Loop-Star and Loop-Tree Decompositions: Analysis and Efficient Algorithms

A new analysis of the spectral properties of Loop-Star and Loop-Tree decompositions is presented in this work. The analysis will shed light on the behavior of these decompositions when used with regular operators such as the magnetic field and the Calderón preconditioned electric field integral operators. This work will explain the ill-conditioning problems reported in literature and will provide a family of efficient algorithms to solve the ill-conditioning and regularizing several Loop-Star/Tree decomposed equations of interest for applications. The theory will be corroborated by numerical results that will show the practical impact of the theoretical developments.