Generic Case of Leap-Frog Algorithm for Optimal Knots Selection in Fitting Reduced Data.

The problem of fitting multidimensional reduced data \(\mathcal{M}_n\) is discussed here. The unknown interpolation knots \(\mathcal{T}\) are replaced by optimal knots which minimize a highly non-linear multivariable function \(\mathcal{J}_0\). The numerical scheme called Leap-Frog Algorithm is used to compute such optimal knots for \(\mathcal{J}_0\)via the iterative procedure based in each step on single variable optimization of \(\mathcal{J}_0^{(k,i)}\). The discussion on conditions enforcing unimodality of each \(\mathcal{J}_0^{(k,i)}\) is also supplemented by illustrative examples both referring to the generic case of Leap-Frog. The latter forms a new insight on fitting reduced data and modelling interpolants of \(\mathcal{M}_n\).