A N-Body Solver for Free Mesh Interpolation

TL;DR

This paper introduces a novel N-body solver for free-mesh interpolation that leverages Gaussian RBF kernel factorization, achieving significant accuracy improvements in 3D function approximation by decoupling sampling and representation errors.

Contribution

It develops a spectral approximation method based on kernel factorization and octree decomposition, enabling high-accuracy free-mesh interpolation with decoupled sampling and representation errors.

Findings

Achieves roughly 5 orders of magnitude improvement in interpolation errors for the 3D Franke function.

Decouples mesh sampling errors from polynomial representation errors.

Demonstrates effectiveness of the method in high-dimensional free-mesh interpolation.

Abstract

Factorization of the Gaussian RBF kernel is developed for free-mesh interpolation in the flat, polynomial limit corresponding to Taylor expansion and the Vandermonde basis of geometric moments. With this spectral approximation, a top-down octree-scoping of an interpolant is found by recursively decomposing the residual, similar to the work of Driscoll and Heryudono (2007), except that in the current approach the grid is decoupled from the low rank approximation, allowing partial separation of sampling errors (the mesh) from representation errors (the polynomial order). Then, it is possible to demonstrate roughly 5 orders of magnitude improvement in free-mesh interpolation errors for the three-dimensional Franke function, relative to previous benchmarks. As in related work on $N$-body methods for factorization by square root iteration (Challacombe 2015), some emphasis is placed on resolution of the identity.