Efficient preconditioners for solving dynamical optimal transport via interior point methods

TL;DR

The study addresses the numerical solution of dynamical optimal transport via the Benamou-Brenier formulation using interior-point methods, where solving Newton-derived saddle-point systems dominates compute time. It proposes a family of preconditioners, highlighting a novel BB-preconditioner that exploits partial commutation in the dual Schur complement to accelerate iterative solves. Numerical experiments show the BB-preconditioner is the most efficient among the proposed options and exhibits near-linear CPU-time scaling with problem size, albeit with some deterioration in the final interior-point steps. The work advances scalable solvers for large-scale dynamical OT, enabling high-resolution discretizations and practical applicability in large 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.