An adaptive partition of unity method for multivariate Chebyshev polynomial approximations

TL;DR

This paper introduces an adaptive partition of unity method for multivariate Chebyshev polynomial approximation, improving speed, flexibility, and robustness over traditional low-rank methods, especially for complex functions and nonrectangular domains.

Contribution

The paper presents a recursive domain splitting approach combined with partition of unity for efficient multivariate polynomial approximation, addressing limitations of low-rank methods.

Findings

Comparable speed to existing software on low-rank functions

Significantly faster on complex functions in serial computation

Reduced sensitivity to coordinate axis alignment

Abstract

Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has mainly focused on low-rank approximation. While this method is very effective for some functions, it is highly anisotropic and unacceptably slow for many functions of potential interest. A method based on automatic recursive domain splitting, with a partition of unity to define the global approximation, is easy to construct and manipulate. Experiments show it to be as fast as existing software for many low-rank functions, and much faster on other examples, even in serial computation. It is also much less sensitive to alignment with coordinate axes. Some steps are also taken toward approximation of functions on nonrectangular domains, by using least-squares polynomial approximations in a manner similar to Fourier extension methods, with promising results.