insight - Computational Complexity - # Convergence rate of lightning plus polynomial approximation for power functions

Core Concepts

The lightning plus polynomial approximation can achieve a root-exponential convergence rate for approximating power functions xα on [0, 1] for any σ > 0, and the choice of σ = 2π/√α achieves the fastest convergence rate among all σ > 0.

Abstract

The paper analyzes the convergence rate of the lightning plus polynomial approximation for approximating power functions xα on [0, 1], where 0 < α < 1.
The key highlights and insights are:
The lightning plus polynomial approximation rN(x) (1.3) based on the tapered exponential clustering of poles (1.4) can achieve a root-exponential convergence rate for any σ > 0:
For σ ≤ 2π/√α, the convergence rate is O(e-σα√N).
For σ > 2π/√α, the convergence rate is O(e-4π2(σ/√N)).
The choice of σ = 2π/√α achieves the fastest convergence rate among all σ > 0, confirming the conjecture in [7].
The analysis leverages the Poisson summation formula and the decay behaviors of Fourier transforms to rigorously establish the exponential convergence rates, which were previously only illustrated through numerical experiments.
The constants in the O-terms are independent of α ∈ (0, 1), σ > 0, and N, ensuring the theoretical results are applicable for a wide range of parameters.
The paper provides a detailed treatment of the quadrature error analysis near x = 0 and for x ∈ [x*, 1], where x* is a carefully chosen threshold, to obtain the uniform convergence rates.

Stats

sin(απ)/απ
sin((1-α)π)/(1-α)π
e-T

Quotes

"There exist coefficients {aj}N1j=1 and a polynomial bN2 with N2 = O(√N1), for which the lightning + polynomial approximation rN(x) (1.3) having tapered lightning poles (1.4) with σ > 0 satisfies: |rN(x) - xα| = O(e-σα√N) for σ ≤ 2π/√α, and O(e-4π2(σ/√N)) for σ > 2π/√α as N = N1 + N2 → ∞, uniformly for x ∈ [0, 1]."

Key Insights Distilled From

by Shuhuang Xia... at **arxiv.org** 04-16-2024

Deeper Inquiries

The techniques developed in the paper for approximating functions with branching singularities on the boundary, such as power functions, can be extended to approximate other types of functions with similar characteristics. Functions with branch cuts, logarithmic singularities, or other types of singularities can also be approximated using similar methods. By adapting the lightning plus polynomial approximation scheme and utilizing tools like the Poisson summation formula and Fourier transforms, it is possible to construct efficient approximation schemes for a broader class of functions with singularities. The key lies in understanding the behavior of the function near the singularities and devising appropriate strategies to capture this behavior accurately in the approximation.

The optimal convergence rate achieved by the lightning plus polynomial approximation has significant implications for practical applications that require high-accuracy function approximations. In fields such as numerical analysis, scientific computing, and engineering, where precise function approximations are crucial for simulations, modeling, and data analysis, having a method that can achieve fast convergence rates is highly beneficial. The ability to approximate functions with branching singularities at a root-exponential convergence rate can lead to more accurate results with fewer computational resources. This can improve the efficiency and reliability of algorithms that rely on function approximations, ultimately enhancing the quality of the outcomes in various applications.

The insights from this work can indeed inspire new approaches for constructing efficient approximation schemes for a broader class of functions with singularities. By understanding the convergence properties of the lightning plus polynomial approximation and the underlying mathematical principles involved, researchers can explore similar techniques for approximating different types of singular functions. This could lead to the development of specialized approximation methods tailored to specific types of singularities, improving the accuracy and efficiency of numerical computations in various domains. Additionally, the rigorous analysis and proof techniques employed in this work can serve as a foundation for further research in function approximation theory, paving the way for advancements in computational mathematics and related fields.

0