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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 va...
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... denote by í µí± í µí°¾,í µí¼ the Euclidean distance between the cell center x í µí°¾ and the midpoint of the edge í µí¼ ∈ Σ í µí°¾ . In Figure 1 the notation is exemplified for a triangular element. ...
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... take again as cell centers x í µí°¾ of the fine mesh the circumcenters of each cell í µí°¾. This construction is illustrated in Figure 1. Note that the partition obtained in this way is indeed admissible. ...
Context 3
... denote by í µí± í µí°¾,í µí¼ the Euclidean distance between the cell center x í µí°¾ and the midpoint of the edge í µí¼ ∈ Σ í µí°¾ . In Figure 1 the notation is exemplified for a triangular element. ...
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The method of brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitation...
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... Furthermore, each subproblem in RBCD is a small-size LP, allowing for flexible resolution choices. Interior-point methods and simplex methods are also revisited and enhanced [22,37,63,69]. In particular, the authors of [69] propose an interior point inspired algorithm with reduced subproblems to address large-scale OT. ...
The optimal transport (OT) problem can be reduced to a linear programming (LP) problem through discretization. In this paper, we introduced the random block coordinate descent (RBCD) methods to directly solve this LP problem. Our approach involves restricting the potentially large-scale optimization problem to small LP subproblems constructed via randomly chosen working sets. By using a random Gauss-Southwell-q rule to select these working sets, we equip the vanilla version of (RBCD0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {RBCD}}_0$$\end{document}) with almost sure convergence and a linear convergence rate to solve general standard LP problems. To further improve the efficiency of the (RBCD0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {RBCD}}_0$$\end{document}) method, we explore the special structure of constraints in the OT problems and leverage the theory of linear systems to propose several approaches for refining the random working set selection and accelerating the vanilla method. Inexact versions of the RBCD methods are also discussed. Our preliminary numerical experiments demonstrate that the accelerated random block coordinate descent (ARBCD) method compares well with other solvers including stabilized Sinkhorn’s algorithm when seeking solutions with relatively high accuracy, and offers the advantage of saving memory.
... The first is a reconstruction operator \bfitR \scrE : \BbbR N \scrT \prime \rightar \BbbR N \scrE , mapping a positive density \bfitrho defined on the cells of the coarse grid to a variable defined on the edges of the finer one. In [38], this reconstruction first lifts the density \bfitrho into the finer space via the operator \bfitJ and then averages the values of adjacent cells. The authors considered two types of averages, a weighted arithmetic mean and a weighted harmonic mean (the latter is not considered in this paper to avoid complicating the exposition). ...
... 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 \Omega , in order to discretize the density \rho and the potential \phi 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 \Omega , in order to discretize the density \rho and the potential \phi 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). ...
... Recently, inspired by the results in [13,21,29], Lavenant [39] proposed a general convergence framework for dynamic 2 without any assumptions on the regularity and the mesh size, which has been applied to the finite element discretization on the surface [40], the mixed finite element discretization [55], and the finite volume scheme [29]. We also mention the work [54] by Natale and Todeschi, where the authors observed numerical instabilities for the finite volume schemes in [28,29] and introduced a new scheme based on a two-level spatial discretization to overcome this issue. ...
A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures) was defined in [44] a` la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein-Fisher-Rao distance, thanks to the static formulation, we show that such an assumption can be removed.
... Recently, there has been a growing interest in using Interior Point Methods (IPMs) [17] in applications that involve optimal transport, in particular for very large scale instances of such problems, see e.g. [33,46,48]. ...
In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal-dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal-dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers.
... 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. ...
... Many methods have been designed for solving OT problems (Aurenhammer et al. 1998, Ling and Okada 2007, Merigot 2011, Cuturi 2013, Gottschlich and Schuhmacher 2014, Schmitzer 2016, Schrieber et al. 2017, Facca and Benzi 2021. A similar idea to the one proposed here, where a sparse version of IPM is used to solve optimal transport problems, was recently proposed in Wijesinghe and Chen (2022), for a particular subset of OT problems, and in Natale and Todeschi (2021) an IPM was applied to solve optimal transport problems involving finite volumes discretization. ...
... In [15] the problem has been discretized using finite volumes, following a "first discretize, then optimize" approach in order to preserve the variational structure, and solved using an Interior Point (IP) method [16,17]. The optimization problem is relaxed by adding a logarithmic barrier function scaled by a parameter µ > 0, that is, −µ 1 0 Ω log(ρ). ...
... From a computational point of view, the most demanding task in [15] is to solve a sequence of saddle point linear systems in the following form ...
... Our problem may in fact be seen as the vanishing viscosity limit of a particular mean-field game. However, their linear algebra approaches do not fit into the discretization scheme used in [15]. Moreover, they lose efficiency for vanishing viscosities, that is when the two problems are more similar. ...
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.
... This work is mainly based on the three published papers [32,104,105], which are respectively presented in Chapter 3, Chapter 4 and Chapter 2. The presentation of these works is not chronological but follows a conceptual order. The latter two works in particular had been here extended. ...
This thesis is devoted to the design of locally conservative and structure preserving schemes for Wasserstein gradient flows, i.e. steepest descent curves in the Wasserstein space. The time discretization is based on variational approaches that mimic at the discrete in time level the behavior of steepest descent curves. These discretizations involve the computation of the Wasserstein distance, an instance of optimal transport problem. The space discretization is based on Two-Point Flux Approximation (TPFA) finite volumes, a well-known methodology particularly suited for the discretization of partial differential equations that present a conservative structure. In order to preserve the variational structure at the discrete level, we follow a first discretize then optimize approach. We start by presenting TPFA discretizations for the Wasserstein distance based on the Benamou-Brenier dynamical formulation. We expose some stability issues related to these discetizations, propose a possible solution to overcome them and derive quantitative estimate on the convergence of the discrete model. To solve the discrete optimization problem, we introduce an interior point strategy. Then, we propose first and second order accurate schemes for Wasserstein gradient flows. At this level, to reduce the computational complexity, we use an implicit linearization of the Wasserstein distance. By taking adavantage of the monotonicity of the upwind reconstruction, we propose a first order scheme which can be efficiently solved with a Newton method and show its convergence towards distributional solutions of the Fokker-Planck equation. In order to higher the accuracy in space, we use a centered reconstruction, which requires a different optimization technique. We use again the interior point strategy for this purpose. Finally, we propose a modified variational BDF2 time discretization and prove its convergence towards Wasserstein gradient flows. Thanks to these new discretizations, we design a second order accurate scheme in both time and space. All our approaches are validated with several numerical results.
Discrete optimal transport problems give rise to very large linear programs (LPs) with a particular structure of the constraint matrix. In this paper, we present a hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for the LP originating from the Kantorovich optimal transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we propose a column generation–like technique to force all intermediate iterates to be as sparse as possible. The algorithm is implemented nearly matrix-free. Indeed, most of the computations avoid forming the huge matrices involved and solve the Newton system using only a much smaller Schur complement of the normal equations. We prove theoretical results about the sparsity pattern of the optimal solution, exploiting the graph structure of the underlying problem. We use these results to mix iterative and direct linear solvers efficiently in a way that avoids producing preconditioners or factorizations with excessive fill-in and at the same time guaranteeing a low number of conjugate gradient iterations. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory use. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method.
History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous.
Funding: F. Zanetti received funding from the University of Edinburgh, in the form of a PhD scholarship.
Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0184 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0184 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
