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Summary: MINIMIZING FLOWS FOR THE MONGE-KANTOROVICH PROBLEM
SIGURD ANGENENT, STEVEN HAKER, AND ALLEN TANNENBAUM
Abstract. In this work, we formulate a new minimizing
ow for the optimal mass trans-
port (Monge-Kantorovich) problem. We study certain properties of the
ow including weak
solutions as well as short and long-term existence. Optimal transport has found a number
of applications including econometrics,
uid dynamics, cosmology, image processing, auto-
matic control, transportation, statistical physics, shape optimization, expert systems, and
meteorology.
This paper has been submitted to SIAM Journal of Mathematical Analysis.
1. Introduction
In this paper, we derive a novel gradient descent
ow for the computation of the optimal
transport map (when it exists) in the Monge-Kantorovich framework. Besides being quite
useful for the eÆcient computation of the transport map, we believe that the
ow presented
here is quite interesting from the theoretical point of view as well. In the present work, we
undertake a study of some of its key properties.
The mass transport problem was rst formulated by Monge in 1781, and concerned nding
the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one
site to another. This problem was given a modern formulation in the work of Kantorovich
[14], and so is now known as the Monge{Kantorovich problem. We recall the formulation of
the Monge{Kantorovich problem for smooth densities and domains in Euclidean space. For
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