A variational finite volume scheme for Wasserstein gradient flows

Cancès, Clément; Gallouët, Thierry; Todeschi, Gabriele

doi:10.48550/arxiv.1907.08305

Cited by 2 publications

(5 citation statements)

References 49 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 [7] 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 [17,30].…”

Section: Nonlocal Continuity Equation Let Us Setsupporting

confidence: 55%

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

Esposito,

Patacchini,

Schlichting

et al. 2019

Preprint

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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 IE). We develop the existence theory for the solutions of the NL 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 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result. ContentsNotation 1. Introduction 1.1. Graph setting with general interactions 1.2. General setting for vertices in Euclidean space 1.3. Relation to the numerical finite-volume upwind scheme 1.4. Comparison with other discrete metrics and gradient structures 1.5. Connections to data science 2. Nonlocal continuity equation and upwind transportation metric 2.1. Weight function 2.2. Action 2.3. Nonlocal continuity equation 2.4. Nonlocal upwind transportation quasi-metric 3. Nonlocal nonlocal-interaction equation 3.1. Identification of the gradient in Finsler geometry 3.2. Variational characterization for the nonlocal nonlocal-interaction equation 3.3. The chain rule and proof of Theorem 3.9 3.4. Stability and existence of weak solutions 3.5. Discussion of the local limit Appendix A. Minkowski norm of the underlying Finsler structure

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Section: Nonlocal Continuity Equation Let Us Setsupporting

confidence: 55%

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

Esposito,

Patacchini,

Schlichting

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…M , or equivalently, that the mobility M (h n ) is integrable and nonnegative, is necessary for the weighted Hilbert spaces to be well-defined. Analogous requirements on the mobility have arisen in recent work by Cancés, Gallouët, and Todeschi [5], in which they consider a fully-implicit time discretization, in the special case that M (h) = h + 1 and h > −1.…”

Section: With This We Can Now Define An Hmentioning

confidence: 80%

“…h gradient flows are 2-Wasserstein gradient flows, and the above method can be interpreted as a semi-implicit variant of the Jordan Kinderlehrer Otto (JKO) scheme [21], in which the Wasserstein distance is approximated by the corresponding weighted H −1 norm at the previous time step [5].…”

Section: With This We Can Now Define An Hmentioning

confidence: 99%

“…Our assumption that the reciprocal of the mobility is integrable is similar to analogous assumptions in recent work on weighted Hilbert space discretizations for 2-Wasserstein gradient flows. In particular, Cancés, Gallouët, and Todeschi [5] consider a fully implicit scheme for M (h) = h + 1 > 0 on a compact domain, which ensures 1/M (h) ∈ L ∞ , hence the reciprocal of the mobility is integrable.…”

Section: φ(Hmentioning

confidence: 99%

“…This approach is inspired by the classical minimizing movements scheme for gradient flows, known as the JKO scheme in the 2-Wasserstein context [2,21]. In this way, our numerical method can be seen as an extension of recent literature using minimizing movement schemes to simulate nonlinear PDEs as gradient flows on metric spaces; see [5][6][7]26] and the references therein. More generally, it builds on the well-known literature using implicit Euler time discretizations to simulate Hilbertian gradient flows, including the H −1 total variation flow mentioned in equation (1.3) above [23].…”

mentioning

confidence: 99%

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A Proximal-Gradient Algorithm for Crystal Surface Evolution

Craig¹,

Jian-guo²,

Lu³

et al. 2020

Preprint

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As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted H −1 norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.

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A variational finite volume scheme for Wasserstein gradient flows

Cited by 2 publications

References 49 publications

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

A Proximal-Gradient Algorithm for Crystal Surface Evolution

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