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A meshless method for some inverse problems associated with the Helmholtz equation. (English) Zbl 1123.65111
Summary: 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.

MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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
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