In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method.

在本文中，我们提出了一种有理的 RBF 插值方法来逼近具有极点或其他奇点的多元函数在逼近域上或附近。该方法基于散点布局，并且对于问题域的几何形状具有灵活性。尽管存在基于 RBF 的理性技术，但新方法允许使用条件正定核作为基函数。特别是，我们使用多谐核并证明有理多谐插值是可扩展的。如果该方法以局部形式实现，则缩放特性产生稳定的算法。为此，我们将有理多谐插值与统一划分方法相结合。足够数量的数值例子合二为一，