一种具有可证明误差界的修正FC-Gram逼近算法
A modified FC-Gram approximation algorithm with provable error bounds
AI总结：
本文提出一种修正的FC-Gram算法，用两点Hermite多项式的显式延拓替代隐式最小二乘周期延拓，从而消除预计算数据需求并首次建立可证明误差界，实验验证了理论收敛速率。
摘要：
FC-Gram三角多项式逼近用于在等距网格上插值非周期函数，由Bruno和Lyon于2010年提出[J. Comput. Phys, 229(6):2009-2033, 2010]。此后，该逼近算法及其进一步改进被广泛用于各种基于偏微分方程问题的数值求解中，并在处理具有挑战性的构型方面取得了令人瞩目的成功。虽然文献中存在大量计算证据证实FC-Gram逼近的快速收敛性，但理论收敛分析一直悬而未决。本文研究了一种修正的FC-Gram算法，其中Gram多项式的基于隐式最小二乘的周期延拓被替换为利用两点Hermite多项式的显式延拓。这一修正带来了两个显著优势——（i）由于延拓是显式已知的，消除了使用计算成本高昂的预计算延拓数据的需要，从而便于无缝改变延拓长度；（ii）允许为修正后的逼近建立可证明的误差界。我们通过各种计算实验表明，数值收敛速率与理论预测的收敛速率一致。
英文摘要：
The FC-Gram trigonometric polynomial approximation of a non-periodic function that interpolates the function on equispaced grids was introduced in 2010 by Bruno and Lyon [J. Comput. Phys, 229(6):2009-2033, 2010]. Since then, the approximation algorithm and its further refinements have been used extensively in numerical solutions of various PDE-based problems, and it has had impressive success in handling challenging configurations. While much computational evidence exists in the literature confirming the rapid convergence of FC-Gram approximations, a theoretical convergence analysis has remained open. In this paper, we study a modified FC-Gram algorithm where the implicit least-squares-based periodic extensions of the Gram polynomials are replaced with an explicit extension utilizing two-point Hermite polynomials. This modification brings in two significant advantages - (i) as the extensions are known explicitly, the need to use computationally expensive precomputed extension data is eliminated, which, in turn, facilitates seamlessly changing the extension length, and (ii) allows for establishing provable error bounds for the modified approximations. We show that the numerical convergence rates are consistent with those predicted by the theory through a variety of computational experiments.
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