A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation

ARTICLE IN PRESS

Engineering Analysis with Boundary Elements 32 (2008) 778–785 www.elsevier.com/locate/enganabound

A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation Chein-Shan Liu Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan Received 31 October 2006; accepted 17 December 2007 Available online 12 February 2008

Abstract We consider an inverse problem for Laplace equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise. r 2008 Elsevier Ltd. All rights reserved. Keywords: Inverse problem; Modified indirect Trefftz method; Laplace equation; Modified collocation Trefftz method

1. The inverse Cauchy problem in a disk The detection of corrosion inside a pipe is very important in engineering applications. In this paper we consider a mathematical modeling of this problem and give an effective numerical algorithm for a method to detect the corrosion by an electrical field in the pipe. Given the Cauchy data uðx; yÞ and Neumann data qu=qnðx; yÞ at the point ðx; yÞ 2 R2 with unit outward normal nðx; yÞ on the accessible external part G1 of a circle with a radius R, we consider an inverse Cauchy problem of the Laplace equation Duðx; yÞ ¼ 0 in two dimensions to find the unknown function uðx; yÞ on the inaccessible internal part G2 of G ¼ G1 [ G2 . The inverse problem is given as follows: 1 1 Du ¼ urr þ ur þ 2 uyy ¼ 0; r r uðR; yÞ ¼ hðyÞ; 0pypbp, ur ðR; yÞ ¼ gðyÞ;

0proR;

0pypbp,

0pyo2p,

(1) (2) (3)

where hðyÞ and gðyÞ are given functions. This problem is for solving the Laplace equation under an overdetermined Cauchy E-mail address: [email protected] 0955-7997/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.12.002

data on a partial circular boundary. Usually, it requires bX1, which is specified by Mera et al. [1] as a necessary condition for the numerically identifiable of the inverse Cauchy problem. However, in the present study we only require bX0:5 for an accurate reconstruction of unknown data. When bo0:5, the numerical results are less accurate. The use of electrostatic image in nondestructive testings of metallic disks leads to an inverse boundary value problem for Laplace equation in two dimensions. In order to detect the unknown shape of the inclusion within a conducting metal we may impose an overdetermined Cauchy data, for example the voltage and current, on the accessible boundary. This amounts to solving an inverse Cauchy problem from some available data on a part of the boundary. This problem is well known to be highly illposed since the work of Hadamard. The inverse Cauchy problems may arise in the steady-state heat conduction inverse boundary value problems; see, e.g., Mera et al. [2]. The situation is that there are many practical engineering application problems where a part of the boundary is not accessible for direct measurements of temperature and heat flux, but both of them are known on the other part. In order to get the whole temperature field of the body one may encounter the inverse Cauchy problems.

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

For the data completion issues in the elliptic inverse problems there are some tasks to recover either the Dirichlet or Neumann boundary data; see, e.g., Berntsson and Elden [3], Azaiez et al. [4], and Leblond et al. [5]. For the applications of the inverse Cauchy problems on the Robin type exchange coefficient one may refer the works by Fasino and Inglese [6], Chaabane and Jaoua [7], Chaabane et al. [8], Slodicka and Van Keer [9], and Lin and Fang [10]. On the other hand, from the superfluous measurements made on the accessible boundary of a domain one may employ the inverse Cauchy problem technique to detect the geometrical singularities; see, e.g., € et al. [11], Kress [12] and Chapko and Kress [13]. A Bruhl recent review of the inverse Cauchy problems has been given by Ben Belgacem and El Fekih [14]. The inverse Cauchy problem is difficult to solve, since its solution, if exists, does not depend continuously on the given data. Because of this ill-posedness, the errors in measured data will be enlarged in the numerical treatment if we do not take this effect into account. Therefore, we must treat this type problem with a suitable numerical algorithm, which compromises the accuracy and stability. Chang et al. [15] have shown that neither the traditional Tikhonov’s regularization method nor the singular value decomposition method can yield acceptable numerical results for the inverse Cauchy problem of Laplace equation when the influence matrix is highly ill-posed. Lesnic et al. [16] and Mera et al. [17] have applied the boundary element method for the solution of inverse Cauchy problems. Their methods are inevitably required for an iterative process to adjust the solution. In this paper we begin with a modified indirect Trefftz method [18,19] in Section 2, and leave the unknown coefficients determined by the partial boundary conditions from the collocation method. The collocation method is a popular method in the engineering computations for direct problems, because the algebraic equations can be easily derived. However, it is seldom used in the inverse problems because the illposedness is not easy to handle by the collocation method. The method of fundamental solutions (MFS), also called the F-Trefftz method, utilizes the fundamental solutions as basis functions to expand the solution, which is another popularly used meshless method. While Jin and Zheng [20] have applied the MFS to solve the inverse problem of Helmholtz equation, Marin and Lesnic [21] have applied the MFS to solve the inverse Cauchy problem associated with two-dimensional biharmonic equation. In order to tackle the ill-posedness of MFS, these authors proposed new numerical schemes with the regularization parameters determined by the L-curve method. Even, our starting point employs a similar meshless method of Trefftz type, a new modification is required in order to get a non-ill-posed linear equations system. Furthermore, our method uses a very simple regularization of the input data by truncating higher modes. It will be seen that our method is much simpler than that of the MFS. Before the application of the modified indirect Trefftz method, we require to perform a regularization of the

779

accessible boundary data, and then we can obtain a new collocation Trefftz method for the inverse Cauchy problem without needing for any iterations. In Section 3 we give examples to demonstrate the new method, and then the conclusions are drawn in Section 4. 2. The collocation Trefftz method The collocation Trefftz method is popularly used in the engineering computations for the direct problems [22]. We are going to report its modification and application for the inverse Cauchy problem. We replace Eqs. (2) and (3) by the following boundary condition: ( hðyÞ; 0pypbp; uðR; yÞ ¼ F ðyÞ ¼ (4) f ðyÞ; bpoyo2p; where f ðyÞ is an unknown function to be determined. If f ðyÞ can be made available, then the data are completed in the whole boundary, and the solution of Laplace equation can be obtained. Therefore, we face the following inverse problem: Inverse problem. To seek an unknown function f ðyÞ under Eqs. (1)–(3). Here, we suppose that both the functions hðyÞ and gðyÞ are L2 integrable on the interval y 2 ½0; bp. Hence, there are expansions of them in terms of the Fourier series: hðyÞ ¼ a¯ 0 þ gðyÞ ¼ c¯ 0 þ

1 X k¼1 1 X

ð¯ak cos ky þ b¯ k sin kyÞ,

(5)

ð¯ck cos ky þ d¯ k sin kyÞ.

(6)

k¼1

The advantage of the above replacement in Eq. (4) is that we have a series expansion of uðr; yÞ satisfying Eqs. (1) and (4):  1   k  r k X r uðr; yÞ ¼ a0 þ ak cos ky þ bk sin ky , (7) R R k¼1 where Z 1 2p F ðxÞ dx, 2p 0 Z 1 2p ak ¼ F ðxÞ cos kx dx, p 0 Z 1 2p bk ¼ F ðxÞ sin kx dx. p 0 a0 ¼

(8) (9) (10)

By imposing the conditions (2) and (3) on Eq. (7) we obtain a0 þ

1 X ½ak cos ky þ bk sin ky ¼ hðyÞ;

0pypbp,

(11)

k¼1 1 X k k¼1

R

½ak cos ky þ bk sin ky ¼ gðyÞ;

0pypbp.

(12)

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

780

Inserting Eqs. (8)–(10) for the coefficients into the above two equations we can obtain a first kind Fredholm integral equation. Liu [23] has applied the regularized integral equation method to solve this type problem. However, we directly view Eq. (7) as an indirect Trefftz method to expand u in terms of the T-complete functions [24]. Eq. (7) is indeed a modification of the Trefftz method, where we take the size of the circle into account. By letting R ¼ 1 we can recover the Trefftz method. Usually, the characteristic length R is not necessary to be 1. 2

cos y0

sin y0

...

cosðmy0 Þ

cos y1 ðcos y1 Þ=R .. .

sin y1 ðsin y1 Þ=R .. .

... ... .. .

cosðmy1 Þ m cosðmy1 Þ=R .. .

cos ym ðcos ym Þ=R

sin ym ðsin ym Þ=R

... ...

cosðmym Þ m cosðmym Þ=R

1

61 6 6 60 6 6. 6. 6. 6 61 4 0

m X

¯ ½ak cos ky þ bk sin ky ¼ hðyÞ,

k¼1

R

(13) (14)

In the above h¯ and g¯ are the regularizations of h and g obtained by the finite terms truncations from Eqs. (5) and (6):

k¼1 N X g¯ ðyÞ ¼ c¯ 0 þ ð¯ck cos ky þ d¯ k sin kyÞ,

(16)

k¼1

where Npm. Eqs. (13) and (14) are then imposed at different collocated points yi on the interval with 0pyi pbp: a0 þ

¯ i Þ, ½ak cos kyi þ bk sin kyi  ¼ hðy

(17)

k¼1 m X k k¼1

Rc ¼ b1 , where c ¼ ½a0 ; a1 ; b1 ; . . . ; am ; bm T is the vector of unknown coefficients. The conjugate gradient method can be used to solve the following normal equation:

R

(21)

where

½ak cos kyi þ bk sin kyi  ¼ g¯ ðyi Þ.

b :¼ RT b1 .

(18)

It can be seen that the basic idea behind the collocation method is rather simple, and it has a great advantage of the

(22)

Inserting the calculated c into Eq. (7) we thus have a semi-analytical solution of uðr; yÞ, uðr; yÞ ¼ c1 þ

(15)

(20)

Corresponding to the uniformly distributed collocated points on the upper partial circle, y0 ¼ 0 is a single collocated point which is supplemented to provide the nth equation. We denote the above equation by

A :¼ RT R;

N X ¯ hðyÞ ¼ a¯ 0 þ ð¯ak cos ky þ b¯ k sin kyÞ,

(19)

where Dy ¼ bp=ðm þ 1Þ, and yi are the collocated points on the partial circle. When the index i in Eqs. (17) and (18) runs from 1 to m we obtain a linear equations system with dimensions n ¼ 2m þ 1:

Ac ¼ b,

½ak cos ky þ bk sin ky ¼ g¯ ðyÞ.

m X

i ¼ 1; . . . ; m,

3 32 3 2 ¯ 0Þ a0 hðy 7 6¯ sinðmy1 Þ 7 a1 7 hðy1 Þ 7 76 7 6 7 76 6 6 7 6 m sinðmy1 Þ=R 7 b1 7 6 g¯ ðy1 Þ 7 7 76 7 7. 76 .. 7 ¼ 6 .. . 76 6 7 6 .. 7 . . 7 76 7 7 76 6 7 6 7 6 ¯ mÞ 7 sinðmym Þ 54 am 5 4 hðy 5 m sinðmym Þ=R bm g¯ ðym Þ

k¼1 m X k

yi ¼ iDy;

sinðmy0 Þ

Under the case of R41, when the Trefftz method will produce unstable solution, the modified one is still stable. The present purpose is directly using the collocation method to find the unknown coefficients in Eq. (7). The series expansions in Eqs. (11) and (12) are well suited in the partial upper domain of y 2 ½0; bp. Hence, the admissible functions with finite terms can be used to determine the unknown coefficients: a0 þ

flexibility to apply to different geometric shapes, and the simplicity for computer programming. Let

m  k X r k¼1

R

½c2k cos ky þ c2kþ1 sin ky.

(23)

When we apply the modified Trefftz method on the inverse problem by using Eq. (21) we are concerned with its stability. In order to observe this stability problem we plot the condition number of A under different number of bases and different R in Fig. 1, which is defined by CondðAÞ ¼ kAkkA1 k.

(24)

The norm used for A is the Frobenius norm. Therefore, we have 1 lmax ðAÞ CondðAÞp pCondðAÞ, n lmin ðAÞ

(25)

where l is the eigenvalue of A. Conventionally, lmax ðAÞ=lmin ðAÞ is used to define the condition number of A. However, we use Eq. (24) to define the condition number of A.

ARTICLE IN PRESS

forms the T-complete functions, and the solution can be expanded by these bases [22,24]: 1 X uðr; yÞ ¼ a0 þ ½ak rk cos ky þ bk rk sin ky.

(27)

k¼1

It is simply a direct consequence of Eq. (7) by inserting R ¼ 1. However, as noted by Liu [18,19] the new formulation is a modified Trefftz method, which takes the characteristic length of the problem domain R into account. According to the Trefftz method we can derive a linear equations system: 2

1 6 61 6 60 6 6. 6. 6. 6 61 4 0

R cos y0 R cos y1

R sin y0 R sin y1

... ...

cos y1 .. .

sin y1 .. .

... .. .

R cos ym cos ym

R sin ym sin ym

... ...

1E+15 1E+14 1E+13 1E+12 1E+11 1E+10 1E+9 1E+8 1E+7 1E+6 1E+5 1E+4

Modified Trefftz method

10

20 m

30

40

1E+10

1E+9 β=0.5 1E+8

1E+7 β=1 1E+6 1.0

1.5

2.0

2.5

3.0

R Fig. 1. The variations of the condition numbers with respect to m in (a), and R in (b).

32

3 ¯ 0Þ hðy 76 7 6 ¯ 7 76 a1 7 6 hðy1 Þ 7 7 7 6 6 7 6 mRm1 cosðmy1 Þ mRm1 sinðmy1 Þ 7 b1 7 6 g¯ ðy1 Þ 7 76 7 7 6 76 . 7 ¼ 6 . 7. .. .. 76 . 7 6 . 7 . . 76 . 7 6 . 7 76 7 6 7 m m 6 ¯ mÞ 7 R cosðmym Þ R sinðmym Þ 7 54 am 5 4 hðy 5 mRm1 cosðmym Þ mRm1 sinðmym Þ bm g¯ ðym Þ Rm cosðmy0 Þ Rm cosðmy1 Þ

781

Trefftz method

0

Condition Number

From Fig. 1(a), where R ¼ 2 and b ¼ 1 were used, we can see that the condition number is increased with respect to m before mo15, and after that it is less sensitive to m and tends to a constant, which is moderately large. Then we fix m ¼ 10, and plot the condition number with respect to R for b ¼ 0:5 and 1. We can see that the condition number is decreased with respect to R when b ¼ 1, and when b ¼ 0:5 the condition number is larger and is less sensitive to R. The value of b=2 measures the fraction of the available data on the whole boundary. When b is smaller, the available data are less, and usually it is more difficult to recover the unknown data on the inaccessible part. Unless otherwise specified, we will fix b ¼ 1 in the following numerical examples. It is known that for the Laplace equation in the twodimensional domain the set  k  1; r cos ky; rk sin ky; k ¼ 1; 2; . . . (26)

Condition Number

C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

Rm sinðmy0 Þ Rm sinðmy1 Þ

a0

3

2

(28)

This equation is different from Eq. (20). Under the same condition of R ¼ 2 and b ¼ 1 we plot the condition number of the above equation with respect to m as shown by the dashed line in Fig. 1(a). It can be seen that after m420 the condition number is increased fast and is much large than that obtained from the modified Trefftz method.

on the boundary data. We use the function RANDOM_NUMBER given in Fortran to generate the noisy data RðiÞ, which are random numbers in ½1; 1. Hence we use the simulated noisy data given by

3. Numerical examples

where yi ¼ ibp=ðm þ 1Þ; i ¼ 0; 1; . . . ; m þ 1, and  is defined as

Before embarking the numerical study of the new method, we are concerned with the stability of modified collocation Trefftz method, in the case when the boundary data are contaminated by random noise, which is investigated by adding the different levels of random noise

^ i Þ ¼ hðyi Þ þ RðiÞ, hðy

 ¼ maxjhðyÞj 

s , 100

(29)

(30)

where s is the percentage of additive noise on the data.

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

782

3.1. Example 1

gðyÞ ¼ 3R2 sinð3yÞ þ 2R cosð2yÞ;

We first consider a simple example with the exact solution

We apply the modified collocation Trefftz method on this example with very high accuracy as shown in Fig. 5, where R ¼ 1 and m ¼ 3 were used. The boundary data as shown in Fig. 5 were plotted for the lower half circle only, where the solid line represents the exact solution, the dashed line represents the numerical solution without noise, and the dashed-dotted line represents the numerical solution under a noise of s ¼ 1. The first two solutions are almost coincident with an L2 error about 7:1  1011 for the numerical solution. For the noisy case, the numerical error was still with an L2 error smaller than 0.26.

(31)

Therefore, the data on the upper half circle with a radius R are given by hðyÞ ¼ R2 cosð2yÞ; gðyÞ ¼ 2R cosð2yÞ;

0pypp, 0pypp.

(32) (33)

We can apply the modified collocation Trefftz method on this example with very high accuracy as shown in Fig. 2 by displaying the absolute error for R ¼ 1 and m ¼ 2, while the L2 error is about 2:4  1015 . We also plotted the numerical errors in Fig. 2 for other cases with m ¼ 4; 7; 10 but with R ¼ 1. It can be seen that the error increases when m increases. In Fig. 3 we compare the exact solution with the numerical solutions under the noises with s ¼ 2 and 5. It can be seen that the numerical solutions are close to the exact solution, which indicates that the present method is robust against the noise, and even whose level was taken up to 5% ðs ¼ 5Þ, the numerical error was still with an L2 error smaller than 0.34. In Fig. 4 we display the influence of b. Under m ¼ 2 we have found that when bX0:5 the numerical solution is coincident with the exact solution very well with the L2 error about in the order of 7:2  1015 . When bo0:5 the numerical solution is less accurate as shown in Fig. 4 by the dashed-dotted line for b ¼ 0:45.

(36)

2 Exact solution Numerical solution with s =2 Numerical solution with s =5

1

f (θ)

u ¼ x2  y2 ¼ r2 cosð2yÞ.

0pypp.

0

-1

3.2. Example 2 -2

Next, we consider an example with the exact solution u ¼ y3  3x2 y þ x2  y2 ¼ r3 sinð3yÞ þ r2 cosð2yÞ,

and the data on the upper half circle with a radius R are given by hðyÞ ¼ R3 sinð3yÞ þ R2 cosð2yÞ;

0pypp,

3

(34)

4

6

7

θ Fig. 3. For Example 1 the exact solution and numerical solutions with noise.

(35) 2

1E-7

Exact Numerical solution with β=0.45

1E-8

Numerical solution with β=0.5

1E-9

m=10

1E-10 m=7

1E-11

f (θ)

Numerical Error

5

1E-12

0

1E-13 1E-14

m=4

1E-15 1E-16

m=2

1E-17 -2

1E-18 3

4

5 θ

6

Fig. 2. Plotting the numerical errors for Example 1.

7

0

2

4 θ

6

Fig. 4. For Example 1 the influence of b.

8

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

783

1.2

ck

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2

ck

Exact solution

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

Numerical with s =0 Numerical with s =1

0.8

f (θ)

0.4

0.0

-0.4

0

100

200

300

200

300

k -0.8 4

5 θ

6

0.7

7

Fig. 5. For Example 2 the exact solution and numerical solutions with and without noise.

0.6 Error of ck

3

0.5 0.4 0.3 0.2

3.3. Example 3

0.1 0.0

In this example we investigate a discontinuous boundary condition on the unit circle: ( 1; 0oyop; F ðyÞ ¼ (37) 1; poyo2p: For this example the exact solution is given by  2 2y uðx; yÞ ¼ arctan . p 1  x2  y2

(38)

Therefore, the data on the upper half unit circle are given by hðyÞ ¼ 1;

0oyop,

(39)

2 ; 0oyop. (40) p sin y It is noted that the function gðyÞ near to y ¼ 0 and p is singular. However, by the odd extensions of h and g there are the following Fourier expansions: gðyÞ ¼

¯ hðyÞ ¼ g¯ ðyÞ ¼

N X 2 ½1  ð1Þk  sin ky, kp k¼1 N X 2 k¼1

p

½1  ð1Þk  sin ky.

(41) (42)

Since this problem is more difficult to solve due to the singularity of g and the discontinuity of h, we have employed m ¼ 153 collocated points in Eq. (20), where h¯ and g¯ were inserted by Eqs. (41) and (42) with N ¼ m. The y0 is fixed to be y0 ¼ p=2. In Fig. 6(a) the computed

0

100 k

Fig. 6. The Fourier coefficients for Example 3 and the numerical error.

Fourier coefficients were plotted. It can be seen that ck X0 and that the convergence of ck is slow after the first few dominant terms. When we compare the above ck with the exact ck obtained by inserting Eq. (37) into Eqs. (8)–(10) and plotted in Fig. 6(b), the numerical errors are plotted in Fig. 6(c), of which the maximum absolute error is smaller than 0.637. It can be seen that both the numerical and exact coefficients have the similar convergent tendency. In Fig. 7(a) we have compared the exact solution with the numerical solutions under s ¼ 0 and 1. It can be seen that the numerical solution without noise is close to the exact solution with an L2 error about 0.077, of which the absolute error is plotted in Fig. 7(b). Due to the above mentioned discontinuity of h, the errors at the two ends are larger than that in the other part. When the noise is including, it can be seen that the present method is also acceptable, which can be against the noise. In principle we can use a smaller N, which means that we are fitting the available data more loosely. In Fig. 7(a) we also compare the numerical result under N ¼ 100 but fixed m ¼ 153 with the exact solution. Its absolute error is plotted in Fig. 7(b), of which the maximum error is smaller than 9  103 . Very interestingly, under N ¼ 100 the L2 error with about 0.033 and the absolute error are both smaller than that under N ¼ 153. It indicates that a

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

784

-0.8

Exact solution

[25,26]:

Numerical with s = 1 and N = 10

uðx; yÞ ¼ cos x cosh y þ sin x sinh y.

Numerical with s = 0 and N = 153

The exact boundary data can be easily derived as follows:

Numerical with s = 0 and N = 100

-0.9

(43)

f (θ)

hðyÞ ¼ cosðcos yÞ coshðsin yÞ þ sinðcos yÞ sinhðsin yÞ, 0pypbp,

-1.0

-1.1

(44)

gðyÞ ¼  cos y sinðcos yÞ coshðsin yÞ þ sin y cosðcos yÞ sinhðsin yÞ þ cos y cosðcos yÞ sinhðsin yÞ þ sin y sinðcos yÞ coshðsin yÞ,

-1.2 3

4

5 θ

6

7

We apply the modified collocation Trefftz method on this example by directly fitting the above data as that done in Examples 1 and 2, whose result is very accurate as shown in Fig. 8, where m ¼ 10 was fixed. The boundary data F ðyÞ as shown in Fig. 8(a) were plotted for the whole circle, where the solid line represents the exact solution, the dashed line represents the numerical solution with b ¼ 1, and the dashed-dotted line represents the numerical

1.8E-2 Numerical with N=153

Numerical Error

1.5E-2

(45)

0pypbp.

Numerical with N=100

1.2E-2 9.0E-3 6.0E-3

Exact solution Numerical solution with β=1

3.0E-3

2

Numerical solution with β=0.8

0.0E+0 3

4

5 θ

6

7

3.4. Example 4

1

0 0

2

3

4

5

6

7

1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11

Numerical solution with β=1

1E-12

Numerical solution with β=0.8

1E-13 1E-14 0

For this example the solution domain is a simple unit disk. To illustrate the accuracy and stability of the new method we consider the following analytical solution

1

θ

Numerical Error

suitable regularization can be achieved by a looser data fitting, which is better than that uses a strict data on the collocation method. Indeed, when we replace the h¯ and g¯ by the exact data h and g given in Eqs. (39) and (40) in the collocation equation (20), we obtain a very bad numerical result. This situation is very different from that by using the collocation method on the direct problem, of which a more accurate fitting of the exact data always leads to a better numerical result. However, for the inverse problem, due to its ill-posedness we must relax the requirement to perfectly match the exact boundary data. Instead, we are matching an inaccurate (not perfect) boundary data to arrive at a better numerical result for the inverse problem. Therefore, we are said that the present modified collocation Trefftz method by matching a finitely truncated Fourier data is a new type regularization of the inverse Cauchy problem.

F (θ)

Fig. 7. Comparing the exact solution and numerical solutions with and without noise in (a), and plotting the numerical errors in (b) for Example 3.

2

4

6

θ

Fig. 8. Comparing the exact solution and numerical solutions with b ¼ 1; 0:8, and plotting the numerical errors in (b) for Example 4.

ARTICLE IN PRESS C.-S. Liu / Engineering Analysis with Boundary Elements 32 (2008) 778–785

solution with b ¼ 0:8. These two numerical solutions are almost coincident with the exact solution, with an L2 error about 4:9  105 for the numerical solution with b ¼ 1, and with an L2 error about 1:4  103 for the numerical solution with b ¼ 0:8. The absolute errors of these two numerical solutions are plotted in Fig. 8(b), with the maximum error smaller than 105 for the numerical solution with b ¼ 1, and with the maximum error smaller than 104 for the numerical solution with b ¼ 0:8. Upon comparing with the numerical results in [25,26], we can claim that the present modified collocation Trefftz method is much better than the numerical methods in [25,26]. 4. Conclusions We have employed a new idea to treat the inverse Cauchy problem by a modified collocation Trefftz method. The given functions on the accessible boundary are assumed to be integrable, and hence there exist the Fourier expansions of the considered data. We have used the numerical examples to explain that the present collocation method by matching a finitely truncated Fourier data is a new type regularization of the inverse Cauchy problem. The new method can provide us a semi-analytical solution in terms of the Trefftz method, which renders a rather compendious numerical implementation to solve the inverse Cauchy problems without needing for any iteration. The new method was found to be accurate, effective and stable. Even we only considered the ill-posed problem in a disk, the new idea used here can be extended to other problem in a more complex region. More importantly, the present study shortens the gap between direct and inverse problems. The difference is just that when we apply the modified Trefftz method on direct problem we need to collocate the given boundary data as precise as possible, but for the inverse problem we need to collocate a less precise data than the given one. References [1] Mera NS, Elliott L, Ingham DB. On the use of genetic algorithms for solving ill-posed problems. Inverse Probl Eng 2003;11:105–21. [2] Mera NS, Elliott L, Ingham DB, Lesnic D. The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium. Int J Numer Methods Eng 2000;49:481–99. [3] Berntsson F, Elde´n L. Numerical solution of a Cauchy problem for the Laplace equation. Inverse Probl 2001;17:839–54. [4] Azaiez M, Ben Abda A, Ben Abdallah J. Revisiting the Dirichletto-Neumann solver for data completion and application to some inverse problems. Int J Appl Math Mech 2005;1:106–21.

785

[5] Leblond J, Mahjoub M, Partington JR. Analytic extensions and Cauchy-type inverse problems on annular domains: stability results. J Inverse Ill-Posed Probl 2006;14:189–204. [6] Fasino D, Inglese G. An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods. Inverse Probl 1999;15:41–8. [7] Chaabane S, Jaoua M. Identification of Robin coefficients by the means of boundary measurements. Inverse Probl 1999;15: 1425–38. [8] Chaabane S, Elhechmi C, Jaoua M. A stable method for the Robin inverse problem. Math Comput Simul 2004;66:367–83. [9] Slodicˇka M, Van Keer R. A numerical approach for the determination of a missing boundary data in elliptic problems. Appl Math Comput 2004;147:569–80. [10] Lin F, Fang W. A linear integral equation approach to the Robin inverse problem. Inverse Probl 2005;21:1757–72. [11] Bru¨hl M, Hanke M, Pidcock M. Crack detection using electrostatic measurements. Math Model Numer Anal 2001;35:595–605. [12] Kress R. Inverse Dirichlet problem and conformal mapping. Math Comput Simul 2004;66:255–65. [13] Chapko R, Kress R. A hybrid method for inverse boundary value problems in potential theory. J Inverse Ill-Posed Probl 2005;13:27–40. [14] Ben Belgacem F, El Fekih H. On Cauchy’s problem. I. a variational Steklov–Poincare´ theory. Inverse Probl 2005;21:1915–36. [15] Chang JR, Yeih W, Shieh MH. On the modified Tikhonov’s regularization method for the Cauchy problem of the Laplace equation. J Mar Sci Technol 2001;19:113–21. [16] Lesnic D, Elliott L, Ingham DB. An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation. Eng Anal Bound Elem 1997;20:123–33. [17] Mera NS, Elliott L, Ingham DB, Lesnic D. An iterative boundary element method for the solution of a Cauchy steady state heat conduction problem. CMES: Comput Model Eng Sci 2000;1:101–6. [18] Liu CS. A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length. CMES: Comput Model Eng Sci 2007;21:53–66. [19] Liu CS. A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain. CMES: Comput Model Eng Sci 2007;20:111–22. [20] Jin B, Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng 2006;195:2270–88. [21] Marin L, Lesnic D. The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation. Math Comput Model 2005;42:261–78. [22] Li ZC, Lu TT, Huang HT, Cheng AHD. Trefftz, collocation, and other boundary methods—A comparison. Numer Methods Par Diff Eq 2007;23:93–144. [23] Liu CS. A meshless regularized integral equation method for Laplace equation in arbitrary interior or exterior plane domains. CMES: Comput Model Eng Sci 2007;19:99–109. [24] Kita E, Kamiya N. Trefftz method: an overview. Adv Eng Software 1995;24:3–12. [25] Hayashi K, Ohura Y, Onishi K. Direct method of solution for general boundary value problem of the Laplace equation. Eng Anal Bound Elem 2002;26:763–71. [26] Delvare F, Cimetie´re A, Pons F. Two iterative boundary element methods for inverse Cauchy problems, In: Brebbia CA, Power H, editors, Boundary elements, vol. XXII, WIT Press; 2000. p. 211–20.