[Paper Review] Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional penetrable scattering problems

Yassine Boubendir, Oscar P. Bruno+2 more|arXiv (Cornell University)|Oct 4, 2013

Electromagnetic Scattering and Analysis21 references4 citations

TL;DR

This paper introduces a new class of regularized combined boundary integral equations for two-dimensional electromagnetic and acoustic scattering by penetrable scatterers with smooth boundaries. By applying regularizing operators—specifically layer potentials with complex wavenumbers or Fourier-based operators—the method transforms transmission problems into uniquely solvable Fredholm equations of the second kind, significantly reducing Krylov-subspace iteration counts and overall computational cost for a given accuracy.

ABSTRACT

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nystrom computational implementation of the new equations), these "regularized combined equations" can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous boundary integral formulations for transmission problems.

Motivation & Objective

• Address the high iteration counts typical in Krylov-subspace solvers for boundary integral equations in penetrable scattering problems.
• Develop boundary integral formulations that are uniquely solvable Fredholm equations of the second kind to improve spectral conditioning.
• Reduce computational cost by minimizing the number of Krylov iterations required for convergence in high-accuracy simulations.
• Introduce regularizing operators based on complex-wavenumber layer potentials and Fourier symbols to stabilize and regularize hypersingular operators.
• Demonstrate the effectiveness of the new formulations through high-order Nyström discretizations and numerical experiments on smooth, two-dimensional penetrable scatterers.

Proposed method

• Construct field representations using combinations of single and double layer potentials acting on regularizing operators to stabilize the integral equations.
• Apply regularizing operators with complex wavenumbers (e.g., $ ilde{S}_{ ilde{ u}} $) to transform hypersingular kernels into compact, well-conditioned operators.
• Use Fourier-based regularizing operators whose symbols match the high-frequency asymptotics of complex-wavenumber layer potentials.
• Formulate the problem as a system of second-kind Fredholm equations in appropriate Sobolev spaces, ensuring unique solvability.
• Implement high-order Nyström methods using global trigonometric polynomial approximations, kernel splitting into singular and smooth parts, and explicit quadrature for logarithmic singularities.
• Leverage known asymptotic expansions of Bessel and Hankel functions to analyze the smoothing properties of the regularized operators.

Experimental results

Research questions

• RQ1 Can regularizing operators be used to transform hypersingular integral operators in transmission problems into compact, well-conditioned second-kind equations?
• RQ2 Do the new regularized combined integral equations lead to a significant reduction in Krylov-subspace iteration counts compared to classical formulations?
• RQ3 What is the spectral behavior of the new integral operators, and how does it affect the convergence rate of iterative solvers?
• RQ4 Can the regularizing operators based on complex-wavenumber layer potentials or Fourier symbols be rigorously shown to yield second-kind equations with improved conditioning?
• RQ5 To what extent do high-order Nyström discretizations of the new equations achieve optimal convergence rates and reduced computational cost for smooth scatterers?

Key findings

• The proposed regularized combined integral equations are uniquely solvable Fredholm equations of the second kind in appropriate Sobolev spaces, ensuring stable and convergent iterative solution methods.
• The use of complex-wavenumber layer potentials as regularizing operators transforms the system matrix into one with improved spectral properties, leading to a dramatic reduction in Krylov-subspace iteration counts.
• Numerical experiments demonstrate that the new formulations require significantly fewer iterations than classical formulations—often by a factor of 2 to 5—for the same accuracy, reducing overall computational cost.
• The difference between the parametric single-layer operator and its Fourier-based regularized counterpart is a third-order smoothing operator, which justifies the regularization's effectiveness in improving spectral conditioning.
• The high-order Nyström implementation achieves spectral accuracy for smooth boundaries, with explicit quadrature rules for logarithmic singularities enabling high precision.
• The method achieves superior efficiency compared to previous formulations, particularly for high-frequency scattering problems, due to the favorable spectral properties induced by the regularizing operators.