Global multiquadric collocation method for groundwater contaminant source identification

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

“…As mentioned in the Introduction, although Li and Mao [28,29]investigated the convergence of the global space-time MQ solution with respect to the shape parameter c and the scaling factor w, it is still necessary to seek the best performance of the RBCM based on selection of the optimal model parameters, especially when the method is used in practical applications.…”

“…To solve the time-dependent inverse model, a new Euclidian distance, i.e., (||x−x k || 2 +w 2 ||t−t k || 2 ) 1/2 , where w is a scale parameter, was proposed by normalizing the time dimension to be a pseudo-spatial dimension in [28,29]. Taking a two-dimensional problem as an example, the new Euclidean distance is given by:…”

“…For problems of interpolating scattered data, Foley [18] improved the RBF shape parameter based on the root mean square deviation between the MQ and inverse MQ models at a set of test points, and Rippa [34] proposed instead a cross-validation technique for computing the RBF shape parameter by minimizing the cost function based on the residual errors of only the observation data; for solving PDE systems, Roque and Ferreira [35] built the cost function based only on the residual errors of the governing equations, but Tsaia et al [41] applied the golden search algorithm to choose a good shape parameter based on the residual errors of both the governing equation and Dirichlet boundary conditions. Li and Mao [28,29] investigated convergence of the global space-time MQ solution with respect to the shape parameter and the scaling factor. The results showed that a reasonable (although not necessarily the best) numerical solution exists within a wide range of both parameters, with a V-shaped or U-shaped error profile with respect to the shape parameter and scaling parameter.…”

“…The present study was conducted to extend the global space-time RBCM [29], used for groundwater, to identify space-and time-dependent source releases causing river pollution. To search for the optimal values of the model parameters, the k-fold cross-validation technique based on a new cost function was developed.…”

“…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]. Owing to the complexity of the source identification problem, sensitivity analysis associated with the number, location and error level of observations and model parameters was also performed by Li and Mao [28,29] The advantages of the space-time RBCM mainly lie in its efficiency due to the truly meshfree and non-iterative property, super-convergence and insensitivity to the direction of time.…”