Numerical recipes: the art of scientific computing
Numerical recipes: the art of scientific computing
Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Lower bounds for norms of inverses of interpolation matrices for radial basis functions
Journal of Approximation Theory
Improved error bounds for scattered data interpolation by radial basis functions
Mathematics of Computation
A unified theory of radial basis functions Native Hilbert spaces for radial basis functions II
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
A new class of radial basis functions with compact support
Mathematics of Computation
Learning from Data: Concepts, Theory, and Methods
Learning from Data: Concepts, Theory, and Methods
A note on the basis set approach in the constrained interpolation profile method
Journal of Computational Physics
Computers & Mathematics with Applications
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This paper investigates multi-quadric radial basis function (RBF) interpolation and its application in the quantum trajectory method (QTM) for wave packet propagation. In the multi-quadric, φ(r ; δ) = (r2 + δ2)1/2, r is the radial distance from the observation point to the origin of the basis function, φ, and δ is known as the shape parameter due to its affect on the functional form of the basis function. The quality of any RBF interpolation scheme is dictated by the choice of this parameter. Many recent studies have investigated a suitable means for obtaining an "optimized" time-independent δ parameter. The purpose of this study, however, is to not only to find this "optimized" shape parameter, but also to analyze its time-dependence in four different dynamical models; the anisotropic harmonic oscillator, the downhill ramp, the uphill ramp, and a harmonic oscillator coupled with a downhill ramp. To obtain the optimized shape parameter at each time step, an algorithm similar to the leave-one-out method of cross-validation is utilized. The results for each of the four models are presented, and the feasibility and necessity of employing a shape parameter optimization algorithm for each of the models is discussed