JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation97195210.1155/2012/971952971952Research ArticleInverse Source Identification by the Modified Regularization Method on Poisson EquationLiXiao-Xiao1GuoHeng Zhen2WanShi Min3YangFan1GuglielmiNicola1School of ScienceLanzhou University of TechnologyLanzhou 730050Chinalut.cn2Institute of EducationLanzhou City UniversityLanzhou 730070China3Department of Fundamental SubjectTianjin Institute of Urban ConstructionTianjin 300384China2012181201220122306201121092011101020112012Copyright © 2012 Xiao-Xiao Li et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper deals with an inverse problem for identifying an unknown source which depends only on one variable in two-dimensional Poisson equation, with the aid of an extra measurement at an internal point. Since this problem is illposed, we obtain the regularization solution by the modified regularization method. Furthermore, we obtain the Hölder-type error estimate between the regularization solution and the exact solution. The numerical results show that the proposed method is stable and the unknown source is recovered very well.

1. Introduction

Inverse source problem is an ill posed problem that has received considerable attention from many researches in a variety of fields, such as heat conduction, crack identification, electromagnetic theory, geophysical prospecting, and pollutant detection. For the heat source identification, there have been a large number of research results for different forms of heat source . To the authors’ knowledge, there were also a lot of researches on identification of the unknown source in the Poisson equation adopted numerical algorithms, such as the logarithmic potential method , the projective method , the Green’s function method , the dual reciprocity boundary element method , the dual reciprocity method [13, 14], and the method of fundamental solution (MFS) . But, by the regularization method, there are a few papers with strict theoretical analysis on identifying the unknown source.

In this paper, we consider the following inverse problem: to find a pair of functions (u(x,y),f(x)) which satisfy the Poisson equation on half unbounded domain as follows:-uxx-uyy=f(x),-<x<+,  0<y<+,u(x,0)=0,  u(x,y)|y  bounded,-<x<+,u(x,1)=g(x),-<x<+, where f(x) is the unknown source depending only on one spatial variable and u(x,1)=g(x) is the supplementary condition. In applications, input data g(x) can only be measured, and there will be measured data function gδ(x) which is merely in L2() and satisfiesg-gδL2(R)δ, where the constant δ>0 represents a noise level of input data.

The problem (1.1) is mildly ill posed, and the degree of the ill posedness is equivalent to the second-order numerical differentiation. It is impossible to solve the problem (1.1) using classical methods. The major object of this paper is to use the modified regularization method to obtain the regularization solution. Meanwhile, the Hölder-type stability estimate between the regularization solution and the exact solution is obtained. In , the authors ever identified the unknown source on the Poisson equation on half band domain using separation of variables. But in this paper, we identified the unknown source on the Poisson equation on half unbounded domain using the Fourier Transform.

This paper is organized as follows. Section 2 analyzes the ill posedness of the identification of the unknown source and gives some auxiliary results. Section 3 gives a regularization solution and error estimate. Section 4 gives several numerical examples including both nonsmooth and discontinuous cases for the problem (1.1). Section 5 ends this paper with a brief conclusion.

2. Some Auxiliary Results

The ill posedness can be seen by solving the problem (1.1) in the Fourier domain. Let f̂(ξ) denote the Fourier transform of f(x)L2() which is defined byf̂(ξ):=12π-e-iξxf(x)dx. The problem (2.2) can now be formulated in frequency space as follows:ξ2û(ξ,y)-ûyy(ξ,y)=f̂(ξ),y>0,  ξR,û(ξ,0)=0,ξR,û(ξ,y)|y  bounded,ξR,û(ξ,1)=ĝ(ξ),ξR. The solution of the problem (2.2) is given byf̂(ξ)=ξ21-e-ξĝ(ξ). So,f(x)=12π-eiξxξ21-e-ξĝ(ξ)dξ. The unbounded function ξ2/(1-e-ξ) in (2.3) or (2.4) can be seen as an amplification factor of ĝ(ξ) when ξ. Therefore, when we consider our problem in L2(), the exact data function ĝ(ξ) must decay. But, in the applications, the input data g(x) can only be measured and can never be exact. Thus, if we try to obtain the unknown source f(x), high-frequency components in the error are magnified and can destroy the solution. In general, for an ill posed problem, the convergence rates of the regularization solution can only be given under prior assumptions on the exact solution; we impose an a priori bound on the exact solution f(x) as follows:f()Hp(R)E,p>0, where E>0 is a constant and ·Hp() denotes the norm in the Sobolev space Hp() defined byf()Hp(R):=(-|f̂(ξ)|2(1+ξ2)pdξ)1/2.

Now we give some important lemmas as follows.

Lemma 2.1.

If x>1, the following inequality: 11-e-x<2 holds.

Lemma 2.2.

As 0<μ<1, one obtains the following inequalities: supξR|(1-11+ξ2μ2)(1+ξ2)-p/2|max{μp,μ2},supξR|ξ2(1-e-ξ)(1+μ2ξ2)|2μ2.

Proof.

Let G(ξ):=(1-11+ξ2μ2)(1+ξ2)-p/2. The proof of the first inequality of (2.8) can be divided into three cases.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M34"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>).

We obtain G(ξ)(1+ξ2)-p/2|ξ|-pξ0-p=μp.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M36"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>).

We get G(ξ)=ξ2μ21+ξ2μ2(1+ξ2)-p/2ξ2-pμ21+ξ2μ2ξ2-pμ2. If 0<p2, the above inequality becomes G(ξ)ξ02-pμ2=μp. If p>2, we get G(ξ)ξ2-pμ2=μ2.

Case 3 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M42"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>).

We obtain G(ξ)ξ2μ2(1+ξ2)-p/2μ2. Combining (2.10) with (2.12), (2.13), and (2.14), we obtain the first inequality equation.

Let B(ξ):=ξ2(1-e-ξ)(1+ξ2μ2),  D(ξ):=ξ21-e-ξ. The proof of the second inequality of (2.8) can also be divided into two cases.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M45"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>).

We obtain D(ξ)D(1μ)2μ2,if  0<μ<1. So, B(ξ)2μ2.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M48"><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:msub><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>).

We obtain D(ξ)2ξ2,B(ξ)2ξ21+ξ2μ22μ2. Combining (2.17) with (2.18), (2.8) holds.

3. A Modified Regularization Method and Error Estimate

We modify (1.1), where a two-order derivation of f(x), is added, that is,-uxx-uyy+μ2fxx(x)=f(x). This is based on the modified regularization method which we learned from Eldén  who considered a standard inverse heat conduction problem and the idea initially came from Weber . This method has been studied for solving various types of inverse problems . We obtain a stable approximate solution of problem (1.1), that is,-uxx-uyy+μ2fxx(x)=f(x),-<x<+,  0<y<+,u(x,0)=0,u(x,y)|y  bounded,-<x<+,u(x,1)=gδ(x),-<x<+, where the parameter μ is regarded as a regularization parameter. The problem (3.2) can be formulated in frequency space as follows:ξ2û(ξ,y)-ûyy(ξ,y)-μ2ξ2f̂(ξ)=f̂(ξ),ξR,  0<y<+,û(ξ,0)=0,ξR,û(ξ,y)|y  bounded,ξR,û(ξ,1)=ĝδ(ξ),ξR. The solution to this problem is given byf̂(ξ)=ξ2(1-e-ξ)(1+ξ2μ2)ĝδ(ξ):=f̂δ,μ(ξ). Sofδ,μ(x)=12π-eiξxξ2(1-e-ξ)(1+ξ2μ2)ĝδ(ξ)dξ.

Note that, for small μ,ξ2/(1+ξ2μ2)   is close to ξ2. On the contrary, if |ξ| becomes large, |ξ2/(1+ξ2μ2)|   is bounded. So, fδ,μ(x) is considered as an approximation of f(x).

Now we will give an error estimate between the regularization solution and the exact solution by the following theorem.

Theorem 3.1.

Suppose f(x) is an exact solution of (1.1) given by (2.4) and fδ,μ(x) is the regularized approximation to f(x) given by (3.5). Let gδ(x) be the measured data at y=1 satisfying (1.2). Moreover, one assumes the a priori bound (2.5) holds. If one selects μ=(δE)1/(p+2), then one obtains the following error estimate: f()-fδ,μ()2δp/(p+2)E2/(p+2)(1+12max{1,(δE)(2-p)/(p+2)}).

Proof.

From the Parseval formula and (2.3), (3.4), (2.6), (2.7), (2.8), (1.2), (2.5), and (3.6), we obtain f()-fδ,μ()=f̂()-f̂δ,μ()=ξ21-e-ξĝ(ξ)-ξ2(1+ξ2μ2)(1-e-ξ)ĝδ(ξ)ξ21-e-ξĝ(ξ)-ξ2(1+ξ2μ2)(1-e-ξ)ĝ(ξ)+ξ2(1+ξ2μ2)(1-e-ξ)ĝ(ξ)-ξ2(1+ξ2μ2)(1-e-ξ)ĝδ(ξ)=ξ2ĝ(ξ)1-e-ξ(1-11+ξ2μ2)+ξ2(1+ξ2μ2)(1-e-ξ)(ĝ(ξ)-ĝδ(ξ))f̂(ξ)(1+ξ2)p/2(1+ξ2)-p/2(1-11+ξ2μ2)+supξR|ξ2(1+ξ2μ2)(1-e-ξ)|ĝ(ξ)-ĝδ(ξ)supξR|(1-11+ξ2μ2)(1+ξ2)-p/2|  f̂(ξ)(1+ξ2)p/2+supξR|ξ2(1+ξ2μ2)(1-e-ξ)|ĝ(ξ)-ĝδ(ξ)max{μp,μ2}E+2μ2δ=max{(δE)p/(p+2),(δE)2/(p+2)}  E+2(δE)-2/(p+2)δ=2δp/(p+2)E2/(p+2)(1+12max{1,(δE)(2-p)/(p+2)}).

Remark 3.2.

If 0<p2,f()-fδ,μ()3δp/(p+2)E2/(p+2)0  as  δ0. If p>2, f()-fδ,μ()2δp/(p+2)E2/(p+2)+δp/(p+2)E2/(p+2)0as  δ0. Hence, fδ,μ(x) can be regarded as the approximation of the exact solution f(x).

Remark 3.3.

In general, the a priori bound E in (2.5) is unknown exactly in practice. But, if we choose μ=δ1/(p+2), we can also obtain f()-fδ,μ()0,as  δ0. This choice is useful in concrete computation.

4. Several Numerical Examples

In this section, we present three numerical examples intended to illustrate the usefulness of the proposed method. The numerical results are presented, which verify the validity of the theoretical results of this method.

The numerical examples were constructed in the following way. First we selected the exact solution f(x) of problem (1.1) and obtained the exact data function g(x) using (2.3) or (2.4). Then, we added a normally distributed perturbation to each data function giving vectors gδ. Finally, we obtained the regularization solutions using (3.4) or (3.5).

In the following, we first give an example which has the exact expression of the solutions (u(x,y),f(x)).

Example 4.1.

It is easy to see that the function u(x,y)=(1-e-y)sinx and the function f(x)=sinx are satisfied with the problem (1.1) with exact data g(x)=(1-e-1)sinx.

Suppose that the sequence {gk}k=0n represents samples from the function g(x) on an equidistant grid and n is even. Then we add a random uniformly perturbation to each data, which forms the vector gδ, that is,gδ=g+ε  randn(size(g)), whereg=g(g(x1),,g(xn))T,  xi=(i-1)Δx,Δx=1n-1,  i=1,2,,n.

The function “randn(·)” generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1. “Randn(size(g))” returns an array of random entries that is of the same size as g. The total noise level δ can be measured in the sense of root mean square error (RMSE) according toδ=gδ-gl2=(1ni=1n(gi-gi,δ)2)1/2. Moreover, we need to make the vector gδ periodical , and then we take the discrete Fourier transform for the vector gδ. The approximation of the regularization solution is computed by using FFT algorithm , and the range of variable x in the numerical experiment is [-10,10].

Example 4.2.

Consider a piecewise smooth source:   f(x)={0,-10x-5,x+5,-5<x0,-x+5,0<x5,0,5<x10.

Example 4.3.

Consider the following discontinuous case: f(x)={-1,-10x-5,1,-5<x0,-1,0<x5,1,5<x10.

From Figures 13, we can see that the smaller the ε, the better the computed approximation fδ,μ(x).

Comparison between the exact solution and its computed approximations with various levels of noise for Example 4.1: (a) p=1, (b) p=2.

Comparison between the exact solution and its computed approximations with various levels of noise for Example 4.2: (a) p=1, (b) p=2.

. Comparison between the exact solution and its computed approximations with various levels of noise for Example 4.3: (a) p=1, (b) p=2.

In Examples 4.2 and 4.3, since the direct problem with the source f(x) does not have an analytical solution, the data g(x) is obtained by solving the direct problem. From Figures 2 and 3, we can see that the numerical solutions are less ideal than that of Example 4.1. It is not difficult to see that the well-known Gibbs phenomenon and the recovered data near the nonsmooth and discontinuities points are not accurate. Taking into consideration the ill posedness of the problem, the results presented in Figures 2 and 3 are reasonable.

5. Conclusions

In this paper, we consider the identification of an unknown source term depending only on one variable in two-dimensional Poisson equation. This problem is ill posed, that is, the solution (if it exists) does not depend on the input data. We obtain the regularization solution and a Hölder-type error estimate. Through the comparison between  and this paper, as the degree inverse problem of the ill posedness of identifying the unknown source dependent only on one variable in two-dimensional Poisson equation is equivalent to the second-order numerical differentiation, we obtain the same error estimate 2δp/(p+2)E2/(p+2)(1+1/2max{1,(δ/E)(2-p)/(p+2)}). According to , this Hölder-type error estimate is order optimal.

Acknowledgment

This research is supported by the Distinguished Young Scholars Fund of Lanzhou University of Technology (Q201015).

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