Adaptive integral method with fast Gaussian gridding

Abstract

The solution of integral equations pertinent to the analysis of electromagnetic scattering problems via the method of moments (MOM) calls for the solution of linear systems of equations of order N, where N is the number of basis functions used in the expansion of the scatterer current. The classical (i.e. nonaccelerated) iterative solution of adequately preconditioned MOM system requires memory and CPU resources of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). The adaptive integral method (AIM) [1] reduces these memory and CPU requirements to O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> )+γO(N) and O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> logN), respectively; these estimates are for surface scatterers. Here, the first and the second terms in the memory requirement estimate are due to the storage of matrices and vectors needed for fast Fourier transform (FFT) and the AIM mapping coefficients, respectively. When using conventional AIM mapping methods, i.e. those based on far-field [2, 3] and moment matching [1], γ scales as O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), where M is the number of AIM mapping coefficients per source/basis function along each dimension. When large M are needed to ensure high accuracies, the γO(N) mapping component of the scheme’s memory requirements often overtakes its O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> ) FFT component, eventually prohibiting the use of AIM for analyzing electrically large problems. In this work, we propose a new technique that incorporates fast Gaussian gridding [4, 5], a recently developed scheme for computing type I nonuniform FFTs (NUFFTs) with low memory requirements, with AIM. In the proposed technique γ is of O(1), that is, it no longer scales with O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) as in classical AIM schemes.