Radial basis function interpolation in the quantum trajectory method: optimization of the multi-quadric shape parameter

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.