Introducing the evaluation condition number: A novel assessment of conditioning in radial basis function methods

Background and Objective: RBF methods are crucial for reconstructing functions from data sites, like solving partial differential equations. However, ill-conditioning is common, and the standard condition number may not fully address it as it focuses solely on the Kernel matrix. Methods: The concept of the “largest manageable scale” is introduced, representing the maximum scale where errors are minimized without amplification. To determine this scale, we propose a cheaply available “evaluation condition number” (EVN) that evaluates the conditioning of the linear system arising from kernel methods, as well as “reliable digits” that indicate how many digits in the function evaluation are accurate. The results are implemented across various changes in scale, types of RBFs, and sizes of kernel matrices. Results and Conclusions: The evaluation condition number enhances solution quality estimation in kernel method linear systems over standard methods. We establish the connection between three conditioning criteria: the standard, effective, and our proposed evaluation condition numbers. Furthermore, equipped with the EVN criterion, we demonstrate the advantage of normalization techniques, such as truncated SVD over full SVD and MATLAB backslash. Our results confirm the proposed approach's effectiveness in maintaining stability and accuracy despite rank loss. We provide the code implementation at: https://doi.org/10.24433/CO.4444506.v2.