A variational finite volume scheme for Wasserstein gradient flows

Cancès, Clément; Gallouët, Thierry; Todeschi, Gabriele

doi:10.1007/s00211-020-01153-9

Cited by 20 publications

(20 citation statements)

References 55 publications

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“…where ν K i ,K j is the unit normal to K i pointing from K i to K j . We refer to the recent work [9] for a variational interpretation of the upwind scheme, which is close to that we propose for the more general equation (1.7). Earlier results in this direction are contained in [21,38].…”

Section: Relation To the Numerical Finite-volume Upwind Schemesupporting

confidence: 55%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

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Section: Relation To the Numerical Finite-volume Upwind Schemesupporting

confidence: 55%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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“…In this case, the Monge-Kantorovich optimal transport cost T (µ, ν) is defined by (4) T (µ, ν) = inf π∈Π(µ,ν)…”

Section: Preliminaries and Motivationmentioning

confidence: 99%

“…The weighted negative homogeneous Sobolev norm has also been considered in connection to maximum mean discrepancy (MMD) which is a technique that can be used to compute a distance between point clouds, and has many applications in machine learning, see [1,19,20]. Authors have also considered this connection in papers focused on computing the W 2 metric, see [4,9,14]. Moreover, the negative homogeneous Sobolev norm has been considered in several applications to seismic image and image processing, see [8,10,31].…”

mentioning

confidence: 99%

On a linearization of quadratic Wasserstein distance

Greengard¹,

Hoskins²,

Marshall³

et al. 2022

Preprint

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This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W 2 . In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = −∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented.

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“…In particular, the continuity equation is discretized on the fine grid whereas the Hamilton-Jacobi equation on the coarse one. Using a discretization that preserves the monotonocity of the discrete Hamilton-Jacobi operator it is possible to show that the value zero for λ is optimal (see [8] for a problem closely related to 2.6), i.e. the discrete Hamilton-Jacobi equation can be saturated.…”

Section: 2mentioning

confidence: 99%

Computation of optimal transport with finite volumes

Natale

Todeschi

2021

ESAIM: M2AN

Self Cite

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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 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.

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A variational finite volume scheme for Wasserstein gradient flows

Cited by 20 publications

References 55 publications

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

On a linearization of quadratic Wasserstein distance

Computation of optimal transport with finite volumes

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