An assessment of numerical and geometrical quality of bases on surface fitting on Powell–Sabin triangulations

Abstract

It is well known that the problem of fitting a dataset by means of a spline surface minimizing an energy functional can be carried out by solving a linear system. Such a linear system strongly depends on the underlying functional space and, particularly, on the basis considered. Some papers in the literature study the numerical behavior and processing of the above-mentioned linear systems in particular cases. There is a special kind of ‘good’ basis –those with local support and constituting a partition of unity– having attractive properties when handling geometric problems and that, as a consequence, have been profusely used in the literature of fitting surfaces. In this work, we study the numerical effect of considering these bases in the quadratic Powell-Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and determine to what extent already known ‘good’ bases also have good numerical properties. Additionally, we introduce an inverse approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.