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

一种新的理论/计算方法，称为广义多时间偏微分方程(GeMPDE)，正在被研究。GeMPDE公式的主要优点是它从根本上实现、加快或极大地简化了物理和工程系统的理论和计算分析，这些系统的特征是动力变化的慢/快速率被分开许多个数量级。使GeMPDE比现有方法更具吸引力的区别特征和智力优点包括：它(在初步结果中证明)能够以完全的一般性和严密性(以前不可能)执行振荡系统的线性化交流分析；它使用多变量的相位和局部频率变量来产生理论上和计算上稳健的系统公式；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的稳健求解器的开发，该解算器自动适应包含振荡和非振荡分量的非线性系统，基于大信号GeMPDE的包络分析，以及应用于周期性强迫系统、耦合振荡器系统和非周期自治系统。在电子电路(PLL、SD调制器)、纳米技术设备(约瑟夫森结阵列和单电子隧道设备)和生物系统(放电神经元网络)上的直接应用/验证；以及使用开放源码方法开发和传播这项研究产生的原型代码。

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.