A multiple-scale MQ-RBF for solving the inverse Cauchy problems in arbitrary plane domain

The method of radial basis function (RBF) is popularly used in the solution of partial differential equations (PDEs). We propose a multiple-scale MQ-RBF method to solve the linear elliptic PDEs and the corresponding inverse Cauchy problems in simply- and doubly-connected domains, where the multiple scales are automatically determined a priori by the collocation points and source points, which play a role of post-conditioner of linear system to determine the unknown expansion coefficients. In the solution of inverse Cauchy problems the multiple-scale MQ-RBF is quite accurate and stable against large noise level up to 10–30%. Even for a case with only a quarter of boundary being imposed over-specified data, the multiple-scale MQ-RBF can still recover 75% unknown data very well.