Efficient preconditioners for solving dynamical optimal transport via interior point methods

TL;DR

This paper develops and tests efficient preconditioners, especially a novel $oldsymbol{B}oldsymbol{B}$-preconditioner, to improve the solution of large saddle point systems in dynamical optimal transport problems solved via interior point methods.

Contribution

The paper introduces a new preconditioner based on partial operator commutation that significantly enhances the efficiency of solving saddle point systems in dynamical optimal transport.

Findings

The $oldsymbol{B}oldsymbol{B}$-preconditioner outperforms other methods in efficiency.

The proposed preconditioner scales nearly linearly with problem size.

Numerical tests confirm the effectiveness of the new preconditioner in large-scale problems.

Abstract

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 $\boldsymbol{B}\boldsymbol{B}$-preconditioner. A series of numerical tests show that the $\boldsymbol{B}\boldsymbol{B}$-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.