Efficient function approximation on general bounded domains using wavelets on a cartesian grid

TL;DR

This paper introduces an efficient wavelet-based extension method for function approximation on general bounded domains, achieving near-linear complexity in 1-D and improved efficiency in higher dimensions.

Contribution

It proposes a novel wavelet extension approach on hypercubes for arbitrary-shaped subsets, with algorithms that significantly reduce computational complexity.

Findings

Achieves $ ext{O}(N)$ complexity in 1-D

Provides algorithms with $ ext{O}(N^{3(d-1)/d})$ complexity in $d$-D

Sparse QR solvers are more time-efficient despite larger coefficients

Abstract

Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients.