Fan YANG(楊帆)Chuli FU(傅初黎i)Xiaoxiao LI(李曉曉)

1.School of Science,Lanzhou University of Technology,Lanzhou 730050,China

2.School of Mathematics and Statistics,Lanzhou University,Lanzhou 730000,China

E-mail:yfggd114@163.com;fuchuli@lzu.edu.cn;lixiaoxiaogood@126.com

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION?

Fan YANG(楊帆)1,2?Chuli FU(傅初黎i)2Xiaoxiao LI(李曉曉)1

1.School of Science,Lanzhou University of Technology,Lanzhou 730050,China

2.School of Mathematics and Statistics,Lanzhou University,Lanzhou 730000,China

E-mail:yfggd114@163.com;fuchuli@lzu.edu.cn;lixiaoxiaogood@126.com

In this paper,we consider the Cauchy problem for the Laplace equation,which is severely ill-posed in the sense that the solution does not depend continuously on the data.A modifed Tikhonov regularization method is proposed to solve this problem.An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained.Moreover,an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained.Numerical examples illustrate the validity and efectiveness of this method.

Cauchy problem for Laplace equation;ill-posed problem;a posteriori parameter choice;error estimate

2010 MR Subject Classifcation35R25;47A52;35R30

1 Introduction

The Cauchy problem for the Laplace equation has been extensively investigated in many practical areas.For example,some problems related to the geophysics[1,2],plasma physics[3], cardiology[4],bioelectric feld problems[5]and non-destructive testing[6].In the Hadamard’s famous paper[7],this problem is frstly introduced as a classical example of ill-posed problems, which shows that any small change of the data may cause dramatically large errors in the solution.Thus,it is impossible to solve this problem using classical numerical methods and requires special techniques,e.g.,regularization.There have been many papers devoted to this subject,such as the Fourier method[8],the central diference method[9],the quasi-reversibility method[10-13],the Tikhonov regularization method[14],the conjugate gradient method[15], the moment method[16-18],the wavelet method[19-20],the mollifcation method[21],the fundamental solutions[22],and etc.

In this paper,we consider the following Cauchy problem for the Laplace equation in a strip domain[23]:

the solution u(x,y)for 0＜x＜1 will be determined from the noisy data φδ(y)which satisfes

Problem(1.1)is severely ill-posed.However,to our knowledge,there are few papers devoted to the error estimates of regularization methods.In this paper,we will consider not just the a priori choice of the regularization parameter for the modifed Tikhononv regularization method, but also the a posteriori choice of the regularization parameter will be given for problem(1.1).

The modifed Tikhonov regularization was based on the Tikhonov regularization method. Skillfully modifed the penalty term of the Tikhonov functional,a better flter which flters the high frequencies of the measured data was obtained.This idea initially came from Carasso,who modifed the flter gained by the Tikhonov regularization method,and the order optimal error estimate was obtained in[24].By this method,Zhao[25]considered backward heat equation,Fu [26]considered the inverse heat conduction problem on a general sideways parabolic equation. Feng[27]used this method to consider the Cauchy problem for the Helmholtz equation.Cheng [28,29]used this method to consider the spherically symmetric inverse problem.Yang[30,31] used this method to consider the identifcation unknown source.

The plan of this paper is as follows.In Section 2 we simply analyze the ill-posedness of the Cauchy problem for Laplace equation and propose the modifed Tikhonov regularization method.In Section 3 we provide some stability and convergence estimates for the Cauchy problem.In Section 4,some numerical examples are proposed to show the efectiveness of this method.Section 5 puts an end to this paper with a brief conclusion.

2 Preliminaries

In this section,we give some auxiliary results which is needed in next section.For g(y)∈L2(R),?g(ξ)denotes its Fourier transform defned by

Let‖·‖denote the norm in L2(R).Then there holds the Parseval formula:

The application of the Fourier transform technique to problem(1.1)with respect to the variable y yields the following problem in frequency space:

It is easy to obtain that the solution of problem(2.3)is

whereμis the regularization parameter.It can be verifed that φμ,δ(x)is the solution of the following equation[32]:

From(2.5),we can see that T1:L2(R)→L2(R)is linear,compact and one-to one.From(2.5), we can also fnd

Due to Parseral formula,we have

From(2.8),we obtain

Note(2.11)and(2.12),we obtain

Due to(2.7),we know

Note(2.13),we obtain

Thus,the approximate solution of problem(1.1)can be given by

In this paper,we always assume that there holds the following the a priori bound:

where E is a positive constant.

3 Choice of Regularization Parameter and Convergence Results

In this section,we consider the a priori and a posteriori choice rule of regularization parameter,respectively,the corresponding convergence estimates are also obtained.We frst give some useful Lemmas.

Lemma 3.2For 0＜x＜1,we have

ProofUsing Parseval formula,the H¨older inequality and Lemma 3.1,we have

3.1 The a priori choice of the regularization parameter and the error estimate for problem(1.1)

Assume the conditions(1.2),(2.16)hold,take

then we have

Theorem 3.3Let u(x,y)given by(2.5)be exact solution of(1.1)and uμ,δ(x,y)given by (2.16)be the modifed Tikhonov regularized approximation of u(x,y).Let φδ(y)be measureddata at x=0 satisfying(1.2)and the a priori condition(2.17)holds.Then there holds the following error estimate:

ProofBecause φμ1,δ(x)is the minimizer of functional(2.6)and note(3.2),we obtain

Due to(1.2),(3.4),(2.17)and(3.5),we have

Combining(3.6),(3.7)with Lemma 3.2,we obtain the fnal estimate:

3.2 The a posteriori choice rule

In this section,Morozov’s discrepancy principle is used as a posteriori choice rule of the regularization parameter,i.e.,choosingμ=μ2as the solution of

To establish existence and uniqueness of solution for equation(3.8),we need the following lemma and remark:

Lemma 3.4Let(μ):=‖φμ,δ-φδ‖,then for δ＞0,there hold

(a)(μ)is a continuous function;

(d)(μ)is a strictly increasing function.

Remark 3.5To establish existence and uniqueness of solution for equation(3.8),we always suppose 0＜δ＜‖φδ‖.

Theorem 3.6Assume the conditions(1.2),(2.17)hold and take the solutionμ=μ2of Eq.(3.8)as the regularization parameter,then there holds the following error estimate:

ProofSince φμ2,δ(x)is the minimizer of functional(2.6),we have

Due to(3.10),(2.17)and(1.2),we obtain

Due to(3.8)and(1.2),we get

By Lemma 3.2,(3.12)and(3.13),we obtain

The proof of Theorem 3.6.is completed.

4 Several Numerical Examples

In this section,we employ several examples to test the properties of problems(1.1).The regularized solution was computed by the Fast Fourier Transform(FFT)and IFFT techniques. We always fx 0≤x≤1,-10≤y≤10.For the exact data function,its discrete noisy version is

The function“randn(·)”generates arrays of random numbers whose elements are normally distributed with mean 0,variance σ2=1,and standard deviation σ=1,“randn(size(φ))”returns an array of random entries that is the same size as φ.In our computations,we always take N=101.The bisection method is used to solve the Equations(3.8).In order to investigate the algorithm,we evaluate the relative error defned by

Example 1It is easy to verify that the function u(x,y)=ex2-y2sin(2xy)is the exact solution of problem(1.1).The exact data functions u(0,y)=0 and φ(x)=ux(0,y)=2ye-y2.

Figs.1-3 give the numerical results computed by both a priori and a posteriori modifed Tikhonov regularization method at x=0.1,0.5,0.8 with ε=0.01,0.05 for Example 1.From these Figures,we can see that the smaller the x,the better the computed approximations.Inadditional,it can be clearly seen that the smaller the perturbation ε,the better the results. Finally,we can see that the a posteriori modifed Tikhonov regularization methods also work well for problem(1.1).

Example 2The function u(x,y)=e-|y|sinx is the exact solution of problem(1.1)with weak singularity,it satisfeswhere y=(y1,y2)∈L2(R2).

As n=2,x=0.2 and ε=0.01,Fig.4 gives the comparison of the exact and regularization solution.These fgures show that the modifed Tikhonov regularization method is also efective for solving problem(1.1)in two dimensional case.

5 Conclusion

In this paper,we proposed an efcient regularization method for solving a classical ill-posed problem-the Cauchy problem for the Laplace equation.Two diferent regularization parameter choice rules are used.For a priori choices of the regularization parameter,we obtain the H¨older type error estimates.Using the Morozov’s discrepancy principle,we give the a posteriori parameter choice rule which only depends on the measured data.For a posteriori choices of the regularization parameter,we also obtain the H¨older type error estimate.Meanwhile,several numerical examples verify the efciency and accuracy of the method.

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?Received August 26,2014;revised January 20,2015.The project is supported by the National Natural Science Foundation of China(11171136,11261032),the Distinguished Young Scholars Fund of Lan Zhou University of Technology(Q201015),the basic scientifc research business expenses of Gansu province college and the Natural Science Foundation of Gansu province(1310RJYA021).

?Corresponding author:Fan YANG.