Nonlocal Approximation of Slow and Fast Diffusion
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909160250015744 |
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| author | Craig, Katy Jacobs, Matt Turanova, Olga |
| author_facet | Craig, Katy Jacobs, Matt Turanova, Olga |
| contents | Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method in the slow, linear, and fast diffusion regimes. A key ingredient of our approach is a novel technique for using the 2-Wasserstein and dual Sobolev gradient flow structures of the diffusion equations to recover the duality relation characterizing the pressure in the nonlocal-to-local limit. Due to the general class of internal energy densities that our method is able to handle, a byproduct of our result is a novel particle method for sampling a wide range of probability measures, which extends classical approaches based on the Fokker-Planck equation beyond the log-concave setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11438 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlocal Approximation of Slow and Fast Diffusion Craig, Katy Jacobs, Matt Turanova, Olga Analysis of PDEs Probability 35A15, 35Q70, 35Q35, 35Q62, 82C22 Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method in the slow, linear, and fast diffusion regimes. A key ingredient of our approach is a novel technique for using the 2-Wasserstein and dual Sobolev gradient flow structures of the diffusion equations to recover the duality relation characterizing the pressure in the nonlocal-to-local limit. Due to the general class of internal energy densities that our method is able to handle, a byproduct of our result is a novel particle method for sampling a wide range of probability measures, which extends classical approaches based on the Fokker-Planck equation beyond the log-concave setting. |
| title | Nonlocal Approximation of Slow and Fast Diffusion |
| topic | Analysis of PDEs Probability 35A15, 35Q70, 35Q35, 35Q62, 82C22 |
| url | https://arxiv.org/abs/2312.11438 |