1. Introduction
The inverse problem of the Laplace equation appears in many engineering and physical areas, such as geophysics, cardiology, seismology, and so on [
1,
2,
3]. It has been widely recognized that the inverse problem for the Laplace equation has a central position in all Cauchy problems of elliptic partial differential equations. The inverse problem of the Laplace equation is seriously ill-posed, where a tiny deviation in the data can cause a large error in the solution [
4]. It is difficult to develop numerical solutions with conventional methods. Some different methods have been researched, such as the quasi-reversibility [
5], Tikhonov regularization [
6], wavelet [
7], conjugate gradient [
8], central difference [
9], Fourier regularization [
10], and mollification [
11,
12,
13] methods.
The main procedure of the mollification method is using the kernel function to construct a mollification operator by convolution with the measurement data. Manselli, Miller [
14], and Murio [
15,
16] constructed mollification operators by using the Weierstrass kernel to solve some inverse heat conduction problems (IHCP). There have been reports on using the Gaussian kernel to solve the Cauchy problem of elliptic equations [
17,
18,
19,
20,
21]. Hào [
22,
23,
24] adopted the Dirichlet kernel and de la Vallée Poussin kernel to solve some kinds of two-dimensional equations; including the two-dimensional Laplace equation. However, the three-dimensional case was not considered, Moreover, the analysis method used for error estimate was does not generalize to the three-dimensional case well.
Our primary interest is to solve the inverse problem of the three-dimensional Laplace equation with non-homogeneous Neumann boundary conditions. In order to guarantee solvability for the inverse problem provided, a regularization method using the bivariate de la Vallée Poussin kernel is presented.
This paper is organized as follows: In
Section 2, the mathematical problem for the three-dimensional Laplace equation and its ill-posedness are illustrated. In
Section 3, we introduce the bivariate de la Vallée Poussin kernel and its properties, following which our mollification regularization method is proposed. In
Section 4, some stability estimate results are given, in the interior
and at the boundary
, under a priori assumptions. The numerical aspect of our proposed method is showed in
Section 5. Concluding remarks are given in
Section 6.
2. Mathematical Problem and the Ill-Posedness Analysis
We give thought to the following inverse problem of the three-dimensional Laplace equation with non-homogeneous Neumann boundary conditions:
where
is three-dimensional Laplace operator; and
,
are given vectors in
. The solution
will be determined by the noisy data
and
in
that satisfy:
were
denotes the error level, and
denotes the
norm [
1].
Note that the solution of the problem (
1) is the sum
of the solutions for the following two problems:
and
Therefore, in order to simplify the process of the Cauchy problem (
1), we only need to solve problems (
3) and (
4), respectively.
For
, the Fourier transform for a variable
is defined by
where
and
.
The inverse Fourier transform for a variable
is defined by
The Parseval equality [
16] is as follows:
Adopting the Fourier transform for the variable
to problems (
3) and (
4), we obtain
and
The solution of problem (
6) is
The solution of problem (
7) is
Note that
and
are unbounded with respect to the variable
; a small perturbation in the measured data
and
may result in a huge deviation in the solution
and
. Therefore, the problems (
3) and (
4) are severely ill-posed.
3. Mollification Method and Regularization Solution
3.1. Mollification Operator
The bivariate de la Vallée Poussin kernel [
22] function is defined by:
where
is called the mollification radius (or mollification parameter).
has the following properties [
22]:
- (1)
is an entire function of exponential type of degree belong to ;
- (2)
;
- (3)
; and
- (4)
is the Fourier transform of
, satisfying:
where
and
For any function
and
,
, we define two-dimensional convolution [
22] by
It is well-known that [
22]
and
We define the mollification operator
by
3.2. Regularization Approximation Solution
Instead of solving the problems (
3) and (
4) with the data
and
, we attempt to re-construct the noisy data
and
by
and
, respectively. We obtain the problems, with the re-constructed data, as follows:
and
The solution for problem (
12) is
or
The solution to problem (
13) is
or
According to (
11) and the properties (2) and (3) of the kernel
, we have the following conclusion:
Remark 1. If and hold, then 4. Parameter Selection and Error Estimates
Lemma 1. For , the following inequalities hold
- (1)
- (2)
- (3)
and
- (4)
Proof. Inequalities (1) and (2) are easy to obtain. From the inequalities
and the Taylor expansion
we can arrive at (3) and (4). ☐
In the next content, we give stability convergence estimates between the exact solution for problems (
3) and (
4) and the regularization approximate solution of problems (
12) and (
13) in
and at the boundary
, respectively. Convergence estimates will be obtained when we choose a suitable regularization parameter
.
4.1. Error Estimates in the Interior
The convergence estimates for the proposed regularization method, in the case of
, will be given in this section, and we obtain the approximation results as following:
Theorem 1. Let and be the exact solution and the approximation solution for problem (1) with the exact input data and mollified data, respectively. Assume the a priori bounds and hold. We have the following estimate: If α is selected aswe havewhere E is a finite positive constant. Proof. From Parseval’s equality (
5) and the properties of double integrals, we have
Here,
(see
Figure 1)
and
It is easy to verify that
,
From Minkowski’s inequality, we have
Using a similar analysis, we can obtain the integral estimates of the other .
Utilizing the inequality
, we have
According to (1) and (2) of Lemma 1, we obtain
Taking parameter
to be
, we arrive at (
16). ☐
Similarly, the error estimate for problem (
4) can be obtained in the following way.
Theorem 2. Let and be the exact solution and approximation solution for problem (4) with the exact input data and mollified data, respectively. Assume that the a priori bounds and hold, we have If α is chosen as in (15), then we have As for our problem (
1), combining the results of Theorems 1 and 2 and the Minkowski inequality, we have the error estimate, as follows:
Theorem 3. Let and be the exact solution and regular solution for problem (1). Assume that condition (2) and hold. Then, we have If α is chosen as in (15), then we have 4.2. Error Estimates at the Boundary
The estimates (
14), (
17), and (
19) give no information about the error estimates at
, as the constraints
and
are too weak for this purpose. Therefore, to ensure stability of the solution
,
at
, we need the Sobolev space
[
1],
where
is defined by
If , then .
Theorem 4. Let and be the exact and regularization solutions, respectively, of problem (3) at . Suppose that the a priori bounds and hold. Then, we have the following inequality If the regular parameter α is selected asthen we havewhere is a positive constant only depending on p. Proof. From Parseval’s equality (
5), we have
where
are same as in Theorem 1. Let
Using the properties of the double integral, and the inequality
, we have
Using a similar method as in Theorem 1 and the monotonicity of the function
, we obtain
If we chose
as
and utilize inequality
, then (
23) can be obtained. ☐
Similar to Theorem 4, the error estimate for problem (
4) can be obtained as follows.
Theorem 5. Let and be the exact and regularization solutions, respectively, for problem (4) at . Suppose that the a priori bounds and hold. Then, we have the following inequality If the regularization parameter α is chosen as in (22), then Thus, as for problem (
1), using the results of Theorems 4 and 5 and the Minkowski inequality, we have the stable error estimate, as follows:
Theorem 6. Let and be the exact and regularization solutions, respectively, for problem (1) at . Suppose that condition (2) and hold. We have the convergence estimate, as follows: If the regularization parameter α is selected as in (22), then Remark 2. In this part, we consider the stable error estimates in the cases and , respectively. In the interior, , the a priori bound for is sufficient, and the convergence estimate converges quickly to zero as . However, for the case , although a stronger a priori bound for is imposed, the error estimate is only of logarithmic type, with order .
5. Numerical Examples
We performed two numerical examples to verify the accuracy and stability of our proposed method. Our tests were carried out in the MATLAB R2014b software.
In the numerical examples, we selected the discrete interval to be
and the measurement data
was obtained as the following
where
The error level
is given by
In the following numerical implementations, we need to take the two-dimensional discrete Fourier transform of the data vector
and the two-dimensional discrete inverse Fourier transform. We take
,
, and fix the reconstructed position
. The a priori mollification parameter
was determined by (
15) and (
22), where
and
. We define the relative error between the exact solution
u and its approximate solution
as:
Example 1. We chose the function as the exact data for problem (3). Example 2. We chose the function as the exact data for problem (4). We use different perturbation noise levels at the boundary
with
and
, respectively, in
Table 1 and
Table 2. Note that the results for the relative error
at
depended on the error level,
, and
p.
Figure 2 and
Figure 3 show the re-constructed solution and exact solution of Example 1, corresponding to noise levels of
and
, with
and
, respectively.
Figure 4 shows the corresponding error between (a) and (b) in
Figure 2 and
Figure 3.
Figure 5 and
Figure 6 show the regularization solution and exact solution of Example 2, corresponding to noise levels of
and
with
and
, respectively.
Figure 7 shows the corresponding error between (a) and (b) in
Figure 5 and
Figure 6.
In the two examples, we note that the methods which we adopted are stable and effective.
6. Conclusions
In this article, we use a regularization method to solve two Cauchy problems for the three-dimensional Laplace equation. Stable approximate estimates are obtained under a priori bound assumptions and an appropriate choice of the regular parameter. Two numerical examples are investigated to verify the stability of our presented method.
We consider stability error estimates in the cases and , respectively. In the interior, , the convergence estimate is , which quickly converges to zero as . However, at the boundary, , the error estimate is of logarithmic type with order . In future work, we hope to find a new a priori assumption method, in order to obtain an error estimation which achieves better results.
Author Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Funding
Thanks to the National Science Foundation of China (11161036). Thanks to the Natural Science Research Foundation of Ningxia Province, China (NR17260)(NR160117).
Acknowledgments
The authors are deeply indebted to the anonymous referees for their very careful reading and valuable comments and suggestions which immensely improved the previous version of our manuscript.
Conflicts of Interest
The authors declare that they have no competing interests.
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