Generalized artificial-time PDE formulations and computational techniques for multi-rate systems

多速率系统的广义人工时间 PDE 公式和计算技术

• 批准号: 0515227
• 负责人: Jaijeet Roychowdhury
• 金额: $ 20万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0515227&HistoricalAwards=false
• 关键词: Generalized; artificial; time; PDE; formulations

A novel theoretical/computational approach, termed the Generalized Multitime Partial Differential Equation (GeMPDE), is being investigated. The chief advantage of the GeMPDE formulation is that it fundamentally enables, speeds up or greatly simplifies theoretical and computational analysis of physical and engineering systems, which feature slow/fast rates of dynamical change separated by many orders of magnitude.Differentiating features and intellectual merits that make the GeMPDE considerably more attractive than existing approaches include: its ability (demonstrated in preliminary results) to perform linearized .AC analysis of oscillatory systems with full generality and rigor (not possible previously); its use of multivariate phase and local frequency variables to yield theoretically and computationally robust system formulations; and its clean separation of disparate rates of variation using separated artificial time variables.Core theoretical and computational tasks being explored include: stochastic GeMPDE analysis, investigation of specialized GeMPDE forms targeting specific application domains, the development of robust GeMPDE-based solvers that adapt automatically to nonlinear systems containing both oscillatory and non-oscillatory components, large signal GeMPDE-based envelope analysis, and applications to periodically forced systems, systems of coupled oscillators and non-periodic autonomous systems.Broader impacts include a strongly linking educational and research components; direct application/validation on electronic circuits (PLLs, SD modulators), nano technological devices (arrays of Josephson junctions and single-electron tunneling devices), and biological systems (networks of firing neurons); and the use of open source methodologies for developing and disseminating prototype code resulting from this research.

一种新的理论/计算方法，称为广义多时间偏微分方程（GeMPDE），正在研究中。GeMPDE公式的主要优点是它从根本上实现、加速或大大简化了物理和工程系统的理论和计算分析，这些系统的特点是动态变化的慢/快速率被许多数量级分开。使GeMPDE比现有方法更具吸引力的区别特征和智力优势包括：能力（初步结果表明）进行线性化。交流分析的振荡系统具有充分的一般性和严谨性（以前是不可能的）;它使用多变量相位和局部频率变量来产生理论上和计算上鲁棒的系统公式;以及使用分离的人工时间变量对不同变化率的清晰分离。正在探索的核心理论和计算任务包括：随机GeMPDE分析，针对特定应用领域的专门GeMPDE形式的研究，基于GeMPDE的鲁棒求解器的开发，其自动适应包含振荡和非振荡分量的非线性系统，基于GeMPDE的大信号包络分析，以及在周期性强迫系统、耦合振荡器系统和非周期性自治系统中的应用。电子电路的直接应用/验证（PLL、SD调制器）、纳米技术器件（约瑟夫森结和单电子隧穿器件阵列）和生物系统（神经元放电网络）;以及使用开源方法来开发和传播这项研究产生的原型代码。