Efficient Preconditioners for Solving Dynamical Optimal Transport via Interior Point Methods

.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 to as the \(BB\)-preconditioner. A series of numerical tests show that the \(BB\)-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.Keywordsoptimal transportBenamou–Brenier formulationinterior-point methodssaddle point problemalgebraic multigrid methodspreconditionersMSC codes35Q9349M4165K1065F0865F50