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On the Computation of Square Roots and Inverse Square Roots of Gram Matrices for Surface Integral Equations in Electromagnetics

Rui Chen, Adrien Merlini, Francesco P. Andriulli

TL;DR

The paper tackles the problem that discretized Gram matrices in EM surface integral equations do not naturally share the spectrum of the underlying operators, complicating spectral methods and preconditioning. It introduces three polynomial-approximation strategies—Taylor, Chebyshev, and Padé—to compute $\sqrt{\mathbf{G}}$ and $\sqrt{\mathbf{G}}^{-1}$ for SPD Gram matrices, including scaling by $\|\mathbf{G}\|_2$ and matrix-function theory from Higham. The authors provide extensive coefficient tables, discuss matrix-spectrum prerequisites, and demonstrate spectrum-revealing normalization on a sphere scattering problem, where the normalized matrix spectrum aligns with the analytic operator spectrum. The results indicate that these approaches yield controllable accuracy, with Padé often performing well for well-conditioned cases, and that the techniques are broadly applicable to standard EM and acoustic integral equations, enhancing spectral analyses and preconditioning workflows.

Abstract

Surface integral equations (SIEs)-based boundary element methods are widely used for analyzing electromagnetic scattering scenarii. However, after discretization of SIEs, the spectrum and eigenvectors of the boundary element matrices are not usually representative of the spectrum and eigenfunctions of the underlying surface integral operators, which can be problematic for methods that rely heavily on spectral properties. To address this issue, we delineate some efficient algorithms that allow for the computation of matrix square roots and inverse square roots of the Gram matrices corresponding to the discretization scheme, which can be used for revealing the spectrum of standard electromagnetic integral operators. The algorithms, which are based on properly chosen expansions of the square root and inverse square root functions, are quite effective when applied to several of the most relevant Gram matrices used for boundary element discretizations in electromagnetics. Tables containing different sets of expansion coefficients are provided along with comparative numerical experiments that evidence advantages and disadvantages of the different approaches. In addition, to demonstrate the spectrum-revealing properties of the proposed techniques, they are applied to the discretization of the problem of scattering by a sphere for which the analytic spectrum is known.

On the Computation of Square Roots and Inverse Square Roots of Gram Matrices for Surface Integral Equations in Electromagnetics

TL;DR

The paper tackles the problem that discretized Gram matrices in EM surface integral equations do not naturally share the spectrum of the underlying operators, complicating spectral methods and preconditioning. It introduces three polynomial-approximation strategies—Taylor, Chebyshev, and Padé—to compute and for SPD Gram matrices, including scaling by and matrix-function theory from Higham. The authors provide extensive coefficient tables, discuss matrix-spectrum prerequisites, and demonstrate spectrum-revealing normalization on a sphere scattering problem, where the normalized matrix spectrum aligns with the analytic operator spectrum. The results indicate that these approaches yield controllable accuracy, with Padé often performing well for well-conditioned cases, and that the techniques are broadly applicable to standard EM and acoustic integral equations, enhancing spectral analyses and preconditioning workflows.

Abstract

Surface integral equations (SIEs)-based boundary element methods are widely used for analyzing electromagnetic scattering scenarii. However, after discretization of SIEs, the spectrum and eigenvectors of the boundary element matrices are not usually representative of the spectrum and eigenfunctions of the underlying surface integral operators, which can be problematic for methods that rely heavily on spectral properties. To address this issue, we delineate some efficient algorithms that allow for the computation of matrix square roots and inverse square roots of the Gram matrices corresponding to the discretization scheme, which can be used for revealing the spectrum of standard electromagnetic integral operators. The algorithms, which are based on properly chosen expansions of the square root and inverse square root functions, are quite effective when applied to several of the most relevant Gram matrices used for boundary element discretizations in electromagnetics. Tables containing different sets of expansion coefficients are provided along with comparative numerical experiments that evidence advantages and disadvantages of the different approaches. In addition, to demonstrate the spectrum-revealing properties of the proposed techniques, they are applied to the discretization of the problem of scattering by a sphere for which the analytic spectrum is known.
Paper Structure (11 sections, 21 equations, 3 figures, 4 tables)

This paper contains 11 sections, 21 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Computation of the Square Roots and Inverse Square Roots of the Gram Matrices
  4. Taylor Series Expansion
  5. Chebyshev Polynomial Expansion
  6. Padé Approximant Expansion
  7. General Comments
  8. Numerical Results
  9. Accuracy of the Computation of the Square Roots and Inverse Square Roots of the Gram Matrices
  10. Application to Spectral Analysis
  11. Conclusion

Figures (3)

  • Figure 1: Relative error of the computation of (a) $\sqrt{\mathbf{G}_{\mathbf{f}, \mathbf{f}}}$, (b) $\sqrt{\mathbf{G}_{\mathbf{g}, \mathbf{g}}}$, (c) $\sqrt{\mathbf{G}_{\lambda , \lambda}}$, and (d) $\sqrt{\mathbf{G}_{\tilde{\lambda},\tilde{\lambda}}}$ using TSE, CPE-1, CPE-2, and PAE for mesh-1, mesh-2, and mesh-3 against the expansion order.
  • Figure 2: Relative error of the computation of (a) $\sqrt{\mathbf{G}_{\mathbf{f}, \mathbf{f}}}^{-1}$, (b) $\sqrt{\mathbf{G}_{\mathbf{g}, \mathbf{g}}}^{-1}$, (c) $\sqrt{\mathbf{G}_{\lambda , \lambda}}^{-1}$, and (d) $\sqrt{\mathbf{G}_{\tilde{\lambda},\tilde{\lambda}}}^{-1}$ using TSE, CPE-1, CPE-2, and PAE for mesh-1, mesh-2, and mesh-3 against the expansion order.
  • Figure 3: Comparison of the singular values of $\mathbf{T}$ and $\tilde{\mathbf{T}}$ with the associated analytic spectrum of $\mathcal{T}$ including a zoom around singular value index 720.0.