Least‐square‐based radial basis collocation method for solving inverse problems of Laplace equation from noisy data

Abstract

AbstractThe inverse problem of 2D Laplace equation involves an estimation of unknown boundary values or the locations of boundary shape from noisy observations on over‐specified boundary or internal data points. The application of radial basis collocation method (RBCM), one of meshless and non‐iterative numerical schemes, directly induces this inverse boundary value problem (IBVP) to a single‐step solution of a system of linear algebraic equations in which the coefficients matrix is inherently ill‐conditioned. In order to solve the unstable problem observed in the conventional RBCM, an effective procedure that builds an over‐determined linear system and combines with least‐square technique is proposed to restore the stability of the solution in this paper. The present work investigates three examples of IBVPs using over‐specified boundary conditions or internal data with simulated noise and obtains stable and accurate results. It underlies that least‐square‐based radial basis collocation method (LS‐RBCM) poses a significant advantage of good stability against large noise levels compared with the conventional RBCM. Copyright © 2010 John Wiley & Sons, Ltd.