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.

翻译：本文研究二次最优传输问题在动态形式下的数值求解，即所谓的Benamou-Brenier公式。当采用内点法求解时，主要计算瓶颈源于牛顿-拉夫逊迭代中产生的大型鞍点线性系统的求解。本文主要目的在于设计高效的预处理子，以便通过迭代法求解这些线性系统。在所提出的预处理子中，我们引入了一种基于双鞍点线性系统对偶舒尔补算子部分交换的方法，称之为$\boldsymbol{B}\boldsymbol{B}$-预处理子。一系列数值试验表明，尽管在内点法最后几步中性能有所下降，$\boldsymbol{B}\boldsymbol{B}$-预处理子是所提方案中效率最高的。事实上，它是唯一一个计算时间随离散问题未知数增长仅略高于线性关系的预处理子。