Stable multiquadric approximation by local thinning

In this paper our concern is the recovery of a highly regular function by a discrete set $X$ of data with arbitrary distribution. We consider the case of a nonstationary multiquadric interpolant that presents numerical breakdown. Therefore we propose a global least squares multiquadric approximant with a center set $T$ of maximal size and obtained by a new thinning technique. The new thinning scheme removes the local bad conditions in order to obtain $\axt$ well conditioned. The choice of working on local subsets of the data set $X$ provides an effective solution. Some numerical examples to validate the goodness of our proposal are given.