Rational approximation with multidimensional scattered data

Abstract

Accurate and efficient rational approximation schemes are presented for interpolating multidimensional scattered data with a novel weighted least-squares procedure including domain decomposition. Two particular representations of the method are formulated and the corresponding algorithms are implemented. Numerical tests on three- and six-dimensional model systems are carried out, demonstrating high efficiency and accuracy. This work was motivated by the need for multidimensional function approximation using irregular grids when solving quantum fluid dynamics equations, and the method should have broader physical applications.

Fast and stable rational RBF-based partition of unity interpolation

“…In the next subsection, following [4], we show that by adding further constraints to the interpolation conditions, we can uniquely determine R (1) and R (2) .…”

“…However, it is a mesh-dependent approach and, as a consequence, extending polynomial approximation in higher dimensions is quite hard (refer e.g. to [2]).…”

“…However, extension of polynomial approximations to high dimensions is a challenging problem (refer e.g. to [2,3]) and, because of their dependence on meshes, it is not easy to implement for complex shaped domains. These are the main reasons why we direct our interest to rational Radial Basis Function (RBF)-based schemes.…”

Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects

“…Unfortunately, differently from RRBF interpolation, the rational polynomial approximation is quite hard to extend in higher dimensions (refer e.g. to [20]).…”

The potential energy surface for spin-aligned Li3(1 4A′) and the potential energy curve for spin-aligned Li2(a 3Σu+)

“…45 However, one of the difficulties in implementing the general use of the MQ interpolant for data sets in higher dimensions and/or for scattered data sets has been in the a priori selection of the parameter ⌬. 46 The accuracy in the interpolated energy is known to vary smoothly as a function of ⌬, but the accuracy of higher order derivatives depends more strongly on ⌬. Salazar and Bell 47 suggested a method for the a priori selection of ⌬ by minimizing the first difference in the interpolated higher order derivatives.…”