arXiv subjects

Shunfeng Yang

Publications and source records attributed to Shunfeng Yang.

4 recordsLinked to original sources

arXiv2026

Lightning Plus Polynomial Approximation: Optimal Root-Exponential Convergence for Singular Functions in Corner Domains

This paper presents a rigorous convergence analysis for the lightning plus polynomial approximation scheme, which employs rational approximations constructed with preassigned tapered, exponentially clustered poles. This pole placement strategy was originally introduced by Trefethen and his collaborators for the resolution of corner singularities. Ample numerical results indicate that this scheme achieves root-exponential convergence, and in particular attains the same optimal convergence rate as the best rational approximation to $x^\alpha$ on $[0,1]$ established by Stahl.% which is conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. In this work, we establish optimal root-exponential convergence for the class of prototype functions of the form $g(z)z^\alpha$ or $g(z)z^\alpha\log z$, where $g$ is analytic on a neighborhood of the sector domain. These results confirm the validity of Conjectures 3.1 and 5.3 stated in [SIAM J. Numer. Anal., 61:2580-2600, 2023], and demonstrate that the choice $\sigma_{\mathrm{opt}} =\frac{\sqrt{2(2 - \beta)}\pi}{\sqrt{\alpha}}$ achieves the theoretically optimal convergence rate $\mathcal{O}\left(e^{-\sqrt{2(2 - \beta)N\alpha}\pi}\right)$. Notably, for the specific case of $\beta = 0$, the scheme recovers Stahl's optimal convergence rate for $x^\alpha$. Furthermore, working within the decomposition framework for corner domains proposed by Gopal and Trefethen, this paper provides a rigorous proof of optimal root-exponential convergence for lightning plus polynomial approximation problems on corner domains, and explicitly derives the optimal pole clustering parameter.

math.NA↗

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 $z^\alpha$ and $z^\alpha\log z$ 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 $z^{\alpha}$ or $z^\alpha\log z$ 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

The root-exponential convergence of lightning plus polynomial approximation on corner domains

This paper builds further rigorous analysis on the root-exponential convergence for lightning schemes approximating corner singularity problems. By utilizing Poisson summation formula, Runge's approximation theorem and Cauchy's integral theorem, the optimal rate is obtained for efficient lightning plus polynomial schemes, newly developed by Herremans, Huybrechs and Trefethen \cite{Herremans2023}, for approximation of $g(z)z^\alpha$ or $g(z)z^\alpha\log z$ in a sector-shaped domain with tapered exponentially clustering poles, where $g(z)$ is analytic on the sector domain. From these results, Conjecture 5.3 in \cite{Herremans2023} on the root-exponential convergence rate is confirmed and the choice of the parameter $\sigma_{opt}=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}}$ may achieve the fastest convergence rate among all $\sigma>0$. Furthermore, based on Lehman and Wasow's study of corner singularities \cite{Lehman1954DevelopmentsIT, Wasow}, together with the decomposition of Gopal and Trefethen \cite{Gopal2019}, root-exponential rates for lightning plus polynomial schemes in corner domains $\Omega$ are validated, and the best choice of lightning clustering parameter $\sigma$ for $\Omega$ is also obtained explicitly. The thorough analysis provides a solid foundation for lightning schemes.

math.NA↗

arXiv2023

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 $x^\alpha$ 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 $x^\alpha$ 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↗