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Anti-Gauss cubature rules with applications to Fredholm integral equations on the square

Patricia Diaz de Alba, Luisa Fermo, Giuseppe Rodriguez

TL;DR

The paper addresses accurate numerical integration on the square $\mathcal{S}$ for functions with endpoint algebraic singularities by factoring the integrand with a Jacobi-type weight and introduces anti-Gauss cubature rules as a direct multivariate construction. It develops a weighted Nyström method built from Gauss and anti-Gauss cubatures, proves that the two Nyström interpolants bracket the true solution and shows that their average improves accuracy, with convergence tied to the best polynomial approximation in weighted spaces. The authors formulate efficient linear solvers for the resulting dense systems, including GMRES-FM for general tensors and GMRES-SK for separable kernels, and establish stability with discretization-size-independent condition numbers. Numerical experiments confirm the opposite-sign errors of Gauss and anti-Gauss rules and demonstrate substantial accuracy gains and computational savings from the averaged Nyström interpolant, including favorable performance for large, low-regularity problems and separable kernels. These results provide a robust framework for solving weighted Fredholm integral equations on the square with boundary singularities, with clear avenues for extension to additional averaged cubature schemes.

Abstract

The purpose of this paper is to develop the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of Fredholm integral equations of the second-kind is also explored. The stability, convergence, and conditioning of the proposed Nyström-type method are studied. The numerical solution of the resulting dense linear system is also investigated and several numerical tests are presented.

Anti-Gauss cubature rules with applications to Fredholm integral equations on the square

TL;DR

The paper addresses accurate numerical integration on the square for functions with endpoint algebraic singularities by factoring the integrand with a Jacobi-type weight and introduces anti-Gauss cubature rules as a direct multivariate construction. It develops a weighted Nyström method built from Gauss and anti-Gauss cubatures, proves that the two Nyström interpolants bracket the true solution and shows that their average improves accuracy, with convergence tied to the best polynomial approximation in weighted spaces. The authors formulate efficient linear solvers for the resulting dense systems, including GMRES-FM for general tensors and GMRES-SK for separable kernels, and establish stability with discretization-size-independent condition numbers. Numerical experiments confirm the opposite-sign errors of Gauss and anti-Gauss rules and demonstrate substantial accuracy gains and computational savings from the averaged Nyström interpolant, including favorable performance for large, low-regularity problems and separable kernels. These results provide a robust framework for solving weighted Fredholm integral equations on the square with boundary singularities, with clear avenues for extension to additional averaged cubature schemes.

Abstract

The purpose of this paper is to develop the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of Fredholm integral equations of the second-kind is also explored. The stability, convergence, and conditioning of the proposed Nyström-type method are studied. The numerical solution of the resulting dense linear system is also investigated and several numerical tests are presented.
Paper Structure (10 sections, 9 theorems, 88 equations, 2 figures, 7 tables)

This paper contains 10 sections, 9 theorems, 88 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Cubature rules
  3. Nyström methods and the averaged Nyström interpolant
  4. Solving the linear systems
  5. The general case
  6. The case of a separable kernel
  7. Numerical results
  8. Approximation of integrals
  9. Numerical solution of Fredholm integral equations of the second-kind
  10. Conclusion and extensions

Key Result

Proposition 2.1

The error of the anti-Gauss cubature scheme A has the following property

Figures (2)

  • Figure 5.1: Graph of the two terms in inequality \ref{['condsign']} for Example \ref{['example1']} (left) and Example \ref{['example2']} (right): $|S_{n_1,n_2}|$ is represented by a continuous line, $\max(|\mathcal{E}^{(1)}_{n_1,n_2}|,|\mathcal{E}^{(2)}_{n_1,n_2}|)$ by a dashed one.
  • Figure 5.2: Error graphs for Example \ref{['test1']}

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 14 more