Feasibility in Chemical Process Design and Simulation

摘要- Lucia - 9409158 科学和工程中许多问题的数学模型的可行性极限都具有光滑转向点或分叉点的特征，这些光滑转向点或分叉点是某些相关雅可比矩阵的奇异点。 这些转向点和分歧点以及解本身可以是真实的或复值的，并且获得它们的许多数值方法可能会经历周期性、非周期性或发散行为。 因此，PI计划通过以下方式更好地理解奇点在定义和确定化学工艺设计和模拟可行性极限方面的作用和效用： 1. 进行数值研究，围绕使用（a）非奇异隐式复合方程系统，以确定转向点和分叉点，（B）在复杂的区域信赖域方法计算这些转向点和分叉点，（c）混合方法，包括近似的隐式方程的一些第一偏导数。 2.开发必要的数学框架，以了解复杂域或扩展信赖域方法的收敛性和收敛速度，以计算可行性极限。 在科学和工程应用中的问题规格通常会产生可行或不可行的模型方程组。 可行的规范总是在两个范数中产生零值极小值，而不可行的规范对应于范数中的非零值极小值和奇点。 关于可行性或不可行性的信息，无论是先验的还是伴随的，对于设计和仿真中的方程求解任务都很重要。 PI计划： 1.进行数值研究：（a）使用扩展信赖域方法确定和计算可行性或不可行性条件;（B）使用复域拟牛顿更新法加速收敛到奇点，以在零空间建立二次模型。 2.研究上述方法的收敛性和收敛速度。

Abstract - Lucia - 9409158 Limits of feasibility for mathematical models of many problems in science and engineering are characterized by smooth turning or bifurcation points, which are the singular points of some associated Jacobian matrix. These turning and bifurcation points, as well as the solutions themselves, can be real or complex-valued and many numerical methods for obtaining them can experience periodic, aperiodic or divergent behavior. Accordingly, the PI plans to build a better understanding of the role and utility of singularity in defining and determining limits of feasibility in chemical process design and simulation by: 1. Conducting numerical studies that revolve around the use of (a) nonsingular implicit composite equation systems to define turning and bifurcation points, (b) trust region methods in the complex domain to compute these turning and bifurcation points, and (c) hybrid methods consisting of approximating some of the first partial derivatives of the implicit equations. 2. Developing the necessary mathematical framework for understanding the convergence and rate of convergence of complex domain or extended trust region methods for calculating limits of feasibility. Problem specifications in science and engineering applications always give rise to either feasible or infeasible sets of model equations. Feasible specifications always give rise to zero-valued minima in the two-norm, while infeasible ones correspond to nonzero-valued minima and singular points in the norm. Information regarding feasibility or infeasiblity, whether a priori or concomitant, is important to equation-solving tasks in design and simulation. The PI plans to: 1. Conduct numerical investigations on the use of (a) extended trust region methods for identifying and calculating conditions of feasibility or infeasibility and (b) accelerating convergence to singular points using complex domain quasi-Newton updates to build a quadratic model in the null space. 2. Study t he convergence and rates of convergence of the methods developed above.