arXiv subjects

Simon B. Adrian

Publications and source records attributed to Simon B. Adrian.

10 recordsLinked to original sources

arXiv2026

Well-conditioned Electric Field Surface Integral Equations using Reflective Generalized Sources

This work uses the generalized source approach to develop a class of well-conditioned integral operators that are free of internal resonance, without the need for combined formulations. The Generalized source integral equations (GSIEs) kernels are obtained by augmenting the conventional electric field integral equation (EFIE) kernel with auxiliary contributions to enhance the rank deficiency of the corresponding moment matrix blocks. This paper presents the first investigation of the spectra of GSIE operators for auxiliary kernels produced by internal scattering convex shields. Using closed-form expressions for concentric circular scatterers and shields, it is shown that, with shield parameters that are suitable for enhanced compressibility, the transverse magnetic (TM)- and transverse electric (TE)-GSIE operators are free of internal proper and quasi-resonances. The auxiliary components are shown to be compact perturbations of their EFIE counterparts. Hence these GSIESs inherit their dense-discretization breakdown, which remains curable via Calder\'on-type preconditioning. The mechanisms that govern the high-frequency breakdown are shown to be influenced by the auxiliary component, leading, in some cases, to greater resilience. These observations are shown to remain valid for moment matrices and GSIEs designed with non-circular stencil shields. For both GSIE-dual and Yukawa-kernel preconditioners, the formulations exhibit favorable spectral properties while maintaining their compressibility. This makes the formulations attractive for the design of fast iterative solvers. The results on the two-dimensional shield-based operators provide a foundation for extending the approach to three-dimensional problems and to auxiliary kernels with broader geometric applicability.

math.NA↗

arXiv2026

An Explicit Higher-Order Dual Basis for a Multiplicatively Calder\'on Preconditioned Electric Field Integral Equation

One of the most effective means to precondition the electric field integral equation (EFIE) discretized with Rao-Wilton-Glisson (RWG) functions is the multiplicative Calder\'on preconditioner employing Buffa-Christiansen (BC) functions as a basis dual to the RWG basis. It results in a formulation that is free from the dense-discretization and the low-frequency breakdown. To generalize the multiplicative Calder\'on preconditioner from the low-order BC and RWG basis to higher orders, we utilize B-spline-based basis functions and establish the first explicit high-order dual basis. It can be regarded as a generalization of the BC functions to arbitrary polynomial degrees and constitutes a fundamental building block for other approaches that rely on a dual basis. Numerical results for the obtained preconditioner demonstrate a low and constant number of generalized minimum residual (GMRES) iterations independent of the number of unknonws and the polynomial degree for canonical and realistic perfectly electrically conducting (PEC) scatterers; a key to enable the full potential of higher-order bases.

math.NA↗

arXiv2026

A Stabilized Multilevel B-Spline-Based Fast Integral Method for the Solution of the Electric Field Integral Equation

We present a multilevel B-spline-based fast integral method for the solution of the electric field integral equation (EFIE), combining fast Fourier transformation (FFT)-compatible kernel interpolation with robust high-order interpolation. Existing FFT-accelerated global Lagrange-based approaches rely on equidistant interpolation points and can, therefore, suffer from Runge-type instabilities at high interpolation orders, limiting robust high-accuracy compression. In contrast, B-splines on equidistant knot vectors overcome these instabilities and enable robust high-order interpolation for accurate matrix compression. Replacing Lagrange interpolation by B-spline interpolation is, however, non-trivial: B-spline coefficients do not coincide with function values at the interpolation points, and the associated sampling matrices can become ill-conditioned. To address these challenges, we introduce a knot-removal stabilization strategy, combined with exact interlevel transfers based on knot insertion, yielding accurate, well-conditioned multilevel interpolation. Moreover, we propose a factorization strategy that preserves the null space of the scalar potential operator up to machine precision and is compatible with low-frequency preconditioning techniques. Numerical results for both canonical and realistic geometries demonstrate robust high-order interpolation without the breakdown observed for Lagrange-based approaches and confirm $\mathcal{O}(N)$ complexity.

math.NA↗

arXiv2022

On a Calderón preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements

We present a Calderón preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calderón schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios.

math.NA↗

arXiv2022

A New Refinement-Free Preconditioner for the Symmetric Formulation in Electroencephalography

Widely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator ill-suited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calderón-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system matrix, which can be efficiently solved via fast iterative techniques. Numerical results for both canonical and realistic head models validate the effectiveness of the proposed formulation.

math.NA↗

arXiv2022

On the Fast Direct Solution of a Preconditioned Electromagnetic Integral Equation

This work presents a fast direct solver strategy for electromagnetic integral equations in the high-frequency regime. The new scheme relies on a suitably preconditioned combined field formulation and results in a single skeleton form plus identity equation. This is obtained after a regularization of the elliptic spectrum through the extraction of a suitably chosen equivalent circulant problem. The inverse of the system matrix is then obtained by leveraging the Woodbury matrix identity, the low-rank representation of the extracted part of the operator, and fast circulant algebra yielding a scheme with a favorable complexity and suitable for the solution of multiple right-hand sides. Theoretical considerations are accompanied by numerical results both of which are confirming and showing the practical relevance of the newly developed scheme.

math.NA↗

arXiv2020

On the Spectral Behavior and Normalization of a Resonance-Free and High-Frequency Stable Integral Equation

The CFIE used for solving scattering and radiation problems, although a resonance-free formulation, suffers from an ill-conditioning that strongly depends on the frequency and discretization density, both in the low- and high-frequency regime, resulting in slow convergence rates for iterative solvers. This work presents a new preconditioning scheme for the CFIE that cures the low- and the high-frequency as well as the dense discretization breakdown. The new preconditioner for the CFIE is based on a spherical harmonics analysis and the proper regularization with Helmholtz-type operators. Numerical results have been obtained to prove the effectiveness of this new formulation in real scenarios.

physics.comp-ph↗

arXiv2020

On Preconditioning Electromagnetic Integral Equations in the High Frequency Regime via Helmholtz Operators and quasi-Helmholtz Projectors

Fast and accurate resolution of electromagnetic problems via the \ac{BEM} is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderón-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties.

physics.comp-ph↗

arXiv2018

On a Refinement-Free Calder\'on Multiplicative Preconditioner for the Electric Field Integral Equation

We present a Calder\'on preconditioner for the electric field integral equation (EFIE), which does not require a barycentric refinement of the mesh and which yields a Hermitian, positive definite (HPD) system matrix allowing for the usage of the conjugate gradient (CG) solver. The resulting discrete equation system is immune to the low-frequency and the dense-discretization breakdown and, in contrast to existing Calder\'on preconditioners, no second discretization of the EFIE operator with Buffa-Christiansen (BC) functions is necessary. This preconditioner is obtained by leveraging on spectral equivalences between (scalar) integral operators, namely the single layer and the hypersingular operator known from electrostatics, on the one hand, and the Laplace-Beltrami operator on the other hand. Since our approach incorporates Helmholtz projectors, there is no search for global loops necessary and thus our method remains stable on multiply connected geometries. The numerical results demonstrate the effectiveness of this approach for both canonical and realistic (multi-scale) problems.

math.NA↗

arXiv2015

On the Hierarchical Preconditioning of the Combined Field Integral Equation

This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory.

math.NA↗