A theorem of insensitivity of the collocation solution to variations of the metric of the interpolation space

Abstract

The collocation approach to the estimation of a field from observed functionals, is known, by examples and simulations, to display a not very strong dependence from the choice of the specific reproducing kernel-covariance function. In. fact the situation is similar to the case of the dependence of least squares parameters on the weight of observations

The paper, after recalling the basic theory according to its deterministic and stochastic interpretation, shows that the variation of the sought solution is infinitesimal with both, the variation of the metric of the intepolation space going to zero and the quantity of information carried by the observations going to hunderd percent on the specific functional of the field that we want to predict. The combined effect of the two gives an infinitesimal of the second order, namely a theorem of “insensitivity” of the solution to the metric of the interpolation space. Different simulations show the action of this particular effect