A Blob Method for Mean Field Control With Terminal Constraints
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908502538059776 |
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| author | Craig, Katy Elamvazhuthi, Karthik Lee, Harlin |
| author_facet | Craig, Katy Elamvazhuthi, Karthik Lee, Harlin |
| contents | In the present work, we develop a novel particle method for a general class of mean field control problems, with source and terminal constraints. Specific examples of the problems we consider include the dynamic formulation of the p-Wasserstein metric, optimal transport around an obstacle, and measure transport subject to acceleration controls. Unlike existing numerical approaches, our particle method is meshfree and does not require global knowledge of an underlying cost function or of the terminal constraint. A key feature of our approach is a novel way of enforcing the terminal constraint via a soft, nonlocal approximation, inspired by recent work on blob methods for diffusion equations. We prove convergence of our particle approximation to solutions of the continuum mean-field control problem in the sense of Gamma-convergence. A byproduct of our result is an extension of existing discrete-to-continuum convergence results for mean field control problems to more general state and measure costs, as arise when modeling transport around obstacles, and more general constraint sets, including controllable linear time invariant systems. Finally, we conclude by implementing our method numerically and using it to compute solutions the example problems discussed above. We conduct a detailed numerical investigation of the convergence properties of our method, as well as its behavior in sampling applications and for approximation of optimal transport maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Blob Method for Mean Field Control With Terminal Constraints Craig, Katy Elamvazhuthi, Karthik Lee, Harlin Optimization and Control Numerical Analysis Systems and Control Analysis of PDEs 35Q35, 35Q62, 35Q82, 65M12, 82C22, 93A16, 49M41, 49N80 In the present work, we develop a novel particle method for a general class of mean field control problems, with source and terminal constraints. Specific examples of the problems we consider include the dynamic formulation of the p-Wasserstein metric, optimal transport around an obstacle, and measure transport subject to acceleration controls. Unlike existing numerical approaches, our particle method is meshfree and does not require global knowledge of an underlying cost function or of the terminal constraint. A key feature of our approach is a novel way of enforcing the terminal constraint via a soft, nonlocal approximation, inspired by recent work on blob methods for diffusion equations. We prove convergence of our particle approximation to solutions of the continuum mean-field control problem in the sense of Gamma-convergence. A byproduct of our result is an extension of existing discrete-to-continuum convergence results for mean field control problems to more general state and measure costs, as arise when modeling transport around obstacles, and more general constraint sets, including controllable linear time invariant systems. Finally, we conclude by implementing our method numerically and using it to compute solutions the example problems discussed above. We conduct a detailed numerical investigation of the convergence properties of our method, as well as its behavior in sampling applications and for approximation of optimal transport maps. |
| title | A Blob Method for Mean Field Control With Terminal Constraints |
| topic | Optimization and Control Numerical Analysis Systems and Control Analysis of PDEs 35Q35, 35Q62, 35Q82, 65M12, 82C22, 93A16, 49M41, 49N80 |
| url | https://arxiv.org/abs/2402.10124 |