Optimization is still an active area of research and an important computational tool in many branches of science and engineering. Often times, there are dramatic differences between the small and large-scale geometry of a given objective function that are important in the optimization process. These geometric differences can be inherent or they can be due to the presence of multiple length scales and can give rise to objective function surfaces that have many, many local stationary points but only one global optimum (sometimes called rough objective function landscapes or the many minima problem). While quadratic Taylor series expansions represent reasonable models of local geometry, they are generally inadequate for describing large-scale geometry, which is usually strongly non-quadratic. Recent observations show that many different objective functions such as potential energies, free energies, least squares functions, and others have large-scale geometry that can be modeled by non-quadratic exponential funnel functions. When built correctly, these funnels are non-convex, possess a unique and easily calculated minimum, and have the capability of describing a wide variety of geometric shapes. This suggests that funnels have the potential for providing computationally useful non-quadratic approximations to the large-scale geometry of many objective functions that can be used in a multi-scale global optimization methodology.

This work takes a completely different view of the development of optimization techniques for multi-scale global optimization with regard to applications where there is a small number of physically important optima but many stationary points and where it is undesirable or computationally prohibitive to find all stationary points. Of particular interest are problems in the phase transitions associated with wax precipitation. The proposed multi-scale methodology alternates between local and global optimization. Recently developed terrain methods are used for ‘small' scale optimization to provide reliable snapshots of the local geometry (i.e., local minima, saddle points, singular points, objective function values, parts of pathways along valleys and ridges, eigen-information, etc.). Funnel approximations of the large-scale geometry, on the other hand, are constructed using averaged information from integrals along terrain paths and novel least square interpolating formulae. Two-way communication between small and large scales is used for accumulating information and updating approximations to large-scale geometry. Small and large multi-scale molecular modeling problems, including Lennard-Jones clusters and more detailed molecular models for n-alkanes, as well as many geometric illustrations are used to elucidate key points and, in particular, to show that funnel geometry is common in molecular modeling and that the terrain/funneling method can be used to intelligently find physically important stationary points on rough energy landscapes.