An adaptive partition of unity method for Chebyshev polynomial interpolation

TL;DR

This paper introduces an adaptive overlapping partition of unity method for Chebyshev polynomial interpolation that improves convergence for functions with nearby singularities by blending local interpolants smoothly.

Contribution

It proposes a new divide-and-conquer algorithm for adaptive overlapping domain splitting using a partition of unity, enhancing spectral convergence for challenging functions.

Findings

Effective for functions with nearby singularities

Accelerates convergence compared to global interpolation

Applicable to solutions of singularly perturbed boundary value problems

Abstract

For a function that is analytic on and around an interval, Chebyshev polynomial interpolation provides spectral convergence. However, if the function has a singularity close to the interval, the rate of convergence is near one. In these cases splitting the interval and using piecewise interpolation can accelerate convergence. Chebfun includes a splitting mode that finds an optimal splitting through recursive bisection, but the result has no global smoothness unless conditions are imposed explicitly at the breakpoints. An alternative is to split the domain into overlapping intervals and use an infinitely smooth partition of unity to blend the local Chebyshev interpolants. A simple divide-and-conquer algorithm similar to Chebfun's splitting mode can be used to find an overlapping splitting adapted to features of the function. The algorithm implicitly constructs the partition of unity over the subdomains. This technique is applied to explicitly given functions as well as to the solutions of singularly perturbed boundary value problems.