Research

• O(N) Integral Equation Direct Solvers:
The unparallel benefits of Integral Equation(IE) methods are eclipsed by the computational costs required to carry-out an EM analysis based on them. If the discretization of IE methods results in N degrees of freedom then the current state-of-the-art iterative methods like Fast Multipole Method(FMM) utilize N_rhs N_iter O(NlogN) computational resources. We have been able to achieve this optimal O(N) complexity in direct solvers for circuit problems which can invert a single dense VIE matrix of size 1 Million by 1 Million in 4 minutes and less than 5 GB of Memory on a single processor running on 3 GHz clock speed. This work has been as selected as the finalist for the Best Student Paper competitions of the top two conferences of EM community namely IMS and APS, to be held in June and July, 2014 respectively. For dynamic cases, where the size of the problem no longer remains in the sub-wavelength regime, for the first time, O(N) complexity has been achieved for iterative and O(NlogN) complexity has been achieved for direct inverse based VIE analysis.

• First-Principles VIE Formulation for Simultaneous Circuit-Scattering Analysis:
Sensitive microwave, RF and VLSI circuits in devices operating in a battle-field, communication satellite and highly radioactive environment are exposed to severe ambient environment. In the prevailing state-of-art methods and commercial tools, there is always a missing link between full-wave circuit and scattering analysis, thus a failure to deal with such problems. Exploiting all the inherent features which a Volume Integral Equation (VIE) has to offer, we have developed a novel first-principles based VIE formulation which essentially bridges the gap between full-wave circuit and scattering analysis thus enabling a simultaneous circuit-scattering analysis. Our paper detailing this project was ranked FIRST in the frequency domain session of the International Microwave Symposium (IMS) 2013.

• O(1) Solution to Low- and High-frequency Breakdowns in IE Methods:
The incorrect solutions (called as breakdown) obtained from the IE analysis of electrically small problems has always remained a mystery. Some credited this to improper discretization techniques while others devised sub-optimal trial and error techniques to partially address this issue. Teamed with other co-authors, the root-cause of this issue was identified to be the finite machine precision. A theoretically rigorous closed form expression for the inverse of an IE system matrix was derived using eigenvalue analysis. This expression also revealed that the breakdown is not a low-frequency phenomenon but can also occur for highly multi-scale examples with more than 7 orders of magnitude different feature details. An O(1) solution to these problems is also proposed and was highly appreciated by the Antenna & Propagation Community in the 2013 IEEE International Symposium on Antennas and Propagation where a part of this project was awarded with an HONORABLE MENTION AWARD.