Multigrid approach to solving the long transportation problem on a regular grid in cost space

Abstract

Multigrid methods were developed to solve partial differential equations. Research has shown that these methods are applicable to a broader range of problems. This thesis investigates the application of multigrid techniques to minimal cost flow problems, specifically the long transportation problem. This research shows that multigrid techniques can be successfully applied to large-scale long transportation problems posed on a three- dimensional, regular grid in cost space. A V-cycle algorithm is developed for the long transportation problem. Analogies to the multigrid components of restriction, interpolation and relaxation are detailed. Performance of the algorithm is discussed, and computational cost is analyzed. Future research is likely to include the development of more sophisticated restriction and interpolation schemes to provide integer-valued flows, and the development of a method to map an irregularly spaced problem to a regular grid, and to map the regular grid solution back to the original problem domain.