A meshless method for some inverse problems associated with the Helmholtz equation

By Bangti Jin, Yao Zheng. 2006

121 citations recorded by the source corpus.

In this paper, a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the Helmholtz equation, including the Cauchy problem. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L-curve method. Numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data, and the convergence and stability of the scheme are investigated. These results show that the proposed scheme is highly accurate, computationally efficient, stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries, and could be considered as a competitive alternative to existing methods for these problems.

Subfields: Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/fd7dcae94a12e52d21b8101727fa2ba771849a0a

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In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation (2009, methodology): “Jin and Zheng used the MFS combining the truncated singular value decomposition method to treat with the Cauchy problem of a Helmholtz equation [29].”
A Highly Accurate MCTM for Inverse Cauchy Problems of Laplace Equation in Arbitrary Plane Domains (2008, methodology): “While Jin and Zheng (2006) have applied the MFS to solve the inverse problem of Helmholtz equation, Marin and Lesnic (2005) have applied the MFS to solve the inverse Cauchy problem associated with a two-dimensional biharmonic equation.”
A survey of applications of the MFS to inverse problems (2011, methodology): “Further applications of the MFS to inverse Helmholtz and modified Helmholtz Cauchy problems in two dimensions can be found in [59].”
Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method (2013, methodology): “Then, Liu [47] found that the excellent property of the modified Trefftz method can be used to modify the MFS by relating the coefficients obtained from these two methods by a linear transformation, which is later known as one of the Laplacian preconditioners and postconditioners [48].”