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A modified FC-Gram approximation algorithm with provable error bounds

Akash Anand, Prakash Nainwal

TL;DR

This paper studies 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, demonstrating the efficacy of this approximation strategy in scientific computation applications.

Abstract

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.

A modified FC-Gram approximation algorithm with provable error bounds

TL;DR

This paper studies 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, demonstrating the efficacy of this approximation strategy in scientific computation applications.

Abstract

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.
Paper Structure (10 sections, 3 theorems, 48 equations, 2 figures, 10 tables)

This paper contains 10 sections, 3 theorems, 48 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. FC-Gram methodology
  3. Extension
  4. Approximation
  5. Continuation of Gram polynomials in FC-Gram
  6. The FC-Gram approximation errors
  7. A modified FC-Gram
  8. Continuation of Gram polynomial using Hermite polynomials
  9. Convergence analysis
  10. Numerical Implementation

Key Result

Theorem 3.1

Let $f\in C^{\infty}\left( [0, 1]\right)$ for some $d\in \mathbb{N}$. For $b\in\mathbb{Q}$ with $b>1$, let $\mathbb{N}_b = \{n\in\mathbb{N} : nb \in \mathbb{N} \text{ and } 2 \mid nb\}$. For all $n \in \mathbb{N}_b$, let $\tau_n(f) := t_{n,nb-n-1}$ be the trigonometric polynomial approximation that for all $n\in\mathbb{N}_b$.

Figures (2)

  • Figure 2: The uniform grid used for the construction of Gram polynomial extensions where $d = 6,~ C = 25,~ Z = 12$ and $E = 25$
  • Figure 3: plots of $\varphi_{\ell}^{\text{LS}}$ (dotted lines) and $\varphi_{\ell}^{\text{H}}$ (solid lines) for $0\leq \ell \leq 4$ with $s=1$, $d= 5$, $C = 25$, $Z= 12$, $E = 25$ and $h = 1/256$

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of \ref{['theorem: main']}