Efficient preconditioners for solving dynamical optimal transport via interior point methods

Enrico Facca (1) , Gabriele Todeschi (2, 3) , Andrea Natale (4, 1) , Michele Benzi (5, 6)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 ISTerre - Institut des Sciences de la Terre
4 LMO - Laboratoire de Mathématiques d'Orsay
5 Department of Mathematics and Computer Science [Emory University]
6 SNS - Scuola Normale Superiore di Pisa

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 BB-preconditioner. A series of numerical tests show that the BB-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.

Keywords: Optimal transport; Benamou-Brenier formulation; Saddle point problem; Algebraic multigrid methods; Preconditioners

Domains: Numerical Analysis [math.NA]; Optimization and Control [math.OC]

Enrico Facca, Gabriele Todeschi, Andrea Natale, Michele Benzi. Efficient preconditioners for solving dynamical optimal transport via interior point methods. 2022. ⟨hal-03766668v1⟩