As IC design scales into the deep submicron and nanometer regimes, electromagnetics-based analysis has become essential. While so-called computational electromagentics (CEM) has found many successful engineering applications, the performance of existing CEM techniques is still inadequate when tackling realistic IC design problems. The analysis and design of next-generation ICs using the most accurate EM-based models results in numerical problems of very large scale, requiring up to billions of parameters to describe them accurately. However, even the state-of-art techniques do not scale well when applied to matrices of large sizes encountered with the analysis and design of next-generation ICs. In this proposal, the PIs will address the problem of full-wave analysis and design for next-generation integrated circuits, considering numerical problems arising from both partial differential equation (PDE) based models and integral equation (IE) based models. The proposed solution techniques hinge on the observation that the matrices underlying the numerical problems or their inverses are ``sparse-banded,'' wherein the matrices parametrizing the models or their inverses have (either exactly or approximately) a sparse, block-banded structure. There exists a general mathematical framework one that includes sparse-banded matrices as a special case called the ``Hierarchical Matrix'' framework, which enables a highly compact representation and efficient numerical computation. The hierarchical matrix framework will form the basis of the techniques proposed for the solution of the underlying numerical problems. The techniques combine an appreciation of the physics underlying the problems with elegant results from matrix theory and sound computational principles. This combined with the increased availability of distributed computing resources offers another rich new avenue of research.
The philosophy underlying the proposed approach is that by combining advances in theory (i.e., understanding) with progress in optimization and numerical linear algebra (i.e., numerical computation), one can realize enormous advances in the state of the art in research. The PIs have considerable experience with incorporating this philosophy in their own educational efforts. The graduate students who participate in the proposed effort will be trained with a broad range of skills, in areas such as electromagnetics, numerical linear algebra, and parallel computing fundamentals. Undergraduate research projects will provide an integrated research experience to students from the sophomore through the senior years. The PIs have a history of collaboration, commitment to teaching, and fostering diversity in the workplace, and are thus well-positioned to involve a diverse population of students in the research and teaching activities envisioned in this proposal.
