Jackson W. Massey, Anton Menshov, Ali E. Yilmaz
J. W. Massey, A. Menshov, and A. Yilmaz, “An element-diagonal preconditioner for the volume electric-field integral equation,” in USNC-URSI. Radio Sci. Meet., Boulder, CO, San-Diego, CA, Jul. 2017.
Publication year: 2017


For bioelectromagnetic (BioEM) problems in the UHF band, the condition number of the method of moments (MoM) system increases with the frequency, resulting in a larger number of iterations required for convergence (∝ solve cost) (J. W. Massey, M.S. Thesis, The University of Texas at Austin, 2015). Preconditioners have been applied to improve the conditioning of the system, such as the iterative near-field preconditioner or sparse-approximate-inverse preconditioner (T. Malas and L. Gu ̈rel, SIAM J. Sci. Comput., 2009), but their benefits must be balanced with the added complexity of filling them and applying them at each iteration.

An element-diagonal preconditioner proposed in (J. Aronsson, M. Shafieipour, and V. Okhmatovski in 28th Annu. Review Progress Applied Comput. Electromagn., 2012) for RWG basis functions for the MoM solution of the combined-field integral equation (CFIE) is extended and applied to the volume electric-field integral equation (V-EFIE). The element-diagonal preconditioner for the V-EFIE is formed by filling the sparse matrix PED whose non-zero entries are located at [i, j] where the basis/testing functions fi and fj have at least one overlapping patch. Using this “near-field” definition, the number of non-zero entries is O(N), specifically < 7N for SWG basis functions and < 11N for volumetric rooftop basis functions, where N is the number of unknowns in the MoM system.

In this article, the impedance matrix-vector product, ZI, is accelerated using the adaptive integral method (AIM) (F. Wei and A E. Yılmaz, IEEE Trans. Antennas Propag., 2014). Results will be shown for problems in the Austin Benchmark Suite for Computational Bio- electromagnetics comparing the MoM system performance with a diagonal preconditioner (PL = diag(Z)) to the element-diagonal preconditioner implemented using either a sparse- direct inverse solution (PL = PED) or as a right-preconditioner (PR = PED) in an inner-outer framework for the FGMRES solver.