Suppressing Low-Frequency Cancellation Errors in Far-Field Computation with the MFIE, Müller, and Multitrace Müller Integral Equations

Technical Summary: Suppressing Low-Frequency Cancellation Errors in Far-Field Computation with MFIE, Müller, and Multitrace Müller Integral Equations

Problem Statement
Boundary integral equations (BIEs), such as the Magnetic Field Integral Equation (MFIE) for Perfect Electric Conductors (PECs) and the Müller equations for dielectric objects, are widely used for electromagnetic scattering modeling. While second-kind formulations like the MFIE and Müller equations are inherently well-conditioned in the dense-mesh and low-frequency regimes, they suffer from a fundamental numerical issue known as low-frequency cancellation errors (or low-frequency catastrophe) when evaluated in finite-precision arithmetic.

This phenomenon arises from the disparate asymptotic scalings of the Helmholtz components of surface currents (irrotational/non-solenoidal, harmonic/global-loop, and solenoidal/local-loop) as the angular frequency $\omega \to 0$. In standard floating-point representations, the smaller magnitude components (typically the non-solenoidal component in PEC scattering) are overwhelmed by dominant components, leading to a loss of significant digits. While these errors may remain negligible in near-field or surface current reconstructions, they critically degrade the accuracy of far-field evaluations. Specifically, the far-field pattern depends on a combination of these components where the non-solenoidal part, despite being asymptotically smaller, contributes at the same order as the solenoidal parts. Standard discretizations fail to preserve the necessary scaling relationships to prevent this loss of precision, resulting in unreliable radiation patterns at low frequencies.

Methodology
The authors propose a robust numerical framework that combines quasi-Helmholtz projectors with a tailored rescaling procedure to suppress these errors while preserving the favorable conditioning of the underlying mixed discretizations.

1. Quasi-Helmholtz Decomposition: The surface currents are separated into non-solenoidal ($j_\Sigma$) and solenoidal ($j_{\Lambda H}$) components using quasi-Helmholtz projectors ($P_\Sigma$ and $P_{\Lambda H}$). This decomposition is stable with respect to mesh refinement.
2. Right Rescaling: An auxiliary unknown is introduced via a rescaling operator $P = (i\omega c_0^{-1} d) P_\Sigma + P_{\Lambda H}$, where $d$ is a characteristic length. This balances the magnitudes of the components, ensuring they are of comparable order ($O(1)$) in the auxiliary variable, thus preventing loss of significant digits during storage.
3. Coefficient-Space Stabilization: To counteract the artificial ill-conditioning introduced by the right rescaling and to balance the excitation vector, the authors adapt a strategy previously used for PMCHWT formulations. The system is first multiplied by the inverse Gram matrix ($G^{-1}$), mapping the operator back to the coefficient space. Subsequently, a left rescaling operator ($P^{-1}$) is applied.
   • For the MFIE, this yields the stabilized system: $(\frac{1}{2}I - P^{-1}G^{-1}K(\kappa_0)P)i = P^{-1}G^{-1}h_{inc}$.
   • For the Müller equation (homogeneous dielectrics) and the Multitrace (MT) Müller equation (composite structures), the approach is extended using block-diagonal extensions of the projectors and rescaling operators.
4. Static Extraction: To ensure the theoretical $O(\omega^2)$ scaling of specific operator blocks is maintained numerically, the authors analytically extract static contributions from the Green's function kernels before quadrature. This prevents quadrature errors from amplifying into spurious low-frequency terms.
5. Far-Field Evaluation: The far field is computed directly from the rescaled auxiliary unknowns using a formulation that respects the cancellation-free representation, avoiding the reconstruction of the physical current which would reintroduce cancellation errors.

Key Contributions

• Robust Formulations for MFIE and Müller: The paper presents the first analogous stabilization methods for the MFIE and Müller equations that do not rely on symmetrization or auxiliary charge equations. This avoids increasing the number of unknowns or complicating the system structure.
• Extension to Composite Structures: The method is successfully extended to the global Multitrace Müller formulation, enabling accurate far-field computation for complex composite dielectric objects with junctions, non-conformal meshes, and high material contrasts.
• Preservation of Conditioning: Unlike many stabilization techniques that degrade the condition number, the proposed method retains the well-conditioned nature of the mixed MFIE and Müller discretizations in the low-frequency and dense-mesh regimes.
• Handling Topology: The approach addresses multiply connected geometries. While the static MFIE operator possesses a topology-induced nullspace on such surfaces, the proposed method ensures that this does not compromise far-field accuracy, as the plane-wave excitation does not couple to the nullspace modes in the far-field evaluation.

Numerical Results
Extensive numerical experiments validate the proposed methods across a broad frequency range (from full-wave to extremely low frequencies, e.g., $10^{-4}$ Hz) and various geometries:

• PEC Scattering: For both a unit sphere (simply connected) and a square torus (multiply connected), the stabilized MFIE (qHP-MFIE) maintains accuracy where classical and mixed MFIE formulations fail at low frequencies.
• Homogeneous Dielectrics: The stabilized Müller formulation (qHP-Müller) outperforms classical and mixed Müller discretizations for spheres and tori with high permittivity contrasts ($\epsilon_r = 100$).
• Composite Dielectrics: The stabilized MT-Müller formulation (qHP MT-Müller) demonstrates robustness for multiscale, high-contrast composite objects with junctions, matching reference solutions from commercial solvers (Altair Feko) down to extremely low frequencies.
• Conditioning: Iteration counts for GMRES solvers remain bounded and often decrease with the proposed regularization compared to the unstabilized mixed formulations, even for multiply connected geometries.

Significance and Claims
The authors claim that their work provides a robust and comparatively simple alternative for electromagnetic scattering simulations spanning from moderately high to extremely low frequencies. By focusing on the MFIE and Müller equations, they offer a solution that avoids the computational overhead and complexity associated with stabilizing first-kind formulations (like EFIE or PMCHWT) or introducing auxiliary equations.

The paper emphasizes that the proposed regularization is particularly advantageous for:

• Composite configurations involving multiple objects and junctions.
• Time-domain simulations, where existing approaches might lead to prohibitively complicated systems.
• Accurate far-field prediction without introducing unnecessary complexity, ensuring that the essential components for far-field computation are preserved from finite-precision cancellation.

The authors note that while the method does not eliminate the intrinsic resonant instability of the MFIE or the topology-induced breakdown at zero frequency for multiply connected surfaces, it effectively decouples these issues from the far-field evaluation, allowing for rapid and accurate solutions using Krylov iterative solvers. Future work is suggested for extending this framework to time-domain equations and mixed PEC-dielectric scatterers.