Analysis and stable inversions of standard quasi-Helmholtz decompositions

This work presents a novel analysis of Loop-Star and Loop-Tree quasi-Helmholtz decompositions. The spectral properties of Loop, Star, and Tree functions are investigated and linked to the conditioning of Helmholtz decomposed Electric, Magnetic, and Calderón Preconditioned Integral Equations. The analysis will explain and quantify the difference in performance of Loop-Star with respect to Loop-Tree decompositions often observed in literature. Moreover, the theoretical framework will provide a new, linear in complexity, semidirect algorithm to invert the Loop-Star quasi-Helmholtz decomposition and to regularize the Loop-Star decomposed Calderón Electric Field Integral Equation. Numerical results are presented to corroborate the theory and to show the effectiveness of the proposed algorithms.