An adaptive partition of unity method for Chebyshev polynomial interpolation
For numerical analysts and users of Chebyshev interpolation, this provides a smooth, adaptive alternative to piecewise interpolation with comparable convergence acceleration.
The paper addresses the slow convergence of Chebyshev polynomial interpolation for functions with nearby singularities by proposing an adaptive partition of unity method that blends local Chebyshev interpolants on overlapping subdomains. The method achieves spectral convergence and global smoothness, demonstrated on explicit functions and singularly perturbed boundary value problems.
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.