Approximation methods for quick evaluation of invariant manifolds during global optimization

The purpose of this paper is to explore potential methods for improving approximation methods of invariant manifolds. The ultimate goal is to produce a set of methods and algorithms that can then be used for rapid evaluation of the invariant manifolds within a global optimization framework. Implemented algorithms must be computationally efficient in terms of both memory storage and execution and provide reasonable accuracy. The techniques explored in this paper build around a baseline cubic convolution method and include analysis of energy projection, polar parameterization, marginalization (integration), linear, fourth-order, and a one-dimension optimal parameter exploration. The result is new insight into the development of efficient and accurate approximation methods, which tend to show that a mix of the above methods could be used to build a better method than baseline cubic convolution.