Feasibility in Chemical Process Design and Simulation

摘要：科学和工程中许多问题的数学模型的可行性极限是由光滑的转向点或分支点来表征的，这些分支点是一些相关雅可比矩阵的奇异点。这些拐点和分岔点，以及解本身，可以是实值或复值的，许多获得它们的数值方法可以经历周期、非周期或发散的行为。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过程设计和模拟的可行性限制方面的作用和效用：围绕使用(a)非奇异隐式复合方程系统来定义转弯和分岔点，(b)复域信赖域方法来计算这些转弯和分岔点，以及(c)由近似隐式方程的一些一阶偏导数组成的混合方法进行数值研究。2. 发展必要的数学框架，以理解复域或扩展信赖域的收敛和收敛速度，以计算可行性的极限。在科学和工程应用中，问题规范总是会产生可行或不可行的模型方程组。可行规范总是在双范数中产生零值最小值，而不可行规范则对应于范数中的非零值最小值和奇异点。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是重要的。PI计划：1；对使用(a)扩展信赖域方法来识别和计算可行或不可行的条件以及(b)使用复域拟牛顿更新来加速收敛到奇异点在零空间中建立二次模型进行数值研究。2. 研究上述方法的收敛性和收敛率。。

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.