Comparing Nested Sequences of Leja and PseudoGauss Points to Interpolate in 1D and Solve the Schroedinger Equation in 9D

摘要

In this article, we use nested sets of weighted Leja points, which have previously been studied as interpolation points, as collocation points to solve a 9D vibrational Schroedinger equation. Collocation has the advantage that it obviates the need to compute integrals with quadrature. A multi-dimension sparse grid is built from the Leja points and Hermite-type basis functions by restricting sparse grid levels i_c using ∑_c g~c(i_c) ≤ H, where g~c(i_c) is a non-decreasing function and H is a parameter that controls the accuracy. Results obtained with Leja points are compared to those obtained with PseudoGauss points. PseudoGauss points are also nested. They are chosen to improve the accuracy of the Gram matrix. With both Leja and PseudoGauss points it is possible to add one point per level. We also compare Lebesgue constants for weighted Leja and PseudoGauss points.