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

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.