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Fast Singular-Kernel Convolution on General Non-Smooth Domains via Truncated Fourier Filtering

Oscar Bruno, Jinghao Cao

TL;DR

The paper tackles fast, high-order evaluation of convolutions with singular kernels on general non-smooth domains by extending the Truncated Fourier Filtering approach. It decomposes the convolution into a near-field singular part treated via a polar, Fourier-based windowing and a far-field smooth part computed with FFT-based convolution on a truncated Fourier representation of the domain indicator. The authors establish convergence properties, present a complete algorithm with pseudocode, and validate the method on disk and drop-shaped domains, showing superalgebraic accuracy and practical efficiency. This yields a robust, grid-based framework for high-order singular-kernel convolutions that avoids geometric reparameterization and handles corners and limited smoothness effectively.

Abstract

The rapid and accurate evaluation of convolutions with singular kernels plays crucial roles in a wide range of scientific and engineering applications. Building on the recently introduced Truncated Fourier Filtering method for smooth kernels, this work presents a fast, high-order numerical methodology that extends the approach to singular kernels and non-smooth domains. The method relies on truncated Fourier expansions of a prescribed order for the characteristic function of the integration domain, as well as expansions for the products of characteristic functions and singular functions.

Fast Singular-Kernel Convolution on General Non-Smooth Domains via Truncated Fourier Filtering

TL;DR

The paper tackles fast, high-order evaluation of convolutions with singular kernels on general non-smooth domains by extending the Truncated Fourier Filtering approach. It decomposes the convolution into a near-field singular part treated via a polar, Fourier-based windowing and a far-field smooth part computed with FFT-based convolution on a truncated Fourier representation of the domain indicator. The authors establish convergence properties, present a complete algorithm with pseudocode, and validate the method on disk and drop-shaped domains, showing superalgebraic accuracy and practical efficiency. This yields a robust, grid-based framework for high-order singular-kernel convolutions that avoids geometric reparameterization and handles corners and limited smoothness effectively.

Abstract

The rapid and accurate evaluation of convolutions with singular kernels plays crucial roles in a wide range of scientific and engineering applications. Building on the recently introduced Truncated Fourier Filtering method for smooth kernels, this work presents a fast, high-order numerical methodology that extends the approach to singular kernels and non-smooth domains. The method relies on truncated Fourier expansions of a prescribed order for the characteristic function of the integration domain, as well as expansions for the products of characteristic functions and singular functions.

Paper Structure

This paper contains 11 sections, 8 theorems, 99 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Window decomposition of the integral
  4. Computation of : windowed singularity
  5. Computation of : smooth integrand.
  6. Algorithm and Pseudocode
  7. Numerical Results
  8. Some results on Fourier series
  9. Decay of Fourier coefficients
  10. Computation of the Fourier series coefficients for x log(x)
  11. Fourier series of the characteristic function of a parametrized domain

Key Result

Lemma 1

Let $\varphi$ denote a piecewise continuous function on $[-\pi,\pi]$, with Fourier series $\varphi(x) = \sum_{k\in\mathbb{Z}}\varphi_{k}e^{\mathrm{i} k\pi}$. Then, the trapezoidal-rule quadrature error is given by

Figures (8)

  • Figure 1: Illustration of a drop shaped domain utilized in the description of the proposed TFF algorithm, with parameterization given by ${\partial\Omega} = \{(3\sin(t/2),-2\sin(t))\ |\ t\in [0,2\pi]\}$.
  • Figure 2: Illustration of the window function $W_1$ with window widths $w_0 = 1/4$ and $w_1 = 1$, in the context of the domain $\Omega$ depicted in Figure \ref{['fig:teardrop']}. In this case, with singularity point $x=(2.5,1)$, the support of the window function intersects a smooth section of the boundary of $\Omega$. For other points $x\in\Omega$, the support may intersect a section of the boundary containing the corner point, or not intersect the boundary at all.
  • Figure 3: Three intersection cases considered in this section.
  • Figure 4: Illustration of the method used when the window function intersects with a non-smooth corner (case (iii))
  • Figure 5: Convergence analysis for $I_2(x)$ with increasing number of equidistant sample points. Target value $2^{12}$ discretization point in each direction. Here we take $\phi(y) = \exp(\mathrm{i} 40 y_1-\mathrm{i} 20y_2)$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 3 more