A Hierarchical Matrix Framework for Electromagnetics-Based Analysis and Design of Next Generation ICs

项目名称：Dan Jiao， Venkataramanan Balakrishnan和Chenk-Kok KohInst: Purdue university标题：下一代集成电路的基于电磁分析和设计的层次矩阵框架摘要随着集成电路设计扩展到深亚微米和纳米级别，基于电磁的分析变得至关重要。虽然所谓的计算电磁学（CEM）已经找到了许多成功的工程应用，但现有的CEM技术在解决实际的集成电路设计问题时仍然不足。使用最精确的基于em的模型分析和设计下一代集成电路会导致非常大规模的数值问题，需要多达数十亿个参数才能准确描述它们。然而，即使是最先进的技术在应用于分析和设计下一代集成电路时遇到的大尺寸矩阵时也不能很好地扩展。在本提案中，pi将解决下一代集成电路的全波分析和设计问题，考虑基于偏微分方程（PDE）模型和基于积分方程（IE）模型产生的数值问题。所提出的解决技术取决于观察到数值问题或其逆的矩阵是“稀疏带状的”，其中参数化模型或其逆的矩阵具有（精确或近似）稀疏的块带状结构。存在一种通用的数学框架，其中包括稀疏带状矩阵作为一种特殊情况，称为“层次矩阵”框架，它可以实现高度紧凑的表示和高效的数值计算。层次矩阵框架将构成为解决潜在数值问题而提出的技术的基础。这些技术结合了对问题背后的物理的欣赏，以及矩阵理论和可靠的计算原理的优雅结果。这与分布式计算资源的可用性增加相结合，提供了另一个丰富的新研究途径。所提出的方法背后的哲学是，通过将理论（即理解）的进步与优化和数值线性代数（即数值计算）的进步相结合，人们可以实现研究中最先进的巨大进步。pi在将这一理念融入他们自己的教育工作方面有相当多的经验。参与该计划的研究生将接受广泛的技能培训，包括电磁学、数值线性代数和并行计算基础。本科研究项目将为学生提供从大二到大四的综合研究经验。这些pi有着合作的历史，致力于教学，并在工作场所促进多样性，因此能够很好地让不同的学生参与到本提案所设想的研究和教学活动中。

Proposal ID: 0702567 PI name: Dan Jiao, Venkataramanan Balakrishnan, and Chenk-Kok KohInst: Purdue UniversityTitle:A hierarchical matrix framework for electromagnetics-based analysis and design of next generation ICsABSTRACTAs 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.