Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

Investigation of regularized techniques for boundary knot method

“…The original linear system Equation (11) is then replaced by the following problem set of Equation (15), where b is ideal noise-free data obtained at the minimized point. The resulting TSVD solution of Equation (15) is given by K and Equation (16).…”

“…It is noted that the truncated singular value decomposition (TSVD) is clearly superior to Gaussian elimination for noisy boundary conditions [14]. On the other hand, the TSVD is also employed in the BKM solution of inverse problems [15,16]. All these studies, however, have mainly focused on the solution accuracy rather than on the solution INVESTIGATION OF REGULARIZED TECHNIQUES FOR BKM 1869 stability without a detailed investigation on the convergence behaviors.…”

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

“…The Landweber-Fridman method and the BEM were used to solve the Cauchy problem for two-dimensional Helmholtz and modified Helmholtz equations with L 2 -boundary data by Marin et al [16]. Jin and Zheng [17] solved some inverse boundary value problems for the Helmholtz equation using the boundary knot method and a SVD regularization and they also extended this method to some inverse problems associated with the inhomogeneous Helmholtz equation, see Jin and Zheng [18]. The numerical solution for the Cauchy problem for two-and three-dimensional Helmholtz-type equations by employing the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method and SVD, was investigated by Marin and Lesnic [19] and Marin [20], and Jin and Zheng [21], respectively.…”

Least‐square‐based radial basis collocation method for solving inverse problems of Laplace equation from noisy data

“…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.…”

“…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][11][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.…”