A variational finite volume scheme for Wasserstein gradient flows
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Numerische Mathematik
A variational finite volume scheme for Wasserstein gradient flows Clément Cancès1 · Thomas O. Gallouët2 · Gabriele Todeschi2 Received: 12 December 2019 / Revised: 19 March 2020 / Accepted: 18 September 2020 / Published online: 8 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou–Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. The scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. It can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that the scheme admits a unique solution whatever the convex energy involved in the continuous problem, and we prove its convergence in the case of the linear Fokker–Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile. Mathematics Subject Classification 49M29 · 35K65 · 65M08 · 65M12
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Clément Cancès [email protected] Thomas O. Gallouët [email protected] Gabriele Todeschi [email protected]
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Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France
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Inria, Project Team Mokaplan, Université Paris-Dauphine, PSL Research University, UMR CNRS 7534, Ceremade, France
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1 A strategy to approximate Wasserstein gradient flows 1.1 Generalities about Wasserstein gradient flows Given a convex and bounded open subset of Rd , a strictly convex and proper energy functional E : L 1 (; R+ ) → [0, +∞], and given an initial density ρ 0 ∈ L 1 (; R+ ) with finite energy, i.e. such that E(ρ 0 ) < +∞, we want to solve problems of the form: ⎧ δE ⎪ ⎨∂t − ∇ · (∇ δρ []) = 0 in Q T = × (0, T ), ∇ δδρE [] · n = 0 on T = ∂ × (0, T ), ⎪ ⎩ 0 (·, 0) = ρ in .
(1)
Equation 1 expresses the continuity equation for a time evolving density , starting from the initial condition ρ 0 , convected by the velocity field −∇ δδρE []. The mixed boundary condition the system is subjected to represents a no flux condition across the boundary of the domain for the mass: the total mass is therefore preserved. It is now well understood since the pioneering works of Otto [34,52,53] that equations of the form of (1) can be interpreted as the gradient flow in the Wasserstein space w.r.t. the energy E [2]. A gradient flow is an evolution stemming from an initial condition and evolving at each time following the steepest decreasing direction of a prescribed functional. Consider the space P() of nonnegative measures defined on the bounded and convex domain with prescrib
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