Section: New Results

Scaling Algorithms and OT

G. Peyré F-X. Vialard L. Chizat B. Schmitzer S. Di Marino

B. Schmitzer has developed a sparse solver based on entropic regularization and numerical methods to solve unbalanced optimal transport (developed by our team in 2015) have been proposed in [37]. The core of the method consists in using the entropy functional as a reguflarizer and a barrier method. This is a generalization of the Sinkhorn method that has been introduced recently by M. Cuturi in numerical optimal transport. One important contribution of this work is to give a unified formulation of unbalanced optimal transport that can address a whole range of possible metrics and encompasses different applications such as Karcher-Fréchet averages, gradient flows, multimarginal unbalanced optimal transport. These two works are essentially based on a log-domain stabilized formulation, an adaptive truncation of the kernel and a coarse-to-fine scheme. This allows to solve large problems where the regularization is almost negligible.

In particular, this scaling algorithm is applied in its gradient flow formulation in the unbalanced case to obtain accurate simulations of the Hele-Shaw model, which models the cancer tumor growth.