An adaptive partition of unity method for multivariate Chebyshev polynomial approximations
This method improves the efficiency and robustness of spectral polynomial approximation for multivariate functions, addressing limitations of existing low-rank approaches.
The authors propose an adaptive partition of unity method for multivariate Chebyshev polynomial approximations that is as fast as existing low-rank methods for many functions and significantly faster for others, while being less sensitive to coordinate axis alignment.
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.