科研之友
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摘要
We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from ((N+l-1)(l-1)) to l. (N + 1), where l is the number of marginal states and N the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to l. (N - 1) unknowns, and cures the insufficiency of the Monge ansatz; i.e., we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context N corresponds to the number of particles, motivating the interest in large N.
- 出版日期2018
