Replacing random samples with gridded data

Suppose f:Rn->R is given by samples { yi = f(Xi) } and you want to extrapolate this to an approximation of f( ). One way to do it is to select grid points surrounding the Xi and ask: what y values should be assigned to the grid points so that the resulting multilinear interpolations at Xi differ from the sample values as little as possible? It is a linear least squares minimization. The only difficulties are at the edges of the data and where the density of samples is low compared to the number of grid points.
Needless to say, this is useful when you want to replace non gridded samples with gridded ones.