集成电路模拟中的数学方法研究

Modern integrated circuit systems are often described by large time-dependent differential equations. The fast simulation techniques of these systems can be regarded as one of the cornerstones in modern advanced industrials. Therefore, the research on mathematic theories and their applications for modern simulation tools is really meaningful and important. The transient simulation software in industrial applications, such as Electronic Design Automation (EDA) process, has always played an important role. The powerful simulation software usually needs reliable support of theoretical analysis. This project, which is in the framework of two key integrated circuit simulation methods of waveform relaxation and model order reduction, focuses on the mechanism of advanced mathematical methods to compute the transient solutions of modern sophisticated large-scale systems and the applications of the new algorithms in the engineering fields. Specifically, based on the waveform relaxation technique, we first study novel numerical approaches about the time and space decomposition (or parallel processing), for solving large or sophisticated differential systems, and provide rapid simulation strategies which may induce new crossover or hybrid algorithms. Second, we study novel approaches for the model order reduction technique for large input-output differential systems in the time-domain, and find convenient formulae for calculating the moments of systems. Moreover, the common principle for the reduced order systems will be also discussed. Based on the fast developments of modern computing equipments and simulation techniques, we believe that the research work in the project will not only bridge the gap between scientific computing and industrial applications but also enrich the study of computational mathematics in our country.

现代集成电路系统一般由大型时间相关微分方程所描述，其快速模拟技术是当代先进工业的基石之一，研究模拟方法的数学理论和应用是有重要意义的。瞬态模拟软件在工业应用领域,如在电子设计自动化过程中,始终扮演重要角色，功能强大的模拟软件需要可靠的计算理论支持。本项目在波形松弛和模型降阶两类关键集成电路模拟方法的框架下，研究现代复杂大型系统瞬态计算的先进方法的数学机理，以及这些新算法在工程领域中的运用。具体地，研究基于波形松弛技术的求解大型或复杂微分系统的时间与空间同时分解或并行处理的新型数值方法，以及研究一些可能诱发的新的交叉型或混合型算法的快速计算理论；研究基于模型降阶技术的大型微分输入输出系统的时间域新型方法，寻求快速计算系统矩的数学公式和探究系统降阶的普适原理。基于现代计算设备性能的不断提高和我国实用模拟技术的现实考虑，本项目研究可以在科学计算与工业应用之间建立桥梁，以丰富我国计算数学研究内容。

现代集成电路系统一般由大型时间相关微分方程所描述，其快速模拟技术是当代先进工业的基石之一，研究模拟方法的数学理论和应用是有重要意义的。瞬态模拟软件在工业应用领域，如在电子设计自动化过程中，始终扮演重要角色，功能强大的模拟软件需要可靠的计算理论支持。本项目在波形松弛和模型降阶两类关键集成电路模拟方法的框架下，研究现代复杂大型系统瞬态计算的先进方法的数学机理，以及这些新算法在工程领域中的运用。具体地，研究了基于波形松弛技术的求解大型或复杂微分系统的时间与空间同时分解或并行处理的新型数值方法，并且研究了一些能够诱发新的交叉型或混合型算法的快速计算理论；研究了基于模型降阶技术的大型微分输入输出系统的时间域新型方法，建立了快速计算系统矩的数学公式和研究了系统降阶的数学原理。本项目为瞬态模拟软件的开发提供了理论依据，在科学计算与工业应用之间建立了桥梁，并且丰富了我国计算数学研究内容。