number is going to occur throughout this paper and will be denoted by.

If we let, then we can derive that satisfy

So we have

Which means that

Then the approximate solution can be given as

Lemma 1. For any, we have

Lemma 2. [26] For, we have

Lemma 3.

where is the unique minimizer of (10) with g instead of.

Proof.

The proposition follows by applying (16) with b replaced by.

Lemma 4.

where

Proof. With the representation

and Lemma 1, we have

3. The Choices of Regularization Parameter a 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.

3.1. The a Priori Choice Rule

Take

we can obtain the following theorem.

Theorem 5. If (2) holds and (7) holds with, is defined by (14) and (23), then

Proof. With Lemma 3, Lemma 4 and (23) we obtain

Moreover, by using Hölder inequality, we have

Formulae (8) implies that

The assertion of the Lemma follows from (25)-(27).

3.2. The a Posteriori Choice Rule

For any, we define

It is apparent that the function is continuous and strictly increasing on and

So we can get the following lemma

Lemma 6. Let g, and satisfy (2) and

for some. Then there is a unique such that

In the following, we denote the unique determined in (31) by. In the next lemma, we consider the behavior of.

Lemma 7. Let g, and satisfy (2) and (30) for.

a)

b)

where

Proof.

a) Let

then

b)

then from Lemma 1

The rest follows from a).

Theorem 8. Suppose that the conditions (2) and (30) hold, the condition (7) hold with, is defined by (14) and (31), then

Proof. By using the triangle inequality we know

So, in terms of Equations (17), (19) and (33), we have

From (26),

Combining (41) and (42), we obtain

The assertion of the theorem follows from (27).

4. Numerical Tests

In this section, we present some numerical tests to check the effectiveness of the method. The discretization knots are. We first get the datum representing values of, and then obtain the perturbation datum as following

where are generated by Function in Matlab. Because the error satisfies the uniform distribution in this paper, so we let

in practical computing. The relative errors are measured by the weighted -norms defined as follows:

All tests are computed by using Matlab and we will also compare the method (M1) with the method in [6] (M2, notate the approximate function as). The perturbed data are given by

where are generated by function in Matlab.

Example [6] It is easy to see that the function and the function are the exact solutions of the problem (1) for any natural number n. In these cases, the condition (7) hold for any. So we have. Firstly, we exhibit influence of various p and N on ac- curacy of numerical solution. The relative errors have been shown in Table 1 with, and fixed. We can see that when N increases and decreases, the relative errors become smaller and when q increases, the rates of convergence become larger.

In Table 2, we give a numerical comparison between M1 and M2 with fixed,. The relative errors are given in Table 2, we can see that the results of M1 are much better than M2.

5. Conclusion

We have proposed a new method to identify the unknown source in the modified Helmholtz equation. Theoretical analysis as well as experience from computations indicates that the proposed method works well.

Table 1

Relative errors for various p and N with

	N = 32	N = 64	N = 128	N = 32	N = 64	N = 128	N = 32	N = 64	N = 128	N = 32	N = 64	N = 128
1e−1	0.0765	0.0862	0.0746	0.0611	0.0423	0.0215	0.0501	0.0392	0.0128	0.0471	0.0382	0.0099
1e−2	0.0213	0.0203	0.0211	0.0078	0.0061	0.0043	0.0062	0.0040	0.0021	0.0054	0.0039	0.0011
1e−3	0.0092	0.0090	0.0072	0.0009	0.0008	0.0007	0.0008	0.0005	0.0005	0.0006	0.0004	0.0001

Table 2

Comparison of M1 and M2

	M1	M2	M1	M2	M1	M2	M1	M2
1e−1	0.0746	0.1131	0.0215	0.0375	0.0128	0.0413	0.0099	0.0578
1e−2	0.0211	0.0584	0.0043	0.0184	0.0021	0.0194	0.0011	0.0233
1e−3	0.0072	0.0612	0.0007	0.0047	0.0005	0.0082	0.0001	0.0103

Acknowledgements

The project is supported by the National Natural Science Foundation of China (No. 11201085).

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