An Adaptive Partition of Unity Method for Multivariate Chebyshev Polynomial Approximations

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.

Local Radial Basis Function Methods: Comparison, Improvements, and Implementation

“…In all applications of the method so far by the current author, the order of magnitude of the term neglected by the MPU method is the same as that of the error and thus has little effect on the overall accuracy of the method. In reference [2], where the MPU method was also applied, the authors came to a similar conclusion and they also argued that the term could safely be neglected.…”

“…The modified PU method was introduced and applied in reference [1] but an analysis of the differences between the standard and modified PU methods was not given. The modified PU method has been subsequently applied in [2] and [3].…”

Preconditioned nonlinear iterations for overlapping Chebyshev discretizations with independent grids

“…using the methods in [1], in order to help capture the boundary layers, as shown in Figure 4.2. For the initial guess of the outer iterations, the boundary condition (4.6) was extended throughout Ω.…”

“…Finally, we remark that an important extension in [1] is to use least-squares approximation rather than interpolation to incorporate nonrectangular (sub)domains. We have been able to write a least-squares (as opposed to collocation) generalization of SNK and test it in one dimension.…”

“…Our interest is in applying the nonlinear preconditioning technique to spectral collocation discretizations in overlapping rectangles or cuboids, leading to globally smooth approximations constructed from a partition of unity [1]. In this context, there is not naturally a single global discretization whose degrees of freedom are partitioned into overlapping sets, because the Chebyshev (or Legendre, or other classical) nodes will not generally coincide within the overlapping regions.…”