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

Abstract

SUMMARYThe 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.

Global space–time multiquadric method for inverse heat conduction problem

“…Cheng and Cabral [17] 358 Z. LI AND X.-Z. MAO and Mao and Li [20] employed the IMQ to solve several types of ill-posed boundary value problems for the Laplace equation and successfully overcome the bad conditioning of the coefficient matrix. Thus, the inverse problems make no difference to the direct problems in aspect of numerical solution procedure.…”

“…As a result, the coefficient matrix obtained by collocation is very ill-conditioned. In this paper, we follow the Mao and Li's method [20] to build an overdetermined system using two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions, similar to the double boundary collocation approach [32], which may provide a kind of 'relaxation' effect to search for a best-fit solution for the system. On the collocation points located in the domain ×(0, T ) as well as the boundary collocation nodes on ×(0, T ), the temperature u(x, t) is required to satisfy the governing equation by substituting Equation (7) into Equation (1)…”

Estimation of river pollution source using the space-time radial basis collocation method

“…This leads to a poorly conditioned coefficient matrix and instability of the solution, especially when the observation data contain measurement errors. Mao and Li [31] introduced the least-square-based RBCM to search for the bestfit solution of the system, to improve the stability of the numerical solution based on double boundary collocation, one for the governing equation and another for the observation and given boundary conditions.…”

“…The best-fit approximation of the solution [α k ] to system (9) is provided by the singular value decomposition with rank reduction technique [31].…”

“…The RBCM is a domaintype numerical method involving the same single-step solution procedures as in direct problems and can be efficiently applied to the inverse problem. Cheng and Cabral [13] and Mao and Li [31] employed the inverse multiquadric (MQ) as a RBF to solve ill-posed inverse boundary problems for the Laplace equation. As for the timedependent inverse problems, the global space-time RBCM was then constructed for inverse and backward heat conduction problems [28] and for estimations of the spatial plume distribution and source release history in groundwater [29].…”