On the Adaptive Cross Approximation for the Magnetic Field Integral Equation

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.