Global Terrain Methods for Chemical Process Simulation

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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.