Efficient least squares approximation and collocation methods using radial basis functions

TL;DR

This paper introduces an efficient least squares collocation method using radial basis functions for function approximation and boundary value problems on irregular domains, leveraging FFT and low-rank updates for computational speed.

Contribution

It extends the AZ algorithm to solve rectangular least squares systems from RBF collocation efficiently, especially for irregular domains and higher dimensions.

Findings

Near optimal log-linear complexity for univariate problems

Faster than direct solvers for higher-dimensional problems

Effective handling of irregular domain boundaries

Abstract

We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains. The method is based on RBFs with centers on a regular grid defined on a bounding box, with some of the centers outside the computational domain. The equation is discretized using collocation with oversampling, with collocation points inside the domain only, resulting in a rectangular linear system to be solved in a least squares sense. The goal of this paper is the efficient solution of that rectangular system. We show that the least squares problem splits into a regular part, which can be expedited with the FFT, and a low rank perturbation, which is treated separately with a direct solver. The rank of the perturbation is influenced by the irregular shape of the domain and by the weak enforcement of boundary conditions at points along the boundary. The solver extends the AZ algorithm which was previously proposed for function approximation involving frames and other overcomplete sets. The solver has near optimal log-linear complexity for univariate problems, and loses optimality for higher-dimensional problems but remains faster than a direct solver.