Abstract
AbstractWe propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix. Classical bounds in terms of Lebesgue constants and convergence rates with respect to the mesh size of the eigen-rational interpolant are indeed comparable with those of classical kernel-based methods. However, the proposed approach takes advantage of rescaling the classical RBF expansion providing more robust approximations. Theoretical analysis, numerical experiments and applications support our findings.
Highlights
Multivariate approximation is one of the most investigated topics in applied mathematics and finds applications in a wide variety of fields
Nowadays many positive definite functions are classified as Radial Basis Functions (RBFs), a new term that appears for the first time on a publication by N
We propose a new approach, namely what we call the eigenrational method, which is extended to work with conditionally positive definite kernels and enables us to define a rational RBF expansion depending exclusively on kernel and data
Summary
Multivariate approximation is one of the most investigated topics in applied mathematics and finds applications in a wide variety of fields. We propose a new approach, namely what we call the eigenrational method, which is extended to work with conditionally positive definite kernels and enables us to define a rational RBF expansion depending exclusively on kernel and data Speaking, it consists in rescaling the classical RBF interpolant, taking into account the eigenvector associated to the largest eigenvalue of the kernel matrix. Despite the fact that from the analysis of Lebesgue functions and error bounds it turns out that the eigen-rational method behaves to classical kernel-based interpolants, weighting the interpolant by means of such eigenvector numerically provides more accurate approximations Such a phenomenon is evident for RBFs characterized by a fast decay, such as the Gaussian.
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