arXiv subjects

Yanghao Wu

Publications and source records attributed to Yanghao Wu.

2 recordsLinked to original sources

arXiv2025

Exact convergence rates of lightning plus polynomial approximation for branch singularities with uniform exponentially clustered poles

This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner (branch) singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the integral representations of $z^α$ and $z^α\log z$ in the slit disk and develop results akin to Paley-Wiener theorem, from which, together with the Poisson summation formula, the root-exponential convergence of the lightning plus polynomial scheme with an exact order for each clustered parameter is established in approximation of prototype functions $z^α$ or $z^α\log z$ on a sector-shaped domain, which includes $[0,1]$ as a special case. In addition, the fastest convergence rate is confirmed based upon the best choice of the clustered parameter. Furthermore, the optimal selection of the clustered parameter is employed in conformal mappings through solving Laplace problems on corner domains, building upon Lehman and Wasow's analysis of corner singularities and incorporating the domain decomposition method proposed by Gopal and Trefethen.

math.NA↗

arXiv2024

On the best convergence rates of lightning plus polynomial approximations

Building on introducing exponentially clustered poles, Trefethen and his collaborators introduced lightning algorithms for approximating functions of singularities. These schemes may achieve root-exponential convergence rates. In particular, based on a specific choice of the parameter of the tapered exponentially clustered poles, the lightning approximation with either a low-degree polynomial basis may achieve the optimal convergence rate simply as the best rational approximation for prototype $x^α$ on $[0,1]$, which was illustrated through delicate numerical experiments and conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. By utilizing Poisson's summation formula and results akin to Paley-Wiener Theorem, we rigorously show that all these schemes with a low-degree polynomial basis achieve root-exponential convergence rates with exact orders in approximating $x^α$ for arbitrary clustered parameters theoretically, and provide the best choices of the parameter to achieve the fastest convergence rate for each type of clustered poles, from which the conjecture is confirmed as a special case. Ample numerical evidences demonstrate the optimality and sharpness of the estimates.

math.NA↗