Computation of optimal transport with finite volumes

Abstract

We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.

Numerical solution of the $ L^1 $-optimal transport problem on surfaces

“…Nevertheless, the pair based on two grids described in [28,29] was found being able to remove the observed oscillations for the L 1 -OT problem in R 2 . Similar solution were adopted in [47] for the solution of the L 2 -OT problem, and in [21,37] for topology optimization problems. Thus, we employ the triangulation T h/2 (Γ h ) generated from T h (Γ) by conformally refining each flat triangle.…”

Efficient Preconditioners for Solving Dynamical Optimal Transport via Interior Point Methods

“…The problem can then be effectively solved by solving the relaxed optimality conditions via the Newton method, and the original unperturbed solution is retrieved by repeating this procedure while reducing the relaxation term to zero. Numerical experiments in [38] show that the total number of Newton iterations remains practically constant, independently of the number of unknowns.…”

“…While different strategies have been proposed to discretize the Benamou-Brenier formulation, most of these rely on staggered time discretization and a space discretization which may be based on finite differences [8,41], finite volumes [29,38,34,32] or finite elements [35,39,34]. Here, we focus on the framework considered in [38], where one uses finite volumes in space with a two-level discretization of the domain Ω, in order to discretize the density ρ and the potential ϕ separately, which alleviates some checkerboard instabilities that may appear when the same grid is used to discretize both variables (a strategy first used in [26,27] when dealing with optimal transport with unitary displacement cost given by the Euclidean distance).…”

“…While different strategies have been proposed to discretize the Benamou-Brenier formulation, most of these rely on staggered time discretization and a space discretization which may be based on finite differences [8,41], finite volumes [29,38,34,32] or finite elements [35,39,34]. Here, we focus on the framework considered in [38], where one uses finite volumes in space with a two-level discretization of the domain Ω, in order to discretize the density ρ and the potential ϕ separately, which alleviates some checkerboard instabilities that may appear when the same grid is used to discretize both variables (a strategy first used in [26,27] when dealing with optimal transport with unitary displacement cost given by the Euclidean distance). This choice is not restrictive since such a framework constitutes a generalization of the finite volume scheme studied in [29,34] (in which a single grid is used for density and potential), and it also contains as special cases the finite difference scheme in [41] (when a single Cartesian grid is used in space), or the discrete transport models on networks studied in [23] (by an appropriate reinterpretation of the space tessellation).…”

“…The computational cost per iteration of these methods is typically low but the total number of iterations strongly depends on the data and the tuning of the parameters, generally increasing with the problem size. In this paper we will consider instead a secondorder approach, an Interior Point (IP) method proposed in [38], where the optimization problem eq. ( 1) is relaxed by adding a logarithmic barrier function (we refer the reader to [51,14,30] for a broad introduction of IP methods).…”