Least square based radial basis collocat
Least-square-based radial basis collocation method for solving inverse problems of Laplace equation from noisy data
Xian-Zhong Mao ∗, † and Zi Li
Research Center for Environmental Engineering and Management, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, People’s Republic of China
SUMMARY
The inverse problem of 2D Laplace equation involves an estimation of unknown boundary values or the locations of boundary shape from noisy observations on over-specified boundary or internal data points. The application of radial basis collocation method (RBCM), one of meshless and non-iterative numerical schemes, directly induces this inverse boundary value problem (IBVP) to a single-step solution of a system of linear algebraic equations in which the coefficients matrix is inherently ill-conditioned. In order to solve the unstable problem observed in the conventional RBCM, an effective procedure that builds an over-determined linear system and combines with least-square technique is proposed to restore the stability of the solution in this paper. The present work investigates three examples of IBVPs using over-specified boundary conditions or internal data with simulated noise and obtains stable and accurate results. It underlies that least-square-based radial basis collocation method (LS-RBCM) poses a significant advantage of good stability against large noise levels compared with the conventional RBCM. Copyright q 2010 John Wiley & Sons, Ltd.
Received 28 November 2008; Revised 26 January 2010; Accepted 31 January 2010 KEY WORDS : inverse problems of Laplace equation; radial basis collocation method; least-square
technique; noise levels
1. INTRODUCTION
In many practical situations, the known boundary data are not complete due to technical difficulties associated with measurement and acquisition of accurate data. For example, a part of boundary is inaccessible to direct measurement, or the presence of measuring devices disturbs the physical process under investigation. Even the location of boundary itself is sometimes unknown resulting
∗ Correspondence to: Xian-Zhong Mao, Research Center for Environmental Engineering and Management, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, People’s Republic of China.
† E-mail: maoxz@sz.tsinghua.edu.cn Contract/grant sponsor: SRF for ROCS, SEM
Contract/grant sponsor: National Water grant; contract/grant number: 2008ZX07423-001
2 X.-Z. MAO AND Z. LI
from special geological features. Therefore, the inverse boundary value problem (IBVP) arises to deal with the estimation of the recovered boundary conditions on unspecified part of the boundary. In order to make the problem solvable, partial accessible boundary may be over-specified; for example, the Dirichlet and the Neumann conditions are simultaneously prescribed, in some cases the auxiliary internal data are necessary for input.
The IBVPs arise from many engineering applications such as heat transfer [1], geophysical prospecting [2], medical imaging and non-destructive testing [3] and acoustic and electromagnetic waves [4]. It is well known that the inverse problems have a typical characteristic of ill-posedness in the sense that a slight error in the input data may produce an enormous change in the output solution that makes them more difficult to deal with. However, the measurement inevitably poses some noises due to the technical and physical difficulties. It is required to develop a numerical method to solve the inverse problem with noisy data input.
Traditional methods, such as finite difference method (FDM), finite element method (FEM), finite volume method (FVM) and boundary element method (BEM), have been proposed for these kinds of inverse problems, which require to combine with special optimization schemes to solve the direct problems through iterative procedure till the convergent solution is obtained. Essentially they need a mesh on the domain to support the solution process, which will cost large consuming time for generating a good quality mesh for complicated geometry, especially in high dimensions. Although the BEM reduces the dimensionality of the problem and partially alleviates the difficulty, it requires the evaluation of singular integrals for using singular fundamental solutions, and employing an iterative BEM based on regularization algorithm to obtain the convergent solutions in the inverse Laplace problems [5, 6].
In recent years, meshfree methods (MMs), including the method of fundamental solutions (MFS) [ 7], the boundary knot method (BKM) [8] and the radial basis collocation method (RBCM) [9], emerged as competitive alternatives to the traditional methods due to their meshfree and non- iterative characteristics for the inverse problems. All of the MMs directly induce the inverse problem to a single-step solution of a system of linear algebraic equations. However, the MFS is a versatile boundary-type meshless numerical scheme and free from treatment of singularities since the source points are always placed outside the domain. It has been successfully applied to the inverse heat conduction problem [10], backward heat conduction problem [11], Cauchy problem associated with Helmholtz-type equation [12] and Cauchy problem of Laplace equation [
13, 14]. Another boundary-type meshfree technique is the BKM which does not require the artificial boundary and approximates the homogeneous equation’s solution by a linear combination of non-singular general solutions of the differential operator instead of singular fundamental solutions in the MFS. BKM has been proposed for the solution of Cauchy problem associated with the inhomogeneous Helmholtz equation [15]. The RBCM is a domain-type numerical method in which the collocation points are randomly distributed in the domain and on the boundary, and the radial basis functions (RBFs) have many different choices. Hon and Wu [16] utilized Hermite–Birkhoff collocation method in which the RBFs were replaced by harmonic functions with shift invariability to determine an unknown boundary. Li [17, 18] used the conical-type RBFs to solve IBVPs of the second-order elliptic equation and biharmonic equation. Cheng and Cabral [19] chose the inverse multiquadric (IMQ) as the RBFs to solve several types of ill-posed IBVPs for the Laplace equation. Compared with MFS and BKM, which are fully dependent on the availability of the fundamental solutions or the general solutions of governing equations, the RBCM seems to be a more general numerical approach for the inverse problems in practical applications.
3 No matter which MM is adopted, the inverse problems can be converted to a large-scale system
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
of linear algebraic equations in which a number of unknown coefficients need to be determined. However, the coefficient matrix is inherently ill-conditioned and the solution is highly sensitive to the noise of measured input data. Many regularization methods are additionally employed to obtain stable solutions, for example, the standard Tikhonov regularization (TR) technique with the L-curve criterion (LC) [10–12], the truncated singular-value decomposition (TSVD) with the LC [15] and three regularization strategies (TR, TSVD and damped singular-value decomposition) under the different choices for the regularization parameter [14]. Although Li [17, 18] and Cheng and Cabral [19] achieved some encouraging results through the RBCM, they did not test the bad conditioning of the coefficient matrix that may cause the unstable results from noisy data input.
In this paper, we apply the least-square-based RBCM to the inverse problems of Laplace equation with noisy data input. In the proposed approach, an over-determined linear system of algebraic equations is constructed and the least-square procedure is naturally necessary to obtain the stable solution. This paper is organized as follows. Section 2 describes the mathematical formulation of inverse Laplace problems. Section 3 presents the least-square-based radial basis collocation method (LS-RBCM), and its adaptation to the problem is discussed. Section 4 tests the proposed methodology and compares with the conventional RBCM using three examples. The numerical examples include the problem of under-specified boundary recovered from internal data, identification of missing boundary condition from over-specified boundary conditions and shape identification problem. And the sensitivity of the LS-RBCM method with respect to the shape parameter and mesh size is investigated. Finally, the conclusion is given in Section 5.
2. INVERSE PROBLEMS FOR THE LAPLACE EQUATION The mathematical formulation of the 2D Laplace equation, as the governing equation for a field is defined by the following expression:
0 2 (1) And partial boundary condition is given by
∇ 2 u=
f is the prescribed boundary condition
2 2 . If the inverse problem holds, it is necessary to add other conditions together with Equation (2) to make Equation (1) identifiable. Two types of inverse problem arise subject to the different conditions.
If the following boundary condition is given:
in which, g 1 , the problem described by Equations (1)–(3) is the
1 is
4 X.-Z. MAO AND Z. LI
2 is under-specified. Equation (3) compensates for the missing boundary
2 . The Cauchy problem for the Laplace equation arises from many fields in science and engineering, such as non-destructive testing [20], steady-state inverse heat conduction [21] and electro-cardiology [22].
1 by physical measurement that inevitably contains certain level noises. The Cauchy problem is very difficult to
f and fluxes g
be solved both numerically and analytically since the solution does not depend continuously on the prescribed boundary conditions [23]. Another inverse problem needs the assistance of the following auxiliary internal information together with Equation (2) to make Equation (1) solvable:
1 to stabilize the solution. The problem presented by Equations (1), (2) and (4) is another commonly used inverse problem in the Laplace equation.
3. LEAST-SQUARE-BASED RADIAL BASIS COLLOCATION METHOD
3.1. The RBCM The RBCM is one of MMs to solve IBVPs, which does not require the discretization of compu-
tational domain and is independent of fundamental or general solution to particular differential operators. In RBCM, the numerical solution for the potential or temperature is approximated by the following linear combination of radial basis functions:
where m is total number of the collocation points distributed in two-dimensional domain; x k k represents the coefficient to be determined by the collocation method using the governing equation and the prescribed boundary conditions; k ( x,x k ) is radial basis function. In the present work, the IMQ is chosen as RBF
( x,x )=( c 2 x−x 2 ) − k 1/2 k k
is usually taken to be the Euclidean metric; c is a constant referred as the shape parameter, which leads to the scheme’s exponential convergence property. Hardy [24], Kansa and Carlson [25] and Golberg et al. [26] provided some useful guidelines of determining the optimal shape parameter.
Taking the Cauchy problem for example, we can use Figure 1 to illustrate how RBCM obtain a system of linear algebraic equations to solve the inverse problem of Equations (1)–(3). As shown
1 (solid line)
2 (dashed line). The geometrical configuration with the distribution of collocation points is shown in Figure 1 too. Different types of collocation points are marked in different symbols that satisfy the corresponding conditions, as listed in Table I.
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
Figure 1. Distribution of collocation points in the conventional RBCM (a) and LS-RBCM (b) for the Cauchy problem.
Table I. Different symbols denoting different collocation points satisfying the corresponding conditions. The symbol
The condition(s) satisfied ×
The governing equation ◦
The Dirichlet conditions △
The Neumann conditions The Dirichlet and the Neumann conditions •
The Dirichlet conditions and governing equation The Neumann conditions and governing equation
The Dirichlet and the Neumann conditions, and governing equation
As shown in Figure 1(a), there are n 1
2 ) shown in × symbol satisfying the governing equation only, n 2 points on the known boundary
1 shown in ◦ symbol satisfying the Dirichlet conditions only and n 3 points on the known
1 shown in △ symbol satisfying the Neumann conditions only. Totally there are m = ( n 1 + n 2 + n 3 ) collocation points in the conventional RBCM. By substituting Equation (5) into the governing equation (1) and the prescribed boundary conditions (2) and (3), and collocating on the corresponding points, we are able to obtain the following well-determined system of linear algebraic equations:
∇ 2 k ( x i , x k ) k = 0, i= 1, . . . , n 1 , x i ∈
k= 1 m
f (x i ), i=n 1 + 1, . . . , n 1 + n 2 , x i ∈ 1 (8)
k= 1
k ( x i , x k ) k = g(x i ), i=n 1 + n 2 + 1, . . . , n 1 + n 2 + n 3 , x i ∈ 1 (9)
k= 1 *n
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Equations (7)–(9) form a linear system of (n 1 + n 2 + n 3 ) equations with m(= n 1 + n 2 + n 3 ) unknown coefficients, which can be written in the following form:
[ A ik ][ k ]=[ b i ]
(10) k ] is the unknown coefficient; [ A ik ] is the resultant square coefficients matrix; [b i ] is the
column vector that contains the prescribed boundary conditions. The coefficients matrix and the right-hand column vector are expressed as
If the square matrix [ A ik ] is non-singular, the solution of Equation (10) can be solved by the Gaussian elimination method [16], the Linpack’s Cholesky decomposition [13] or LU decompo- sition solver [17].
However, the coefficient matrix [ A ik ] in Equation (10) obtained by the simple collocation method is not usually well-conditioned and such a strong-form method can provide unstable solutions,
7 especially when the input contains large noise. Kee et al. [27] believed that such oscillation is
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
caused by the strong requirement for the approximated field variable to satisfy governing equation and boundary conditions exactly at all the nodes. They have further found that the Neumann boundary condition is the cause of the instability. It could be due to the ‘strong’ requirement of satisfaction of the Neumann boundary condition in the strong formulation.
To provide a kind of ‘relaxation’ effect, we employ the double boundary collocation approach [ 28] and the double use of boundary collocations [8] to construct an over-determined system for Equation (10) and combine with least-square method to stabilize the solutions for the 2D inverse Laplace problems. The double boundary collocation approach [28] requires that the governing equation is collocated at all the collocation points in the domain, including on the boundary, namely the collocation points on the boundary satisfy not only the boundary conditions, but also the governing equation. The resultant coefficients matrix is still well-determined in [28]. The double use of boundary collocations [8] requires the Dirichlet conditions and the Neumann conditions collocating on the same nodes of the boundary. Compared with the conventional RBCM, two types of collocation points are used in the present RBCM: on total m points in the whole domain (shown
1 (shown the double use of boundary collocations. Hence by the governing equation collocating on m points
1 , we are able to have an over-determined system of linear algebraic equations. With the new collocation approach, [ A ik ] becomes a rectangular coefficients matrix in the system of linear algebraic equations (10), which has m unknown coefficients to be determined in ( m+ 2n) linear equations system. Equation (10) is an over-determined system with m +2n>m. This aims to relax the ‘strong’ requirement of satisfaction of the governing equation and boundary conditions to improve the conditioning of the coefficient matrix and the solution accuracy. For an over-determined system (10), the least-square method can provide a best-fit approximation to the solution.
3.2. Least-square technique using the singular-value decomposition (SVD) with rank reduction k ] in (10) can be determined by minimizing L 2 norm using the
least-square method in the following over-determined system:
(11) The least-square problem of Equation (10) can be solved using QR-algorithm [27]. In the present
process, the singular-value decomposition (SVD) is applied to find the solution to the least-square problem. The matrix [ A ik ] is decomposed using the SVD method as follows:
where the column vectors u k ,v k k are the
1 m− 1 m .
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Using the SVD of matrix A, it is straightforward to show the ordinary least-squares solution to Equation (10).
Because the over-determined system (10) from the discrete ill-posed problem is still ill-conditioned and the condition number shown as
CDN = 1
is extremely large, the error in the directions corresponding to smaller singular values is greatly magnified and overwhelms the information contained in the directions corresponding to larger
b[ 29]. Hence, we should filter out enough noise without losing too much information in the computed solution by finding a good regularization parameter that controls how much filtering is introduced.
In our computation, the SVD with rank reduction [30] is employed to regularize the solution and the regularization parameter is in essence the tolerance whose value defaulted in the Matlab code is
(15) where max[size( A)] is the maximum one of (m +2n) and m; norm( A) is the 2-norm of matrix
max[size( A)]·norm( A)·eps
1 ; eps is the floating-point relative accuracy.
r+ 1 m and obtain the regularization solution of Equation (11).
The objective of this paper is to illustrate the advantage of the LS-RBCM in aspects of stability and accuracy against larger noise level in contrast with the conventional RBCM. For comparison, the square coefficient matrix [ A ik ] resulted from the conventional RBCM is also solved using the solution (16) in the numerical examples.
The capability of the LS-RBCM to resist the noises will be demonstrated numerically in the following sections. And accuracy and stability of the LS-RBCM are also discussed.
4. NUMERICAL EXAMPLES
For simplicity, we use random noise simulating the measurement noise in the numerical examples. The simulated noisy data is generated by
i = i [ 1+e ·rand(i )]
i is the exact data; rand(i ) is the random number chosen between [−1, 1] and e is the noise level. The computed results show that the behavior of the proposed method against noise
9 level is similar for different random noises. For comparison, we choose only one group of random
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
noise for each example. The accuracy of the numerical solution to verify the efficiency and the stability of the proposed method is measured by the following relative root mean square error (RRMSE):
where {x j } is the set of test points; N is total number of test points; u a ( x j ) and u n ( x j ) are the analytical solution and numerical solution at the test points, respectively. In our computation, we take N = 21 and {x j } is uniformly distributed on the boundary.
4.1. Problem with under-specified boundary condition and internal data Onishi [31] investigated the ill-posed Laplace problem of potential flow on a square domain
[ 0, 3]×[0, 3], which has the following exact solution: u(x , y) = x 2 − y 2 (19) and the Dirichlet conditions prescribed on the left, top, and right sides of boundary as follows: u( 0, y) = −y 2 , u(x , 3) = x 2 − 9, u( 3, y) = 9− y 2 (20) No boundary condition is given on the bottom side. Obviously, the problem is not identifiable
unless sufficient measurement data in the interior points are supplied. The internal potential values on four points are given based on the exact solution
u( 1, 1) = 0, u( 1, 2) = −3, u( 2, 1) = 3, u( 2, 2) = 0 (21) for compensation to numerically identify the Dirichlet condition on the bottom side of boundary
(22) Cheng and Cabral [19] investigated the numerical example, and reported that the accuracy of RBF
u(x ,
0) = x 2
solution is somewhat better than the FEM solution in [31] with the relatively sparse mesh 7×7. No noise was considered for input data.
In the present work, the simulated noises are imposed with different levels from 10 − 3 to 0.4 on the internal potential values in Equation (21). For comparison, the distribution of collocation points for the conventional RBCM and LS-RBCM is shown in Figure 2 and the collocation points are distributed on a 7×7 grid too. The collocation points are represented by different types of symbols satisfying the corresponding conditions as listed in Table I. It also needs to select a proper value c for IMQ function in Equation (6), and we choose c = 4 as Cheng and Cabral [19] recommended.
The recovered boundary conditions u(x, 0) using the conventional RBCM and LS-RBCM with different noise levels are plotted in Figure 3. The results show that when noise level is smaller than 0.01, the solutions of both methods agree very well with the exact ones, and when the noise level goes to 0.1, 0.2 and 0.4, the LS-RBCM solutions perform much better than the ones of conventional RBCM. When the noise level is up to 40%, the RRMSE of the LS-RBCM solution is about 18.1%, which can be acknowledged as an acceptable error in many practical applications;
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Figure 2. Distribution of collocation points in the conventional RBCM (a) and LS-RBCM (b).
Exact solution 8 8 Exact solution e=0 e=0 e=0.001
u(x,0)
4 u(x,0) 4
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 (a)
(b)
Figure 3. The recovered u(x, 0) with different noise levels in the conventional RBCM (a) and LS-RBCM (b).
whereas the error from the conventional RBCM is greater than 48.8%, and the approximation curve is deteriorated, as shown in Figure 3(a).
Figure 4 shows the RRMSE distribution of both methods against different noise levels. An attractive phenomenon is observed that there exists a critical point at noise level e = 0.01, which fully shows different advantages for both methods in different noise ranges: when the noise level e<0.01, the solutions of conventional RBCM are better; whereas when the noise level e>0.01, the solutions of LS-RBCM are much better than the ones of conventional RBCM.
We also observe from Figure 4 that the input with small noise (less than 10 − 4 ) can match the ‘strong’ requirement to satisfy governing equation and boundary conditions exactly at all the nodes in the conventional RBCM. Hence the solution is not sensitive to the noise, and the accuracy of
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
RBCM
RRMSE(u(x,0))
Noise level e
Figure 4. The RRMSE of recovered u(x, 0) against noise level in the conventional RBCM and LS-RBCM.
the solution depends only on the method of RBCM, not the noise. When the noise level is greater than 10 − 3 , the RRMSE is proportional (very sensitive) to the noise level, and the accuracy depends on the noise only. Under the large noise-level condition, this ‘strong’ requirement may cause the solution distorted as shown in Figure 3(a).
However, LS-RBCM provides ‘relaxation’ effect for the requirement, which aims to search for
a best-fit approximation to the solution in a new-constructed over-determined system. The solution of LS-RBCM is not sensitive to the noise till e = 0.01, and if noise level is greater than 0.01, the RRMSE is proportional (sensitive) to the noise level. Fortunately, the gradient of RRMSE curve is relatively smaller, and the solution is still a best-fit one as shown in Figure 3(b). The numerical solutions from the input with larger noise are under-estimated since it might be caused by bias of the random noise chosen only if four points are used.
It illustrates that ‘relaxation’ effect in LS-RBCM strengthens the stability of the solution and improves the accuracy to resist the large noise levels, but somewhat sacrifices the accuracy of the numerical solution within the smaller noise levels. The over-determined system improves ill-conditioning of the coefficient matrix and the condition number defined by Equation (14)
decreases from 1.68×10 11 in conventional RBCM to 9.39×10 9 in LS-RBCM. It is interesting to note that there exists an error threshold for both methods beyond which the RRMSE increases with the noise-level increase. Theoretically, the accuracy of the method is measured by the truncated error, which depends on the radial basis function, mesh size and shape parameter. The conventional RBCM possesses high accuracy due to the ‘strong’ requirement to satisfy governing equation and boundary conditions exactly at all the nodes. As the noise level of the input increases, both methods search for a best-fit solution through regularization technique for Equation (10). If the noise is less than the truncated error, the accuracy of the solution is dominated by the truncated error, which is shown as an error threshold in Figure 4. If the noise is greater than the truncated error, the accuracy of the solution is influenced by the noise level, which leads to the RRMSE increase with the noise-level increase.
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9 8 Exact solution
e=0.2 5 e=0.4
u(x,0) 4
Figure 5. The recovered u(x, 0) with eight internal data in LS-RBCM.
In order to test the sensitivity of the solution to the input information, we further provide the internal data on four additional points as follows:
1.5, 1) = 1.25 (23) The input is also contaminated by the noise. Hence, we have four additional collocation points.
u( 1, 1.5) = −1.25, u(
1.5, 2) = −1.75, u( 2, 1.5) = 1.75, u(
It requires that four additional linear algebraic equations are included in the system (10), whereas the number of unknown coefficients remains constant (still based on the 7×7 grid). The numerical solutions and the RRMSE distribution are shown in Figures 5 and 6, respectively. The RRMSE of recovered u(x, 0) with eight interior nodes input is always smaller for all noisy levels. The RRMSE curve shifts 30–40% downward as shown in Figure 6 and the recovered u(x, 0) is better fit to the exact one as shown in Figure 5. It means the better the solution of LS-RBCM is with more input information at the same noise level, even the input is contaminated with larger noise. When the noise level is 40%, the RRMSE is reduced to 13.8 from 18.1%. It is quite encouraging for practical applications. The targeted solution can be achieved at the cost of more collecting data.
Although many researchers, for example, Hardy [24], Kansa and Carlson [25] and Golberg et al. [ 26], provided some useful guidelines for determining the optimal shape parameter c, it is of interest to investigate the convergence of the LS-RBCM solution with respect to the value. The RRMSE of recovered u(x, 0) based on 7×7 grid is shown in Figure 7. It is clear to illustrate that the shape parameter c ranging from 0.2 to 8 leads to almost exponential convergence for the numerical solution u(x, 0) in LS-RBCM, and the RRMSE keeps stable when c is larger than
8, which should be the optimal region. Obviously, c = 4 in above discussion is not optimal one. Also we investigate the convergence of the solution obtained from the LS-RBCM with respect to the size of collocation mesh. The results are shown in Figure 8. It illustrates that the RRMSE of estimated u(x, 0) tends to decrease very quickly when the mesh size increases from m = 3×3 to m= 13×13, then keeps stable when the mesh size is greater than 15×15. It is pointed out that
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
10 0 4 internal nodes
8 internal nodes
RRMSE(u(x,0))
Noise level e
Figure 6. The RRMSE of recovered u(x, 0) with four and eight internal data in LS-RBCM.
RRMSE(u(x,0))
Figure 7. The RRMSE of recovered u(x, 0) in LS-RBCM with different value c.
the results in Figure 8 are obtained for a fixed shape parameter c = 4, which should not be the optimal one for different mesh sizes.
4.2. Problem with over-specified boundary condition Lesnic et al. [5] considered the following ill-posed Cauchy problem of heat conduction on a square
domain [0, 1]×[0, 1] with the over-specified boundary condition on partial boundary. The exact solution of the Laplace equation is
u(x , y) = cos(x) cosh(y)+sin(x) sinh(y)
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RRMSE(u(x,0))
Figure 8. The RRMSE of recovered u(x, 0) in LS-RBCM with different collocation mesh m.
The boundary condition is missing on one side of the domain. It assigns the Dirichlet condition, the Neumann condition, and both the Dirichlet and the Neumann conditions on the bottom, top and right sides of the boundary based on the exact solution (24), respectively.
u(x , 0) = cos(x)
*u(x, 1)
= cos(x) sinh(1)+sin(x) cosh(1)
*y u( 1, y) = cos(1) cosh(y)+sin(1) sinh(y)
*u( 1, y)
=− sin(1) cosh(y)+cos(1) sinh(y)
*x
The Cauchy problem arises to recover the following Dirichlet and the Neumann conditions on the left-side boundary:
u( 0, y) = cosh(y)
*u( 0, y)
= sinh(y)
*x
Young et al. [13] used the double precision MFS to solve the temperature and heat flux distri- butions on the missing boundary. It showed that the results had good agreement with the exact solution when boundary conditions were imposed with noise level from 10 − 5 to 10 − 3 , and when larger noise 0.01 was used, regularization was necessary. Cheng and Cabral [19] showed the high accuracy using RBCM on different grid sizes from 6×6 to 12×12, but no noise was considered.
In the present work, 9×9 grid is used for collocation points. The distribution of collocation points in the conventional RBCM and LS-RBCM is shown in Figure 9. The collocation points are marked by different types of symbols as listed in Table I to satisfy the corresponding conditions.
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
Figure 9. Distribution of collocation points in the conventional RBCM (a) and LS-RBCM (b).
In this example, we impose the simulated noise on the Cauchy boundary conditions (27) and (28). The shape parameter in IMQ is chosen as c = 4.
The recovered solution u(0, y) and flux u x ( 0, y) on the missing boundary with different noise levels using both methods are plotted in Figures 10 and 11. The RRMSE distribution against the different noise levels is shown in Figure 12. Similar phenomenon is observed as in Example 1, but the critical noise value moves to 0.001. When the noise level is smaller than 0.001, the solutions u( 0, y) and flux u x ( 0, y) on the missing boundary of the conventional RBCM are better than the ones of LS-RBCM; whereas when the noise level is greater than 0.001, the solutions and flux of LS-RBCM are much better. When noise level reaches 20%, the RRMSEs of the temperature and heat flux on the left-side boundary in the conventional RBCM are about 14.7 and greater than 100%, and numerical curve is distorted as shown in Figure 10; while errors reduce to only 6.9 and 22.3% for the solution and flux in LS-RBCM, respectively, as shown in Figure 11.
Compared with Example 1, the Cauchy boundary problem poses high ill-conditioning of the coefficient matrix [ A ik ] , which may result in the solution more sensitive to the noise. In the conventional RBCM, the over-specified boundary condition with very slight noise can match the ‘strong’ requirement to satisfy boundary conditions exactly at the collocation points. We observe from Figure 12 that the RRMSE of the solution and its flux are almost proportional (very sensitive) to the noise level even when the noise level is as small as 0.0001. The accuracy of the method depends on the noise level. However, the solution of the LS-RBCM is not sensitive to the noise level till e = 0.001 and the flux is not sensitive to the noise level till e = 0.004 due to ‘relaxation’ effect. Only when noise level is greater than 0.002 (0.005), the RRMSE of the temperature (flux) is proportional (sensitive) to the noise level. The gradients of RRMSE curves of the solution and its flux are expected to be smaller in LS-RBCM. And the RRMSE curve moves 55% downward of the temperature and 90% downward of heat flux when the noise level e>0.005.
The condition number is improved from 3.56×10 19 to 5.64×10 17 in LS-RBCM. The over- determined system can provide a best-fit approximation not only to the solution but also to the flux under the over-specified conditions with larger noise. Hence for a highly ill-conditioned Cauchy problem, it works as that ‘relaxation’ effect strengthens the stability of the solution and improves
16 X.-Z. MAO AND Z. LI
Exact solution
Exact solution
u(0,y)
∂x(0,y) ∂u/
Figure 10. The recovered u(0, y) (a) and u x ( 0, y) (b) with different noise levels in the conventional RBCM.
Exact solution
1.2 Exact solution
1.3 u(0,y)
∂x(0,y) 0.6 ∂u/
Figure 11. The recovered u(0, y) (a) and u x ( 0, y) (b) with different noise levels in LS-RBCM.
the accuracy to resist the noise levels, but somewhat sacrifices the accuracy of the numerical solution within the smaller noise levels.
In order to test the sensitivity of the solution to the input measurement information, we add eight points as input on the Cauchy boundary, namely 17 points uniformly distributed on the boundary as measurement input data. And the input is also contaminated by the noise. It requires that 16 additional linear algebraic equations (eight satisfying the Dirichlet condition and another eight satisfying the Neumann condition) are included in the system (10), whereas the unknown coefficients remain constant based on the 9×9 grid. The numerical solution, flux and their errors distribution are shown in Figures 13 and 14. Although the RRMSE of the solution and flux from more input information are still proportional to the noise level after a certain value, the gradients of the curves become smaller. Hence the recovered solution u(0, y) and its flux u x ( 0, y) are improved much better, especially when the noise level is larger than 0.001 for u(0, y), and 0.01 for the flux
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
∂x(0,y)) ∂u/
RRMSE(u(0,y))
RRMSE(
10 0 10 0 (a)
Noise level e
(b)
Noise level e
Figure 12. The RRMSE of recovered u(0, y) (a) and u x ( 0, y) (b) against noise level in the conventional RBCM and LS-RBCM.
Exact solution e=0
1.2 Exact solution e=0
1.5 e=0.001
e=0.001
e=0.01 1 e=0.1 e=0.01 e=0.1 1.4 e=0.2
e=0.2
0.8 1.3 ∂x(0,y)
u(0,y)
Figure 13. The recovered u(0, y) (a) and u x ( 0, y) (b) with 17 boundary nodes input in LS-RBCM.
u x ( 0, y). That means, the better the solution of LS-RBCM is with more input information, even the input is contaminated with larger noise. When the noise level is up to 20%, the RRMSEs are reduced from 6.9 to 2.9% for u(0, y), and from 22.3 to 12.3% for the flux u x ( 0, y). The numerical solution and its flux are better fit to the exact ones with the more measured information input on the Cauchy boundary, as shown in Figure 13.
We investigate the convergence of the solution with respect to the shape parameter c and the size of collocation mesh. The results are shown in Figures 15 and 16. Similar results are shown as in Example 1 that the shape parameter c ranging from 0.2 to 2 leads to almost exponential convergence for the numerical solution u(0, y) and flux u x ( 0, y), then RRMSE keeps stable when
c is larger than 2 on the mesh m = 9×9. And the RRMSE of estimated u(0, y) and flux u x ( 0, y) decrease quickly when the mesh size increases from m = 3×3 to m = 8×8, then keeps stable when the mesh size is greater than 8×8 for a fixed shape parameter c = 4.
18 X.-Z. MAO AND Z. LI
9 boundary nodes 9 boundary nodes 17 boundary nodes
17 boundary nodes
∂x(0,y)) ∂u/
RRMSE(u(0,y))
RRMSE(
10 0 10 0 (a)
Noise level e
(b)
Noise level e
Figure 14. The RRMSE of recovered u(0, y) (a) and u x ( 0, y) (b) with 9 and 17 boundary nodes input in LS-RBCM.
u(0,y) ∂u/∂x(0,y)
Figure 15. The RRMSE of recovered u(0, y) and u x ( 0, y) in LS-RBCM with different value c.
4.3. Shape identification problem Hon and Wu [16] investigated the following ill-posed problem for shape identification, which
often arises in practical applications of non-destructive testing. The domain of interest is an upper half-plane. It is assumed that observation can be only conducted on a finite segment {−
u(x , 0) = x 2 − 0.31x −1.089
*u(x, 0)
= 0.1x +2.19
*y
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
u(0,y) ∂u/∂x(0,y)
Figure 16. The RRMSE of recovered u(0, y) and u x ( 0, y) in LS-RBCM with different collocation mesh m.
The problem is to search for the location of the internal shape that satisfies zero potential condition with the over-specified boundary condition on a segment of the boundary. The exact solution of the Laplace equation is
1.1) 2 + 0.1(x −0.1)(y −1.1)+0.1 (33) The problem is solved in finite domain [−2, 2]×[0, 2] due to the instability of the Cauchy
u(x , y) = (x −
0.1) 2 −( y−
condition and the constraint of finite range of the available data. The exact positions of the trajectories where the potential value u(x, y) = 0 is satisfied are
y= 1.095+0.05x +1.001 0.1098−0.2x + x 2 (34) y= 1.095+0.05x −1.001 0.1098−0.2x + x 2 (35)
Hon and Wu [16] tested the sensitivity of solution subject to noisy boundary data from 10 − 5 to
10 − 4 . Eleven data points were used for collocation on the Cauchy boundary [−0.5, 0.5]. When the noise reached 10 − 4 , the shapes became unidentifiable, and the solution failed. Cheng and Cabral [ 19] re-examined the numerical experiment using the RBCM on 9×9 grid, and collocated for the Dirichlet and the Neumann conditions each at five points on the Cauchy boundary. They found that when noise level was up to 0.01, the solution of lower trajectory started to deteriorate, but failed for the upper one. And they recovered the shape much improvingly with auxiliary information with one internal point. Young et al. [13] applied the MFS to investigate the problem, and collocated for the Dirichlet and the Neumann conditions each at six points on the Cauchy boundary. The
zero potential result showed good accuracy when low noise level (10 − 5 to 10 − 3 ) was considered. However, for large values of noise level such as 0.01, regularization was necessary since the MFS would not give promising results in the upper branch.
In the present work, 9×9 grid is used for collocation points. The distribution of collocation points in the conventional RBCM and LS-RBCM is shown in Figure 17. The collocation points are
20 X.-Z. MAO AND Z. LI
Figure 17. Distribution of collocation points in the conventional RBCM (a) and LS-RBCM (b).
marked by different types of symbols (as listed in Table I) to satisfy the corresponding conditions. We impose the simulated noise on the Cauchy boundary conditions (31) and (32). The shape parameter in IMQ is chosen as c = 4.
In our computation, we use Newton’s iteration method to search for the locations of the zero potential trajectories after the solution is obtained. The zero potential trajectories with different noise levels using the conventional RBCM and LS-RBCM are plotted in Figure 18, and the RRMSE distribution of the position along the different noise levels is plotted in Figure 19. The results show that we recover the lower branch very well, but there is a slight deviation for the upper one for both methods, and we cannot recover the upper branch when noise level is 0.01 in the computational region. And also we observe in Figure 19, LS-RBCM has much better results for both branches
when noise is smaller e<10 − 4 ; however, when the noise goes to 0.01, the error tends to be of the same magnitude for both methods till the computation fails to recover the upper trajectory at the noise level e = 0.0063. The RRMSE of recovered solution on the top and bottom sides of the boundary in LS-RBCM is better than that in the conventional RBCM within small noise levels, as shown in Figure 20. The phenomenon is totally different from Examples 1 and 2. Why does LS-RBCM obtains better results than the conventional RBCM within small noise levels?
The Cauchy problem on a finite segment of boundary is extremely ill-posed. The input data with small errors will cause the solution to deteriorate rapidly. In the present work, a finite domain [− 2, 2]×[0, 2] is chosen as computational region to represent an upper half-plane physical domain, in which no boundary conditions are prescribed except a finite segment of the boundary with the Cauchy conditions. This is a very special case, which seems to be a problem without any constraint of boundary conditions on the open boundary. In the numerical procedure, only on a finite segment of the boundary it needs to match ‘strong’ requirement to satisfy the Cauchy conditions. If the noise level is greater than 0.0063, for example, e = 0.01, little information can recover the upper branch of zero potential trajectory in the computational domain [−2, 2]×[0, 2]. Actually if we take a look the range outside of [−2, 2]×[0, 2], there still exists an upper branch at the range y>2.
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
Exact solution
Exact solution
1 e=0.002 1 e=0.002
Figure 18. The recovered trajectories in the conventional RBCM (a) and LS-RBCM (b).
Upper for RBCM
Lower for RBCM
RRMSE
Noise level e
Figure 19. The RRMSE of recovered trajectories in the conventional RBCM and LS-RBCM.
In the conventional RBCM, the Dirichlet and the Neumann conditions are satisfied on two types of collocation points (total 10 points) as shown in Figure 17(a), whereas the Dirichlet and the Neumann conditions are satisfied simultaneously on the same five collocation points in LS-RBCM, as shown in Figure 17(b). The conventional RBCM can match ‘strong’ requirement to satisfy the Dirichlet and the Neumann conditions exactly on the corresponding nodes of the boundary; however, the Cauchy conditions in the problem are constrained only in a finite segment {− instability of the Cauchy problem and the requirement not ‘strong’ for the physical problem. The LS-RBCM might match the requirement of the governing equation and the Cauchy conditions better for the physical problem due to the Cauchy conditions satisfying on the same points of the
22 X.-Z. MAO AND Z. LI
Top for RBCM
Bottom for RBCM
RRMSE
Noise level e
Figure 20. The RRMSE of recovered u(x, 0) (bottom side) and u(x, 2) (top side).
finite segment. This may be the reason why the RRMSE of both recovered trajectories and the RRMSE of the solutions on the top and bottom sides of boundary in LS-RBCM are much better than that in the conventional RBCM when e<0.001, then tends to the same RRMSE as the noise level increases till fails to recover the upper trajectory, as shown in Figures 19 and 20. In the conventional RBCM, the solutions of the top and bottom boundaries are not sensitive to the noise when e<0.002; however, the error of the solution on the top side boundary is as large as 15%, and the RRMSE of the solution on the top and the bottom sides of boundary is smaller than the one in LS-RBCM when e>0.003. The reason that the solutions on the bottom boundary and the lower trajectory are recovered much better is because the Cauchy boundary is a part of the bottom side of boundary.
We use four more points on the Cauchy boundary, namely total nine points are uniformly distributed on the boundary. The input is also contaminated by measurement noise. It requires that eight additional linear algebraic equations (four satisfying the Dirichlet condition, and another four satisfying the Neumann condition) are included in the system (10), whereas the unknown coefficients keep constant (still based on the 9×9 grid). The recovered trajectories of both branches are plotted in Figure 21, and the RRMSE distribution of the positions of the trajectories is shown in Figure 22. The RRMSE of the recovered solution on the top and the bottom sides of boundary is plotted in Figure 23. It is very encouraging that, only with four additional points as input, we successfully produce a reasonable approximation for the solution, and can recover the trajectories of both branches when the noise level goes to 0.1. The results are improved much better when
the noise is greater than 10 − 4 . Also we observe that when the noise level is less than 0.004, the recovered shape on the lower branch is better than the upper one, and when the noise level is greater than 0.004, both trajectories hold the same error level, increasing with noise increase. The recovered positions of both trajectories are independent on the noise when e<0.004, then proportional to the noise level, but the recovered positions still agree very well till e = 0.05 as shown in Figure 21. The solutions on the top and bottom sides of computational boundary are not sensitive to the noise when e<0.01, then proportional to the noise level. Compared with the result
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
1.5 Exact solution e=0 e=0.001
1 e=0.01 e=0.05 e=0.1
Figure 21. The recovered trajectories with nine boundary nodes input in LS-RBCM.
Upper for 5 boundary nodes Upper for 9 boundary nodes Lower for 5 boundary nodes Lower for 9 boundary nodes
RRMSE
Noise level e
Figure 22. The RRMSE of recovered trajectories with five and nine boundary nodes input in LS-RBCM.
from five points input, the errors of the solution and the position of trajectories on both sides drop dramatically when e>0.001, as shown in Figures 22 and 23.
We investigate the convergence of the solution with respect to the shape parameter c and the size of collocation mesh. The results are shown in Figures 24 and 25. The similar results are shown that the shape parameter c ranging from 2.0 to 4.5 leads to almost exponential convergence for the recovered trajectories in LS-RBCM for both sides, then the RRMSE keeps almost stable when c is larger than 4.5 on the mesh m = 9×9. And the RRMSE of the recovered trajectories in LS-RBCM for both sides tends to decrease when the mesh size increases from m = 3×3 to m = 9×9, then keeps stable when the mesh size is greater than 11×11 for a fixed shape parameter.
24 X.-Z. MAO AND Z. LI
Top for 5 boundary nodes Top for 9 boundary nodes Bottom for 5 boundary nodes Bottom for 9 boundary nodes
Noise level e
Figure 23. The RRMSE of u(x, 0) (bottom side) and u(x, 2) (top side) in LS-RBCM.
Upper Lower
RRMSE
Figure 24. The RRMSE of recovered trajectories in LS-RBCM with different value c.
5. CONCLUSION
In this paper, we present an efficient numerical technique LS-RBCM for solving ill-posed IBVPs. The method builds an over-determined linear system in which the double use of boundary colloca- tions method [8] is employed as well as the double boundary collocation approach [28]. And the system is transformed to a rank-deficient least-square problem to find a best-fit approximation to the solution. The LS-RBCM can obtain the stable solution for ill-posed IBVPs with large noise-level input because the least-square procedure ‘relaxes’ the ‘strong’ requirement of satisfaction of the
LS-RBCM FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION
Upper Lower
RRMSE
Figure 25. The RRMSE of recovered trajectories in LS-RBCM with different collocation mesh m.
governing equation and the Neumann boundary conditions. Three numerical examples demonstrate that LS-RBCM has better performance for the ill-posed problem with large noisy input compared with the conventional RBCM. It illustrates that the ‘relaxation’ effect from an over-determined algebraic equations system strengthens the stability of the solution and improves the accuracy to resist the noise, but somewhat sacrifices the accuracy of the numerical solution when the noise is smaller than the truncated error of the method. The numerical sensitivity shows that the shape parameter c in IMQ leads to almost exponential convergence of the solution in the LS-RBCM till an optimal value. The capability of the LS-RBCM to resist the noise is a potentially powerful tool for practical inverse problems in engineering applications.
ACKNOWLEDGEMENTS
The project is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China (The Project-sponsored by SRF for ROCS, SEM.), and National Water Grant (NO. 2008ZX07423-001). We would like to thank anonymous reviewers for their thorough reviews and constructive remarks.
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