Implementation of positive definite kernels in preCICE

Using preCICE
1

Hi,

I have a question regarding the implementation of RBF-based data mapping in preCICE. From my understanding, when using positive-definite (PD) kernels, the polynomial part in the interpolant (as described in [1], as well as in Bernhard Gatzhammer’s thesis, among other references) can be omitted, since the interpolation problem is uniquely solvable in that case.

However, it seems that the polynomial contribution is still computed, or at least I haven’t come across any mention indicating otherwise. Is this done to ensure proper reconstruction of displacements and similar quantities, or is there another reason for including it?

Thank you in advance for your response.

Reference:
[1] Efficient Partition-of-Unity Radial-Basis-Function Interpolation for Coupled Problems, David Schneider and Benjamin Uekermann (2025)

2

The answer can be found in Sec 2.1 of the mentioned article: they guarantee exact representation of linear data. Note that we anyway (by default) dont merge the polynomial columns into the kernel matrix. So practically speaking, we dont modify the “guaranteed solvability” for our spd kernels. In fact, it might happen that has a scenario where our cpd kernels lead to an ill-conditioned matrix purely due to of the vertex layout, i.e., by default we dont guarantee this for cpd kernels (but for spd kernels we do). In practice, however, that’s rarely the case.

Apart from the given reason and more from a pragmatic standpoint: the polynomial greatly enhances the interpolation quality in many cases and does not induce notable computational overhead.