Radial basis functions (RBFs) play an important role in function interpolation, in particular in an arbitrary set of interpolation nodes. The accuracy of the interpolation depends on a parameter called the shape parameter. There are many approaches in literature on how to appropriately choose it as to increase the accuracy of interpolation while avoiding instability issues. However, finding the optimal shape parameter value in general remains a challenge. In this work, we present a novel approach to determine the shape parameter in RBFs. First, we construct an optimisation problem to obtain a shape parameter that leads to an interpolation matrix with bounded condition number, then, we introduce a data-driven method that controls the condition of the interpolation matrix to avoid numerically unstable interpolations, while keeping a very good accuracy. In addition, a fall-back procedure is proposed to enforce a strict upper bound on the condition number, as well as a learning strategy to improve the performance of the data-driven method by learning from previously run simulations. We present numerical test cases to assess the performance of the proposed methods in interpolation tasks and in a RBF based finite difference (RBF-FD) method, in one and two-space dimensions.

翻译：径向基函数（RBF）在函数插值中，尤其是在任意插值节点集上，扮演着重要角色。插值的精度取决于一个称为形状参数的参数。文献中有许多关于如何适当选择该参数以提高插值精度同时避免不稳定性问题的方法。然而，一般而言，寻找最优形状参数值仍然是一个挑战。在本工作中，我们提出了一种确定径向基函数中形状参数的新方法。首先，我们构建一个优化问题以获得能产生具有有界条件数的插值矩阵的形状参数；随后，我们引入一种数据驱动方法，通过控制插值矩阵的条件数来避免数值不稳定的插值，同时保持很高的精度。此外，我们提出了一种回退程序来强制条件数的严格上界，以及一种学习策略，通过从先前运行的模拟中学习来提升数据驱动方法的性能。我们通过数值测试案例，在一维和二维空间中，评估所提方法在插值任务以及基于径向基函数的有限差分（RBF-FD）方法中的性能。