# Efficient preconditioners for solving dynamical optimal transport via interior point methods

arXiv: https://arxiv.org/abs/2209.00315 · HTML: https://mlsanity.com/paper/2209.00315/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2022.

## Built on by (most cited first, 2 of 2)

- [Learning Discrete Diffusion of Graphs via Free-Energy Gradient Flows](https://mlsanity.com/paper/2604.11311/) (2026)
- [Generalized Schr"odinger Bridge on Graphs](https://mlsanity.com/paper/2602.04675/) (2026)

## Abstract (arXiv)

In this paper we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer as boldsymbolBboldsymbolB-preconditioner. A series of numerical tests show that the boldsymbolBboldsymbolB-preconditioner is the most efficient among those presented, despite a performance deterioration in the last steps of the interior point method. It is in fact the only one having a CPU-time that scales only slightly worse than linearly with respect to the number of unknowns used to discretize the problem.