Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

Abstract

This article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore-Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi-analytical and local radial basis function collocation method using the non-singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy-to-program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

“…The first two methods need to determine the fictitious boundary and the source intensity factor resulting from the source singularity, while the latter two methods can be used for numerical approximation directly. Recently, these schemes have been successfully applied to heat and mass transfer [34,35], acoustics [36], elastic mechanics [37,38], inverse problems [39,40] and other aspects. For details, see [41] and references therein.…”