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

It is well known that the problem of fitting a dataset by using 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 specific cases. The bases that have local support and constitute a partition of unity have been shown to be interesting in the frame of geometric problems. In this work, we investigate the numerical effects of considering these bases in the quadratic Powell-Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and assess whether the already known 'good' bases also possess favorable numerical properties. Additionally, we introduce an inverse optimization approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.