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Scaling Optimized Hermite Approximation Methods

Hao Hu, Haijun Yu

TL;DR

This work tackles the long-standing question of scaling in Hermite spectral methods by introducing a rigorous error-analysis framework that partitions the $L^2$ projection error into spatial truncation, frequency truncation, and Hermite-spectral components. By relating the spatial and frequency domains through a scaling factor $\beta$, the authors show that the optimal $\beta$ balances the two truncation errors, recovering geometric or doubled convergence rates for broad function classes and clarifying the sub-geometric behavior observed in pre-asymptotics. The paper also establishes that properly scaled Gauss--Hermite quadrature can achieve optimal rates comparable to domain-truncated methods, addressing claims of suboptimality in the literature. Overall, the framework provides practical guidelines for selecting scaling to improve accuracy in Hermite-based approximations and explains key convergence phenomena using a dual space perspective grounded in Fourier analysis and uncertainty principles.

Abstract

Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the $L^2$ projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

Scaling Optimized Hermite Approximation Methods

TL;DR

This work tackles the long-standing question of scaling in Hermite spectral methods by introducing a rigorous error-analysis framework that partitions the projection error into spatial truncation, frequency truncation, and Hermite-spectral components. By relating the spatial and frequency domains through a scaling factor , the authors show that the optimal balances the two truncation errors, recovering geometric or doubled convergence rates for broad function classes and clarifying the sub-geometric behavior observed in pre-asymptotics. The paper also establishes that properly scaled Gauss--Hermite quadrature can achieve optimal rates comparable to domain-truncated methods, addressing claims of suboptimality in the literature. Overall, the framework provides practical guidelines for selecting scaling to improve accuracy in Hermite-based approximations and explains key convergence phenomena using a dual space perspective grounded in Fourier analysis and uncertainty principles.

Abstract

Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

Paper Structure

This paper contains 23 sections, 13 theorems, 82 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basics on Hermite functions
  3. An intuitive explanation of our idea
  4. How our results help better understand the Hermite approximation
  5. Projection error in $L^2$ norm
  6. Definition of projection
  7. The duality of $u$ and $\mathcal{F}[u]$ when approximating by Hermite functions
  8. Projection error in $L^2$ norm: a description without proof
  9. Projection error for a specific class of Gaussian-type functions
  10. Proof of \ref{['thm:L2-proj']}
  11. Projection error with higher-order derivatives and interpolation error
  12. Inverse inequalities
  13. Projection error with higher-order derivatives
  14. Interpolation error
  15. Application: optimality of scaled Gauss--Hermite quadrature
  16. ...and 8 more sections

Key Result

Lemma 2.1

\newlabellem:cn recurr0 If $e^{-m x^2} e^{i k x}=\sum\limits_{n=0}^{\infty} c_n \widehat{H}_n$, then $\{c_n\}$ satisfy

Figures (4)

  • Figure 1: Errors for different quadrature rules to integrate $u=e^{-x^2}\frac{1}{1+x^8}$. The number of quadrature nodes N varies from 10 to 400.
  • Figure 2: Interpolation error without scaling and with optimal scaling for $u=e^{-x^8}$.
  • Figure 3: Error $\|u - u_N\|$ in solving model equation \ref{['eq:model-problem']} with exact solution $u = 1/(1+x^2)$ and scaling factor $\beta=1, \beta = 10/\sqrt{N}$. N ranges from 200 to 300.
  • Figure 4: Error $\|u - u_N\|$ in solving model problem \ref{['eq:model-problem']} with exact solution $u = 1/(1+x^2)^h$ for different values of parameter $h$.

Theorems & Definitions (17)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • Proof 1
  • Proof 2
  • Lemma 3.1
  • Lemma 3.2
  • ...and 7 more