The theory of Wasserstein gradient flows in the space of probability measures provides a powerful framework to study dissipative partial differential equations (PDE). It can be used to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium. However, many PDE are not gradient flows, and hence the theory is not immediately applicable. In this work we develop a straightforward entropy regularised splitting scheme for degenerate non-local non-gradient systems. The approach is composed of two main stages: first we split the dynamics into the conservative and dissipative forces, secondly we perturb the problem so that the diffusionis no longer singular and perform a weighted Wasserstein ``JKO type'' descent step. Entropic regularisation of optimal transport problems opens the way for efficient numerical methods for solving these gradient flows. We illustrate the generality of our work by providing a number of examples, including the Regularized Vlasov-Poisson-Fokker-Planck equation , to which our results applicable .
In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local , degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation .
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The theory of Wasserstein gradient flows in the space of probability measures provides a powerful framework to study dissipative partial differential equations (PDE). It can be used to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium. However, many PDE are not gradient flows, and hence the theory is not immediately applicable. In this work
In this paper,
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straightforward entropy regularised splitting scheme for degenerate
natural operator-splitting variational scheme for a general class of
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non-gradient systems. The approach is composed of two main stages: first we split the dynamics into the conservative and dissipative forces, secondly we perturb the problem so that the diffusionis no longer singular and perform a weighted Wasserstein ``JKO type'' descent step. Entropic regularisation of optimal transport problems opens the way for efficient numerical methods for solving these gradient flows. We
, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We
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Regularized
kinetic Fokker-Planck equation and the (regularized)
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, to which our results applicable