Global Terrain Methods for Chemical Process Simulation

摘要：机构：罗德岛大学提案编号：0113091本研究的目的是通过动态地揭示地形的最小二乘函数的本质特征来找到物理模型的所有相关解和奇异点。 中心思想来自于寻找智能的方法来上下移动最小二乘函数的景观。 研究的基础在于以下基本观察：驻点在两次连续可微下光滑连通;谷、脊、壁架等，为这种连通性提供了一个自然和有用的特征;山谷，山脊等，可以表征为一组水平曲线上的约束最小值的集合;牛顿矢量场的自然流动倾向于沿着这些独特的地形特征。这些观测和构造使用化学工程模型和数学基准来说明。基于利用这些观测结果的方法学被称为全局地形算法（GTA）。并且由连续的下坡序列组成（即，方程求解计算）和上坡运动（即，预测-校正计算）。 下坡运动到奇点或解决方案是建立使用可靠的，规范减少（复域）的信赖域方法。 另一方面，上坡运动必然是一个奇异点，并使用上坡牛顿式预测步骤结合间歇性校正步骤定义的相邻极值的梯度范数在当前水平集的最小二乘函数。 初始起始点是任意的，而定义从一个静止点到另一个静止点的运动的后续子问题的起始点是沿着适当确定的特征向量。 这些特征向量被计算或近似，提供关于谷、脊、壁架等的知识，为进一步的下坡或上坡运动提供良好的初始化，并且可以被认为是Sridhar和Lucia的特征分解（或鞍点理论）的推广。 碰撞的可行区域和切断的山谷和山脊的边界也被认为是。 由于不动点的连通性在问题求解过程中展开，有限的连通性被揭示出来，并被用来定义GTA的终止准则。 类似的思想也适用于化工过程优化，即用其他目标函数代替最小二乘函数。

ABSTRACTPI: Angelo Lucia Institution: University of Rhode IslandProposal Number: 0113091The aim of this research is to find all relevant solutions and singular points to physical models by dynamically uncovering the essential features of the terrain of the least-squares function. The central idea comes from finding intelligent ways of moving both up and down the landscape of the least-squares function. The foundation of the research rests on following fundamental observations:Stationary points are smoothly connected under twice continuous differentiability;Valleys, ridges, ledges, etc., provide a natural and useful characterization of this connectedness;Valleys, ridges, etc., can be characterized as a collection of constrained minima over a set of level curves; andThe natural flow of Newton-like vector fields tends to be along these distinct features of the landscape.These observations and conjectures are illustrated using both chemical engineering models and mathematical benchmarks.The methodology based on exploiting these observations results in algorithms called Global Terrain Algorithms (GTA) and consist of successive sequences of downhill (i.e., equation-solving computations) and uphill movements (i.e., predictor-corrector calculations). Downhill movement to either a singular point or solution is established using reliable, norm-reducing (complex domain) trust region methods. Uphill movement, on the other hand, is necessarily to a singular point and uses uphill Newton-like predictor steps combined with intermittent corrector steps defined in terms of neighboring extrema in the gradient norm on the current level set for the least-squares function. Initial starting points are arbitrary while starting points for subsequent subproblems defining movement from one stationary point to another are along appropriately determined eigenvectors. These eigenvectors are calculated or approximated, provide knowledge about valleys, ridges, ledges, etc., give good initiations for further downhill or uphill movement, and can be considered a generalization of the eigendecomposition (or saddle point theory) of Sridhar and Lucia. Collisions with boundaries of the feasible region and severed valleys and ridges are also considered. As the connectedness of stationary points unfolds during problem-solving, limited connectedness is revealed and used to define the termination criterion for the GTA. Similar ideas are applicable to chemical process optimization be replacing the least-squares function with some other objective function.