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），正在研究。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，这些系统的特点是慢/快的动态变化速率被许多数量级分开。使GeMPDE比现有方法更具吸引力的不同特征和智力优点包括：它能够（在初步结果中证明）执行线性化。振荡系统的交流分析具有充分的普遍性和严谨性（以前不可能）；它使用多变量相位和局部频率变量产生理论和计算鲁棒的系统公式；它使用分离的人工时间变量将不同的变化率清晰地分离出来。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的鲁棒解算器的开发，该解算器可自动适应包含振荡和非振荡成分的非线性系统，基于大信号GeMPDE的包络分析，以及周期性强制系统，耦合振荡器系统和非周期自治系统的应用。更广泛的影响包括将教育和研究部分紧密联系起来；电子电路（锁相环，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）的直接应用/验证；以及使用开源方法来开发和传播由本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.