Controlling the condition number of multiquadric RBF matrices via poisson disk sampling

Abstract Radial Basis Function (RBF) interpolation is a powerful tool for approximating functions from scattered data. However, the resulting interpolation matrix, especially when using the multiquadric (MQ) function, is often ill-conditioned, affecting numerical stability. This work investigates the relationship between the geometric distribution of interpolation centers and the condition number of the MQ matrix. Essentially, we impose a Poisson disk sampling constraint, which ensures a minimum distance \((h)\) between points. Using asymptotic spectral analysis and matrix perturbation theory, we derive approximate expressions for the eigenvalues and the condition number \((\kappa(A))\) as functions of the number of points \((N)\), the shape parameter \((c)\), and the characteristic distance \((r_0)\). We show that the condition number can be minimized, reaching a constant value independent of \((N)\), when the average distance between points satisfies a specific functional relationship with \((c)\) and \((N)\). Numerical experiments in \((\mathbb{R}^2)\) confirm the theoretical predictions, demonstrating that an adaptive choice of the Poisson disk minimum distance effectively controls the growth of ill-conditioning, unlike fixed or arbitrary choices.