On the best convergence rates of lightning plus polynomial approximations

TL;DR

This work analyzes lightning-plus-polynomial approximations for singular functions, focusing on $x^\alpha$ on $[0,1]$. It develops two pole-distribution models—tapered and uniform exponential clustering—and proves rigorous root-exponential convergence with exact orders, linking the convergence to optimal choices of the clustering parameter $\sigma$ and to a low-degree polynomial basis with $N_2=\mathcal{O}(\sqrt{N_1})$. By leveraging Poisson summation and Paley–Wiener type arguments, the authors derive uniform, sharp quadrature-error bounds that underpin the convergence rates and confirm a conjectured optimal rate for the tapered scheme. The results provide a solid theoretical justification for the LP approach and quantify how to select clustering to achieve fastest convergence, with numerical evidence validating the sharpness and optimality of the bounds.

Abstract

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.

Figures (12)

• Figure 1: Convergence rates of the LPs with $\sigma=2\pi$ and $\sigma=2\sqrt{2}\pi$ compared with lightning with $\sigma=\pi$ and $\sigma=\sqrt{2}\pi$ for $\sqrt{x}$Herremans2023, respectively. The lightning approximations (cf. Trefethen2021) are equipped with $N =N_1$, while the LPSs with $N = N_1 + N_2$ and $N_2={\rm ceil}(1.3\sqrt{N_1})$. The best rational approximation of $\sqrt{x}$ based on solving a nonlinear approximation problem with free poles was studied by Vjačeslavov 1974Approximation.
• Figure 1: Quadrature errors $\|I(x)-S(x)\|_{\infty}$ of the rectangular rules for $I(x)$ with various values of $\alpha$ and $h=8\pi^2\alpha$$(\sigma_1=\frac{2\sqrt{2}\pi}{\sqrt{\alpha}})$, $h=4\pi^2\alpha$$(\sigma_2=\frac{4\pi}{\sqrt{\alpha}})$, $h=\pi^2\alpha$$(\sigma_3=\frac{\pi}{\sqrt{\alpha}})$, where $N=N_1+N_2$ and $N_1=10:10:500$, $N_2={\rm ceil}(1.3\sqrt{N_1})$.
• Figure 1: The contours for integrals in \ref{['ConvertIntegralToBrokenLine_Uniform']}. The poles nearest to the real line of $\overline{f}(u,x)$ are $\dot{u}_0$ and $\dot{u}_1$.
• Figure 1: The locations of $u_L$ and $u_R$, which divide the integral path $[h-2ia,+\infty-2ia)$ into three parts $[h-2ia,u_L],\ [u_L,u_R]$ and $[u_R,+\infty-2ia)$ satisfying \ref{['bounds of Im(t-2ia)']}, \ref{['monotonicity on h-baru']}, \ref{['monotonicity on baru-infty']}, \ref{['eq:lowerboundforsqruh']} and \ref{['eq:upperboundforsqruh']}, where $\hat{u}$ lies between the points $u_L$ and $u_R$, that is, $v_L=\Re(u_L)<\Re(\hat{u})=\hat{v}<\Re(u_R)=v_R$. Additionally, $\Re(\sqrt{u_0})=\Re(\sqrt{\hat{u}})$.
• Figure 2: Convergence rates of $r_{N}(x)$ for $x^\alpha$ with various values of $\alpha$ and $\sigma=\frac{\pi}{\sqrt{\alpha}}\ (h=\pi^2\alpha)$, $\sigma=\frac{2\pi}{\sqrt{\alpha}}\ (h=4\pi^2\alpha)$ and $\sigma=\frac{3\pi}{\sqrt{\alpha}}\ (h=9\pi^2\alpha)$, where $\mathcal{G}=4^{1+\alpha}\sin(\alpha\pi)$ and $N=N_1+N_2$, $N_1=4:4:100$, $N_2={\rm ceil}(1.3\sqrt{N_1})$.

Theorems & Definitions (23)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.1
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Proof 1
• Theorem 3.3
• Proof 2