Numerical optimal transport is an important area of research, but most problems are too large and complex for easy computation. Because continuous transport problems are generally solved by conversion to either discrete or semi-discrete forms, I focused on methods for those two.I developed a discrete algorithm specifically for fast approximation with controlled error bounds: the general auction method. It works directly on real-valued transport problems, with guaranteed termination and \emph{a priori} error bounds.I also developed the boundary method for semi-discrete transport. It works on unaltered ground cost functions, rapidly identifying locations in the continuous space where transport destinations change. Because the method computes over region boundaries, rather than the entire continuous space, it reduces the effective dimension of the discretization.The general auction is the first relaxation method designed for compatibility with real-valued costs and weights. The boundary method is the first transport technique designed explicitly around the semi-discrete problem and the first to use the shift characterization to reduce dimensionality. No truly comparable methods exist.The general auction and boundary method are able to solve many transport problems that are intractable using other approaches. Even where other solution methods exist, my tests indicate that the general auction and boundary method outperform them.
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The boundary method and general auction for optimal mass transportation and Wasserstein distance computation
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