Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations
Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations
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DOI：
10.1007/s10884-018-9659-x
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发表时间：
2016-08
影响因子：
1.3
通讯作者：
S. Chow;L. Dieci;Wuchen Li;Haomin Zhou
中科院分区：
文献类型：
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作者：
S. Chow;L. Dieci;Wuchen Li;Haomin Zhou
我们提出了一种新的半离散化方案，通过利用概率密度空间中 2-Wasserstein 度量的梯度流结构来近似非线性 Fokker-Planck 方程。我们通过有限图离散化基础状态，并在离散概率空间中定义离散 2-Wasserstein 度量。基于这样的度量，我们引入了离散自由能的梯度流作为半离散化方案。我们证明该方案保持了自由能的耗散性，并以指数耗散率收敛到离散吉布斯测度。我们通过几个数值示例展示了这些属性。
We propose a new semi-discretization scheme to approximate nonlinear Fokker–Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric in the space of probability densities. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric in the discrete probability space. Based on such metric, we introduce a gradient flow of the discrete free energy as semi discretization scheme. We prove that the scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential dissipation rate. We exhibit these properties on several numerical examples.