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Benamou-Brenier and other continuous numerical methods

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Optimal Transport for Applied Mathematicians

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 87))

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Abstract

In this chapter we present some numerical methods to solve optimal transport problems. The most famous method is for sure the one due to J.-D. Benamou and Y. Brenier, which transforms the problem into a tractable convex variational problem in dimension d + 1. We describe it strongly using the theory about Wasserstein geodesics (rather than finding the map, this method finds the geodesic curve μ t ). Two other classical continuous methods are presented: the Angenent-Hacker-Tannenbaum method based on the fact that the optimal maps should be a gradient and that removing non-gradient parts decreases the energy and the Loeper-Rapetti method based on the resolution of the Monge-Ampère equation. Both require smooth and nonvanishing densities and special domains to handle boundary data (a rectangle or, better, a torus). In the discussion section, we briefly present some discrete and semi-discrete methods, based on linear programming and Voronoi cells.

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Notes

  1. 1.

    We choose to denote by ϱ the interpolating measures, in order to stress the “continuous” flavor of this method (typically, ϱ is a density, and μ a generic measure).

  2. 2.

    This is typical of 1-homogeneous functionals (the fact that the result is independent of the reference measure).

  3. 3.

    The Uzawa algorithm is actually more general than this: it can handle inequality constraints of the form \(\varphi ^{i}(x) \leq 0\), with the Lagrangian \(L(x,\lambda ) = f(x) +\sum _{i}\lambda ^{i}\varphi ^{i}(x)\) and \(\lambda \in (\mathbb{R}_{+})^{k}\), but we prefer to stick to the equality constraints for simplicity of exposition.

  4. 4.

    We sometimes hear that the simplex method is exponential: this is a worst-case estimate. Also, other solvers for linear programming problems exist, for instance, interior point methods, and their computational cost can be way better than exponential. However, in practice they are all too expensive.

  5. 5.

    We switch back to the language, of economists, who prefer to maximize utility rather than minimizing costs.

  6. 6.

    Observe that this assumption on the values u ij being integer, corresponding to integer costs on each edge in the network interpretation, appears quite often and simplifies the complexity of other algorithms as well. For instance, in [245] a polynomial bound on the cost of a network simplex algorithm is given under this assumption, obtaining \(O(\min (kh\log (kC),kh^{2}\log k)\) for a network with k nodes, h edges, and maximal cost equal to C.

  7. 7.

    See also Example 1.6 in [177] for a first genesis of these ideas.

  8. 8.

    The computation of the gradient, and its continuity, could have been simplified by looking at the subdifferential; see [24].

  9. 9.

    The pictures in the next pages have been kindly provided by B. Lévy and are produced with the same algorithms as in [209].

  10. 10.

    This is the semidiscrete method coming from this idea, implemented in [114]. There is also a continuous counterpart, i.e., a PDE on the Kantorovich potentials \(\psi _{\varepsilon }\). It has been studied from the theoretical point of view in [66], where Bonnotte proved a nontrivial well-posedness result, based on the application of the Nash-Moser implicit function theorem around \(\varepsilon = 0\) (once we are on \(\varepsilon > 0\), the equation is more regular). Later, the same author also started numerical implementations of the time discretization of this equation in [65]. Note that his continuous method recalls in some aspects both the AHT flow described in Section 6.2 (as it starts from the Knothe map, is continuous in time but discretized for numerical purposes, and evolves by imposing prescribed image measure at every time) and the LR Newton’s iterations of Section 6.3 (as the underlying PDE is based on a linearization of the Monge-Ampère equation).

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Santambrogio, F. (2015). Benamou-Brenier and other continuous numerical methods. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_6

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