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On the Adaptive Cross Approximation for the Magnetic Field Integral Equation | IEEE Journals & Magazine | IEEE Xplore

On the Adaptive Cross Approximation for the Magnetic Field Integral Equation


Abstract:

We present an adaptive cross approximation (ACA) strategy for the magnetic field integral equation (MFIE), where an application of the standard ACA strategy can suffer fr...Show More

Abstract:

We present an adaptive cross approximation (ACA) strategy for the magnetic field integral equation (MFIE), where an application of the standard ACA strategy can suffer from early convergence, in particular, due to block-structured interaction matrices associated with well-separated domains of the expansion and testing functions. Our scheme relies on a combination of three pivoting strategies, where the active strategy is determined by a convergence criterion that extends the standard criterion with a mean-based random-sampling criterion; the random samples give rise to one of the pivoting strategies, while the other two are based on (standard) partial pivoting and a geometry-based pivoting. In contrast to other techniques, the purely algebraic nature and the quasi-linear complexity of the ACA for electrically small problems are maintained. Numerical results show the effectiveness of our approach.
Published in: IEEE Transactions on Antennas and Propagation ( Volume: 72, Issue: 12, December 2024)
Page(s): 9366 - 9377
Date of Publication: 24 October 2024

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I. Introduction

Surface integral equations, such as the electric field integral equation (EFIE), magnetic field integral equation (MFIE), or combined field integral equation (CFIE), are a popular choice to solve electromagnetic scattering and radiation problems. Standard discretizations via the boundary element method (BEM) lead, however, to dense system matrices, resulting in complexity for computing and storing system matrices as well as for the cost of a single matrix-vector product, where N is the number of unknowns. To reduce the computational complexity, a plethora of so-called fast methods has been developed, such as the fast multipole method (FMM) [1], multilevel fast multipole algorithm (MLFMA) [2], [3], the adaptive integral method (AIM) [4], panel clustering [5], multilevel matrix decomposition algorithm (MLMDA) [6], [7], or precorrected-FFT [8]. For electrically small problems, a popular choice is the adaptive cross approximation (ACA) compression of -matrices, first developed for scalar problems in [9] and later investigated for the EFIE [10], because it allows to purely algebraically construct and store -matrices [11], [12], [13] in complexity by suitably sampling the rows and columns of the matrices of the well-separated interactions. Lately, the ACA is also used in the context of -matrix representations [14], [15]. For an extensive discussion of low-rank matrix factorization methods, see [16].

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