14. Convergence of a fully discrete variational scheme for a thin-film equation

Osberger, Horst; Matthes, Daniel

doi:10.1515/9783110430417-015

Cited by 10 publications

(11 citation statements)

References 30 publications

(73 reference statements)

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“…Another common approach is to leverage structural similarities between (3) and equations from fluid dynamics to develop particle methods [14,27,30,36,43,48,57,60,88,92]. Until recently, the key limitation of such methods has been developing approaches to incorporate diffusion.…”

Section: Classical Numerical Pde Methodsmentioning

confidence: 99%

Primal Dual Methods for Wasserstein Gradient Flows

et al. 2021

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Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

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Section: Classical Numerical Pde Methodsmentioning

confidence: 99%

Primal Dual Methods for Wasserstein Gradient Flows

et al. 2021

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“…Another common approach is to leverage structural similarities between (1.3) and equations from fluid dynamics to develop particle methods [14,25,28,34,41,46,55,58,81,85]. Until recently, the key limitation of such methods has been developing approaches to incorporate diffusion.…”

Section: Introductionmentioning

confidence: 99%

Primal dual methods for Wasserstein gradient flows

Carrillo

Craig

Wang

et al. 2019

Preprint

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Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Using the JKO scheme, along with the Benamou-Brenier dynamical characterization of the Wasserstein distance, we reduce computing the solution of these evolutionary PDEs to solving a sequence of fully discrete minimization problems, with strictly convex objective function and linear constraint. We compute the minimizer of these fully discrete problems by applying a recent, provably convergent primal dual splitting scheme for three operators [100]. By leveraging the PDE's underlying variational structure, our method overcomes traditional stability issues arising from the strong nonlinearity and degeneracy, and it is also naturally positivity preserving and entropy decreasing. Furthermore, by transforming the traditional linear equality constraint, as has appeared in previous work, into a linear inequality constraint, our method converges in fewer iterations without sacrificing any accuracy. Remarkably, our method is also massively parallelizable and thus very efficient in resolving high dimensional problems. We prove that minimizers of the fully discrete problem converge to minimizers of the continuum JKO problem as the discretization is refined, and in the process, we recover convergence results for existing numerical methods for computing Wasserstein geodesics [88]. Finally, we conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our numerical method.

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“…Our method avoids these computationally intensive procedures by the approximation of the energy in the discrete setting. Finally, note that gradient-flow-based Lagrangian methods for higher-order, drift diffusion and Fokker-Planck equations have recently been proposed in [29,30,28,22].…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of a Particle Method for Gradient Flows

Carrillo,

Huang,

Patacchini

et al. 2015

Preprint

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We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via nonoverlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.with superlinear growth at infinity and H(0) = 0. Furthermore, h(λ) := λ d H λ −d is convex and non-increasing on (0, ∞).

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14. Convergence of a fully discrete variational scheme for a thin-film equation

Cited by 10 publications

References 30 publications

Primal Dual Methods for Wasserstein Gradient Flows

Primal Dual Methods for Wasserstein Gradient Flows

Primal dual methods for Wasserstein gradient flows

Numerical Study of a Particle Method for Gradient Flows

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