# A fast approach to optimal transport: The back-and-forth method

## TL;DR

This paper introduces a fast iterative method for solving optimal transportation problems with convex costs, achieving efficient computation on large spatial grids with exponential convergence.

## Contribution

The paper proposes a novel back-and-forth iterative approach that significantly reduces computational complexity for large-scale optimal transport problems with convex costs.

## Key findings

- Able to solve large grid problems in minutes
- Uses linear storage space and near-linear time per iteration
- Achieves exponential convergence rate

## Abstract

We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n points, we compute the optimal map using O(n) storage space and O(n log(n)) operations per iteration, with an approximately exponential convergence rate. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096x4096 and 384x384x384 in a matter of minutes.