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Analysis of the Boundary Knot Method for 3D Helmholtz-Type Equation
Numerical solutions of the boundary knot method (BKM) always perform oscillatory convergence when using a large number of boundary points in solving the Helmholtz-type problems. The main reason for this phenomenon may contribute to the severely ill-conditioned full coefficient matrix. In order to obtain admissible stable convergence results, regularization techniques and the effective condition number are employed in the process of simulating 3D Helmholtz-type problems. Numerical results are tested for the 3D Helmholtz-type equation with noisy and non-noisy boundary conditions. It is shown that the BKM in combination with the regularization techniques is able to produce stable numerical solutions.
During the past two decades, meshless methods have attracted great attention in the area of computational science and engineering. The boundary-type meshless methods, which used the boundary data only, is a hot topic in computational mechanics and computational mathematics. It can be divided into two types, that is, the week formulation based on boundary integral equations and the force collocation formulation. The first type is formed by combining the moving least-square approximation with boundary integral equation methods. Excellent researches include the boundary node method [1, 2], local boundary integral equation method (LBIE) [3–6], improved boundary node method [7–11], and boundary element-free method (BEFM) [12–15]. These methods have been successfully applied to many engineering problems; take the potential problem and elasticity problem as examples [16, 17]. The second type includes the method of fundamental solutions (MFS) [18, 19], boundary knot method (BKM) , boundary collocation method , and modified method of fundamental solutions [22–26]. They are very popular in solving problems governed by homogeneous equations. The BKM and MFS are truly meshless since integration is no more needed as in the week formulation meshless method based on boundary integral equations. In particular, the BKM performs especially well for solving the Helmholtz-type problems [27, 28]. However, the convergence of the BKM as well as that the MFS is often unstable when the number of boundary points becomes large. The main reason of instability is largely due to the ill-conditioned full interpolation matrix [29–31].
As is known to all, regularization techniques are often used to deal with ill-conditioned cases encountered in inverse problems. For instance, a comprehensive comparison using the MFS and various regularization techniques have been made by Wei et al. in the study of inverse Cauchy problems . The Tikhonov regularization (TR) method is used in the MFS solution of inverse source problems . The truncated singular value decomposition (TSVD) is employed in the BKM solution of inverse problems [34, 35]. For direct problems, the singular value decomposition (SVD) is applied to resolve the ill-conditioning of the MFS . However, Chen et al.  reexamined the results given in  and showed that there is no difference between using the direct Gaussian solver and the SVD for noise-free boundary conditions. They also demonstrated that the TSVD is clearly superior to Gaussian elimination for noisy boundary conditions. Recently, Wang et al.  have made a comprehensive study on regularization techniques in the BKM solution for direct problems in 2D cases.
For both direct and inverse problems, it is well known that the resulting matrices of the boundary-type meshless methods can be expressed in the standard form . Clearly, we should not rely solely on the condition number to predict accuracy of the computed solution of all practical ill-conditioned systems. Most importantly, the solution accuracy has an obvious dependence on the right-hand side vector . So the effective condition number (ECN), which is related to the right-hand side vector , is introduced to replace the condition number in the estimation of conditioning of the global interpolation methods [38–40]. In particular, the authors in  propose a new ECN, which will be used in this paper, to more accurately evaluate the conditioning of the resultant matrix in the solution process of the MFS.
It is the purpose of the present paper to provide the stability analysis of the 3D Helmholtz-type equations by using the BKM. To obtain a stable numerical scheme, various regularization techniques are employed for solving the resulting discretized system of equations. Then the relationship between the solution accuracy and the ECN of a linear system is investigated when using the BKM. Finally, we combine the BKM with various regularization techniques to examine Helmholtz problems under noisy and noise-free boundary conditions.
2. The BKM
We consider the time harmonic waves of frequency which leads to the 3D boundary value problems for the Helmholtz-type equation where , is the Laplace operator, means the wave number, and are the known functions, denotes the 3D domain and its boundary with Dirichlet boundary and Neumann boundary , and represents the unit outward normal.
The nonsingular general solutions of the homogeneous Helmholtz operator and homogeneous modified Helmholtz operator are, respectively, given by with representing the Euclidean norm distance between two points and . Since there is no singularity in nonsingular general solutions, all collocation and source points can be placed on the physical boundary simultaneously.
The basic theory of the BKM lies in that the solution of Helmholtz-type boundary value problems can be approximated by where denotes the total number of boundary points and is the unknown coefficients. By collocating (3) on Dirichlet and Neumann boundary conditions, we have
Equation (4) can be written in the following matrix system: where is an interpolation matrix, is the number of points on Dirichlet boundary and the rest on Neumann boundary , is the unknown coefficients, and is the known boundary data. We notice that the BKM produces a highly ill-conditioned and dense matrix system when a large number of boundary points are used. For more details, we refer readers to [29, 30].
3. Measurement of the Coefficient Matrix Conditioning
The interpolation matrix in (5) can be decomposed by the singular value decomposition (SVD) where and are orthogonal matrices, , where denotes the identity matrix and is a diagonal matrix with diagonal elements , where , are called singular values of . The condition number of a nonsingular square matrix is defined by . Throughout this paper, the condition number can be stated as .
For practical applications, the boundary data may be disturbed by some noise. Clearly, we should not rely solely on the condition number to predict accuracy of the computed solution of all practical ill-conditioned BKM systems since the right-hand side vector is excluded, particularly for the noisy boundary conditions. In many applications, is problem-dependent but fixed. Under these conditions, as an alternative tool to estimate the accuracy of the BKM, we consider the effective condition number which is described in [27, 41] as where depends on the right-hand side vector . For more details about other types of ECN, we refer readers to [39, 40, 42].
4. Regularization Methods
Based on the SVD, we briefly present three commonly used regularization techniques under two parameter choices, that is, the TSVD, the TR, and the damped singular value decomposition (DSVD) under parameter choice of the -curve criterion (LC) and the GCV .
4.1. Regularization Techniques
TSVD. The fundamental theory of the TSVD lies in that the TSVD solution is often used to obtain a better estimate of the least squares solution. It is given by approximating a rank- full matrix by a rank matrix in which only the largest singular values are retained In this instance, we can replace matrix in (5) by , which has a well defined null space of dimension spanned by the right singular value vectors, . The original linear system (5) is then replaced by the following problem set of (11), where is ideal noise-free data obtained at the minimized point. The resulting TSVD solution of is given by where is a regularization parameter.
TR. The TR is a direct method in that the TR replaces the linear system (5) by the minimization problem where is a regularization parameter.
Based on the SVD, we can express (13) in terms of where is the rank of matrix and are the Wiener weights.
DSVD. A relatively less known regularization technique which is based on the SVD is the DSVD. Here, instead of using the filter factors in the TR, one introduces a smoother cut-off by means of filter factors defined as and these filter factors decay slower than the TR filter factors and thus, in a sense, introduces less filtering.
The suitable value of the regularization parameter is chosen by the LC or the GCV in this paper.
4.2. Regularization Parameters
LC. The LC, which is considered as the most convenient graphical tool for analysis of discrete ill-posed problems, is defined as Note here that the -curve is a continuous curve when the regularization parameter is real in the TR and the DSVD. In numerical computation, the point with maximum curvature will be searched as the corner of the -curve. For the regularization methods with a discrete regularization parameter, such as in the TSVD, a finite set of points will be obtained and interpolated by a spline curve. The point on the spline curve with the maximum curvature is then chosen as the desirable regularization parameter.
The distinctive feature of the LC lies in that the regularized solution varies with the regularization parameter .
GCV. To obtain the optimal value of the regularization parameter, the GCV is used to minimize the functional where is defined as The statistical method GCV is a predictive mean-square error criteria, in the sense that it estimates the minimizer of residual function We note that is defined for both continuous and discrete regularization parameters.
5. Numerical Examples
In order to compare with the BKM with no regularization techniques, numerical results are given by using six regularization methods, that is, GCV-TR, LC-TR, GCV-DSVD, LC-DSVD, GCV-TSVD, and LC-TSVD.
Some random noise to the boundary conditions is added by where . Here, denotes the noise level and the random number generator “Rand” is used to produce random numbers in . The root mean square error (RMSE) is used in the following cases .
5.1. Helmholtz Case on a Unit Cubic Domain
Here, we consider the 3D Helmholtz equation under boundary condition , . The corresponding wave number is .
Figure 1 displays the RMSE versus the boundary point number on a unit cubic domain . From which we notice that convergence curve of the BKM without regularization techniques, No regu., is oscillatory when the boundary point number becomes large. This phenomenon contradicts with the traditional opinion: a more accurate fitting of the exact data always leads to a better numerical results . The reason may contribute to the highly ill-conditioned interpolation matrix (≈1020) which is shown in Figure 2. Besides, we note that the effective condition number, ECN, is almost half the size of the condition number.
Using TSVD, DSVD, or TR under the regularization parameter LC, the accuracy of BKM seems to be even worse. On the other hand, the regularization techniques TSVD, DSVD, and TR under the regularization parameter GCV obtain more stable results. After the boundary point number , the convergence curves have no oscillatory phenomenon. The GCV-TR method performs the best in this case. This is similar to the 2D cases .
Once noise is added on the boundary, it is seen from Table 1 that the ECN decreases while the RMSE increases using boundary point number . We observe a sharp drop in the ECN as a tiny amount of noise is added, which leads to a poor accuracy with . This means that condition of the interpolation system in this case is worse than the 2D cases . Even though all runs in Table 1 have exactly the same condition number, completely different errors and ECNs are observed in cases with higher noise levels. At the same time, the drop of ECN has the same rate with the increase of the RMSE. More specifically, one order drop of the ECN corresponds with one order increase of the RMSE. This means that the ECN is a superior choice to the condition number for problems with noisy boundary conditions.
From Table 1, we observe that the relation between ECN and RMSE is more strict than the relation observed in 2D spaces . As more noise is added, the ECN is small enough to indicate that the BKM solution will not be accurate enough. This example is a good indication of the relationship between the accuracy of the BKM and the ECN because only the right-hand vector is altered with all other factors, including the ill-conditioned matrix , staying constant.
For different fixed noise levels, we notice that the GCV based regularization techniques and LC-TSVD perform better than the other cases. Among which the GCV-TR and LC-TSVD give better solution accuracy with two-decimal precision than the one without regularization techniques. However, the other two LC based regularization techniques do not perform well. We point that the GCV base regularization techniques are less accurate with one-decimal precision than the one without regularization.
5.2. Helmholtz Case on a Unit Spherical Domain
In order to see the effect of the physical domain on solutions, we consider the 3D Helmholtz equation on a unit spherical domain . Consider the following: under boundary condition , , with corresponding wave number .
For noise added on the boundary data, we observe from Table 2 that the ECN decreases while the RMSE increases for fixed boundary point number . Obviously, the condition number is irrelevant with the noisy data. Although the condition number in case 1 is larger than the one in this case, the ECN in case 1 is smaller than the one in this case. This phenomenon again proves that the condition number has no relation to the ECN. More interestingly, we find that the ECN almost remains the same with nonnoisy case as a tiny amount of noise is added. The corresponding solution accuracy is similar with the nonnoisy case . This means that condition of the interpolation system in this case is better than the previous case (Table 1). The main reason for the difference between case 1 and case 2 may be contributed to two reasons, that is, the different physical domain and the boundary value problems.
A sharp drop in the ECN is find as the noise is added, which leads to solution accuracy with . As more noise is added, the solution accuracy gets even worse. Once regularization techniques are introduced, we find that the GCV based regularization techniques have better results than the LC based regularization techniques. This is similar to case 1 (Figure 1). If there is no noise added on the boundary, the GCV based regularization techniques have much more accurate solutions than the one without regularization techniques .
5.3. Modified Helmholtz Case on a Unit Spherical Domain
In this case, we consider the 3D modified Helmholtz equation under boundary condition , , with wave number .
Figure 3 displays the RMSE versus the boundary point number on a unit spherical domain . Similar to case 1, we find that the convergence curve of the BKM without regularization techniques is oscillatory when the boundary point number becomes large. The result obtained using the LC-DSVD or LC-TR is unacceptable while the LC-TSVD shows a much smoother convergence curve. Even though the GCV based methods have oscillate convergence, they are much better than the one without regularization technique.
For boundary point number with noisy boundary conditions, it is seen from Table 3 that the ECN decreases while the relative average error increases. We observe a sharp drop in the ECN as a tiny amount of noise is added. However, the solution is still accurate with . A larger noise added leads to unacceptable result . This shows that this case is similar to case 2 but less sensitive than case 1. By this point, we can believe that the spherical domain is less sensitive than the cubic domain for problems with noisy boundary conditions.
It is noted that the condition number of the cubic domain case is worse than the spherical domain case. We also observe from Figure 4 that the condition number and ECN are larger than those in Figure 2. Similarly, the relation still works here. For different fixed noise levels, we also find that the GCV based regularization techniques give better accuracy than the one with no regularization technique. Different from the cubic domain case, the LC-TSVD does not perform well in this case while the other two LC based regularization techniques give better results.
5.4. Modified Helmholtz Case on a Unit Cubic Domain
In this case, we consider the 3D modified Helmholtz equation under Dirichlet boundary condition with corresponding wave number .
For fixed boundary point number , Table 4 gives the variation of condition number, ECN, and numerical results of the BKM under noisy and noise-free boundary conditions. We find that the boundary point number in this case is larger than case 1, but the numerical solution in case 1 is better than the one in this case . On the other hand, the condition number in case 1 is smaller than the one in this case . This phenomenon shows that the character of the Helmholtz equation is better than the modified Helmholtz equation.
Similar to case 1, We observe a sharp drop in the ECN as a tiny amount of noise is added. It leads to unacceptable numerical solutions with . For different fixed noise levels, we also find that the GCV-DSVD and GCV-TSVD regularization techniques give better accuracy than the one with no regularization technique with about one-decimal precision. Different from the previous three cases, the LC-TSVD performs well in this case while the GCV-TR regularization techniques give unacceptable result for noise .
In this paper, the stability analysis of the BKM has been investigated for 3D Helmholtz-type problems. Coupled with three regularization techniques and two algorithms for selecting regularization parameters, we are able to overcome the numerical instability induced from the ill-conditioned BKM interpolation matrix. On the other hand, the effective condition number is introduced to scale the ill-conditioned BKM interpolation matrix.
From the numerical results obtained in the previous section, we observe that the physical domain and boundary value problems have effect on numerical solutions. But this point has no relation to our conclusion; that is, the GCV based regularization techniques perform well for both noisy and noise-free boundary conditions. Although the LC based regularization techniques are often used for solving inverse problems with noisy boundary conditions, they are not acceptable for problems tested in this study. It should be noted that the condition of the BKM interpolation system is more dependable on the solution domain for 3D cases than 2D cases. More interestingly, the highly ill-conditioned interpolation system is less sensitive to the noisy boundary conditions than the less ill-conditioned one.
Besides, numerical results show that the character of the Helmholtz equation is better than the modified Helmholtz equation. Theoretical investigation is under way.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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