Fisher information regularization schemes for Wasserstein gradient flows
Fisher information regularization schemes for Wasserstein gradient flows
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DOI：
10.1016/j.jcp.2020.109449
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Wuchen Li;Jianfeng Lu;Li Wang
中科院分区：
文献类型：
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作者：
Wuchen Li;Jianfeng Lu;Li Wang
我们提出了一个计算Wasserstein梯度流的变分格式。该方案建立在Jordan-Kinderlehrer-Otto框架与Benamou-Brenier的动态制定的二次Wasserstein度量，并增加了正则化的Fisher信息。这个正则化可以从能量分裂的角度推导出来，并且与薛定谔桥问题密切相关。它改善了变分问题的凸性，并自动保持解的非负性。因此，它允许我们应用序列二次规划来解决子优化问题。我们进一步节省计算成本，表明没有额外的时间插值是需要在底层的动态制定的Wasserstein-2度量，因此，问题的维度大大降低。给出了多孔介质方程、非线性Fokker-Planck方程、聚集扩散方程和Derrida-Lebowitz-Speer-Spohn方程的数值算例。这些例子证明了所提出的方案的简单性和稳定性。
We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan–Kinderlehrer–Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fisher information. This regularization can be derived in terms of energy splitting and is closely related to the Schrödinger bridge problem. It improves the convexity of the variational problem and automatically preserves the non-negativity of the solution. As a result, it allows us to apply sequential quadratic programming to solve the sub-optimization problem. We further save the computational cost by showing that no additional time interpolation is needed in the underlying dynamic formulation of the Wasserstein-2 metric, and therefore, the dimension of the problem is vastly reduced. Several numerical examples, including porous media equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and Derrida-Lebowitz-Speer-Spohn equation, are provided. These examples demonstrate the simplicity and stableness of the proposed scheme.