Efficient least squares approximation and collocation methods using radial basis functions

TL;DR

An efficient method for the approximation of functions using radial basis functions (RBFs) is described, and this to a solver for boundary value problems on irregular domains, which has near optimal log-linear complexity for univariate problems, and loses optimality for higher-dimensional problems but remains faster than a direct solver.

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.

Figures (10)

• Figure 1: Schematic description of the centers and collocation points for approximation on $[-1,1]$. The radial basis functions are periodized on the larger interval $[-T,T]$ and their centers are equispaced on $[-T,T]$ as well. The collocation points are equispaced on $[-1,1]$, in such a way that they form a subset of a larger equispaced grid on $[-T,T]$. Finally, oversampling by a factor $s$ is achieved by considering $s$ shifted grids. These grids are grouped together into the matrix $A$. The figure shows the case $s=2$.
• Figure 2: We approximate the periodic function $f(x) = \sin \left( \left[ \frac{N}{5} \right] \pi x \right)$, with $T=1$ and $s=3$. Although the function is simple, it becomes increasingly oscillatory with $N$ and, hence, the approximation difficulty is constant. The purpose is to illustrate stability of the method for large $N$, which is confirmed in the left panel: the error remains roughly constant as $N$ increases. The right panel shows the computing time. The efficient solver outperforms a direct solver once $N > 150$.
• Figure 3: Comparison of the rank of system matrix $A$ and $A - AZ^*A$. The left panel shows the rank of the two matrices with different $N$. The rank of $A$ increases with $N$, while the rank of $A-AZ^*A$ remains constant. The right panel shows the singular values of the two matrices with different $N$. The spectrum of $A-AZ^*A$ consists only of a so-called plunge region up to $N=10$, which is why the AZ algorithm is efficient.
• Figure 4: We approximate the non-periodic function: $f(x) = \sin \left( \frac{N}{5} x \right)$ in $[-1,1]$, with $T=1.5$ and $s=2$. This function becomes increasingly oscillatory with $N$. The purpose is to illustrate stability of the method for large $N$, which is confirmed in the left panel: the error remains roughly constant as $N$ increases. The right panel shows the computing time. The AZ algorithm exhibits near linear complexity.
• Figure 5: Periodic function approximation: $f(x,y) = \sin \left( \left[ \frac{N}{10} \right] \pi (x+y) \right)$ in $\Omega_{\rm B}$ , $T^x = T^y=1$, $s^x = s^y=2$. Left: max error. Right: calculating time. Blue star line: backslash in MATLAB. Red circle line: efficient solver.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 4.1: coppe2020az
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof