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, with a CPU-time scaling only slightly more than linearly with respect to the number of unknowns used to discretize the problem.

翻译：本文研究了二次最优传输问题在动态形式（即Benamou-Brenier公式）下的数值求解方法。采用内点法求解时，主要计算瓶颈源于牛顿-拉夫逊格式所产生的大型鞍点线性系统的求解。本文旨在设计高效预处理方法，通过迭代法求解这些线性系统。在提出的预处理方法中，我们引入了一种基于鞍点线性系统对偶舒尔补算子部分交换性的方法，并将其称为$\boldsymbol{B}\boldsymbol{B}$-预处理子。一系列数值实验表明，$\boldsymbol{B}\boldsymbol{B}$-预处理子是所有方法中效率最高的，其CPU时间随问题离散化未知量数量的增长仅略高于线性关系。