Practical Evaluation Condition Assessment for Meshless Kernel-Based Methods

Kernel-based methods, particularly the Radial Basis Function (RBF) method, face an ongoing challenge in determining the optimal scale or shape parameter. This parameter plays a crucial role in the conditioning of the linear system, which directly affects the Power Function component of the error. The condition number for the evaluation of functions (the so called evaluation condition number), first introduced by Lyche and Peña in 2004 [1], serves as a valuable tool for assessing the conditioning of RBF methods. In this research, we propose using this evaluation condition number as a cost-effective means to assess the conditioning of the linear system in meshless methods, with a specific focus on RBF methods. Our study evaluates the impact of variations in the kernel matrix, particularly changes in the scale parameter, within the context of basis adjustments. The evaluation condition number quantifies the minimum number of accurately represented digits in function evaluations. Furthermore, we compare the accuracy and stability of two solvers: the widely used MATLAB backslash operator and truncated SVD. For more details on this study, please refer to [6]. The code and data can be accessed at: https://doi.org/10.24433/CO.4444506.v2