2017/12/31 by Gero Friesecke, Friesecke, G., Daniela Vögler +1 · 1 citation
Engineering · Mathematics · #49J40 #49K30 #49S05 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Hydrocarbon exploration and reservoir analysis #Markov Chains and Monte Carlo Methods #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1801.00341
openalex publication_date 2017/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \tbinomN+ℓ-1ℓ-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.