In this article we present a new adaptive algorithm for solving 2D interpolation problems of large scattered data sets through the radial basis function partition of unity method. Unlike other time-consuming schemes this adaptive method is able to efficiently deal with scattered data points with highly varying density in the domain. This target is obtained by decomposing the underlying domain in subdomains of variable size so as to guarantee a suitable number of points within each of them. The localization of such points is done by means of an efficient search procedure that depends on a partition of the domain in square cells. For each subdomain the adaptive process identifies a predefined neighborhood consisting of one or more levels of neighboring cells, which allows us to quickly find all the subdomain points. The algorithm is further devised for an optimal selection of the local shape parameters associated with radial basis function interpolants via leave-one-out cross validation and maximum likelihood estimation techniques. Numerical experiments show good performance of this adaptive algorithm on some test examples with different data distributions. The efficacy of our interpolation scheme is also pointed out by solving real world applications.

在本文中，我们提出了一种新的自适应算法，该算法通过统一方法的径向基函数划分来解决大型分散数据集的2D插值问题。与其他耗时的方案不同，此自适应方法能够有效地处理域中密度变化很大的分散数据点。通过在可变大小的子域中分解基础域，从而确保每个域中有适当数量的点，可以实现此目标。这些点的定位是通过有效的搜索过程完成的，该过程取决于正方形单元中域的分区。对于每个子域，自适应过程会识别由一个或多个级别的相邻小区组成的预定义邻域，这使我们能够快速找到所有子域点。进一步设计了该算法，用于通过留一法交叉验证和最大似然估计技术来最佳选择与径向基函数插值相关的局部形状参数。数值实验表明，该自适应算法在一些数据分布不同的测试示例中具有良好的性能。通过解决实际应用也可以指出我们的插值方案的功效。