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.