Boundary knot method for some inverse problems associated with the Helmholtz equation. (English) Zbl 1085.65104

Jin, Bangti; Zheng, Yao

Int. J. Numer. Methods Eng. 62, No. 12, 1636-1651 (2005).

Summary: The boundary knot method is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill-posed 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 both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems.

MSC:

65N21	Numerical methods for inverse problems for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R30	Inverse problems for PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs

Keywords:

boundary knot method; inverse problem; Helmholtz equation; truncated singular value decomposition; L-curve method; radial basis function; collocation; ill-posed Cauchy problem; regularization method; numerical results; stability

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