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Spectral approximation of convolution operators of Fredholm type

Xiaolin Liu, Kuan Deng, Kuan Xu

TL;DR

This work develops a numerically stable spectral framework for Fredholm-type convolution operators using Legendre polynomial expansions on canonical intervals. It introduces a convolution matrix $R$ built via a four-term recurrence that yields an $ ext{O}(M^2)$ construction and an $N$- and $r$-independent evaluation cost, enabling fast Fredholm convolutions and a spectral method for Fredholm convolution integral equations. The approach supports eigenvalue and pseudospectrum computations and offers stability analyses and strategies for handling different domain ratios $r$, including a Volterra-type relation when $r=1$. Empirically, the method delivers significant speedups over existing techniques, demonstrates high accuracy on challenging examples, and provides a practical tool for convolution-based modeling and spectral analysis with potential extensions to singular kernels and fractional operators.

Abstract

We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such a matrix approximation. When used for computing the Fredholm convolution of two given functions, such approximations produce the convolution more rapidly than the state-of-the-art methods. The proposed approximation also leads to a spectral method for solving the Fredholm convolution integral equations and enables the computation of eigenvalues and pseudospectra of Fredholm convolution operators, which is otherwise intractable with existing techniques.

Spectral approximation of convolution operators of Fredholm type

TL;DR

This work develops a numerically stable spectral framework for Fredholm-type convolution operators using Legendre polynomial expansions on canonical intervals. It introduces a convolution matrix built via a four-term recurrence that yields an construction and an - and -independent evaluation cost, enabling fast Fredholm convolutions and a spectral method for Fredholm convolution integral equations. The approach supports eigenvalue and pseudospectrum computations and offers stability analyses and strategies for handling different domain ratios , including a Volterra-type relation when . Empirically, the method delivers significant speedups over existing techniques, demonstrates high accuracy on challenging examples, and provides a practical tool for convolution-based modeling and spectral analysis with potential extensions to singular kernels and fractional operators.

Abstract

We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such a matrix approximation. When used for computing the Fredholm convolution of two given functions, such approximations produce the convolution more rapidly than the state-of-the-art methods. The proposed approximation also leads to a spectral method for solving the Fredholm convolution integral equations and enables the computation of eigenvalues and pseudospectra of Fredholm convolution operators, which is otherwise intractable with existing techniques.
Paper Structure (9 sections, 8 theorems, 53 equations, 9 figures, 1 algorithm)

This paper contains 9 sections, 8 theorems, 53 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Constructing the convolution matrices
  3. Relation to the convolution operator of Volterra type
  4. Numerical examples
  5. Speed in computing Fredholm convolutions
  6. An example for demonstrating the Weierstrass approximation theorem
  7. Convolution integral equation
  8. Pseudospectrum
  9. Conclusion

Key Result

Lemma 2.1

\newlabellemma10 For $n\geq 1$, where $C$ is the integration constant.

Figures (9)

  • Figure 1: Error growth when recursing for $R$ corresponding to $f_{39}$ using \ref{['R_m', 'n+1']}, which is stable in the region that is to the left of the red borderline.
  • Figure 1: Entrywise magnitude of $\hat{R}$ and $R$.
  • Figure 1: Computational time versus $M$ for various choices of $N$ and $r$.
  • Figure 2: Partition of the nonzero region by the borderline of stability (red) and the line of dependence (blue).
  • Figure 2: Computational time versus $N/M$ for various choices of $M$ and $r$.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Theorem 2.3
  • Proof 3
  • Lemma 2.4
  • Proof 4
  • Theorem 2.5
  • Proof 5
  • ...and 4 more