Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914093245399040 |
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| author | Xu, Yewei Li, Qin |
| author_facet | Xu, Yewei Li, Qin |
| contents | Wasserstein gradient flows have become a central tool for optimization problems over probability measures. A natural numerical approach is forward-Euler time discretization. We show, however, that even in the simple case where the energy functional is the Kullback-Leibler (KL) divergence against a smooth target density, forward-Euler can fail dramatically: the scheme does not converge to the gradient flow, despite the fact that the first variation $\nabla\frac{δF}{δρ}$ remains formally well defined at every step. We identify the root cause as a loss of regularity induced by the discretization, and prove that a suitable regularization of the functional restores the necessary smoothness, making forward-Euler a viable solver that converges in discrete time to the global minimizer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization Xu, Yewei Li, Qin Numerical Analysis Optimization and Control 65J20, 35Q49 Wasserstein gradient flows have become a central tool for optimization problems over probability measures. A natural numerical approach is forward-Euler time discretization. We show, however, that even in the simple case where the energy functional is the Kullback-Leibler (KL) divergence against a smooth target density, forward-Euler can fail dramatically: the scheme does not converge to the gradient flow, despite the fact that the first variation $\nabla\frac{δF}{δρ}$ remains formally well defined at every step. We identify the root cause as a loss of regularity induced by the discretization, and prove that a suitable regularization of the functional restores the necessary smoothness, making forward-Euler a viable solver that converges in discrete time to the global minimizer. |
| title | Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization |
| topic | Numerical Analysis Optimization and Control 65J20, 35Q49 |
| url | https://arxiv.org/abs/2509.13260 |