Solving high-dimensional Hamilton-Jacobi-Bellman equation with functional hierarchical tensor
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913915676393472 |
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| author | Tang, Xun Sheng, Nan Ying, Lexing |
| author_facet | Tang, Xun Sheng, Nan Ying, Lexing |
| contents | This work proposes a novel numerical scheme for solving the high-dimensional Hamilton-Jacobi-Bellman equation with a functional hierarchical tensor ansatz. We consider the setting of stochastic control, whereby one applies control to a particle under Brownian motion. In particular, the existence of diffusion presents a new challenge to conventional tensor network methods for deterministic optimal control. To overcome the difficulty, we use a general regression-based formulation where the loss term is the Bellman consistency error combined with a Sobolev-type penalization term. We propose two novel sketching-based subroutines for obtaining the tensor-network approximation to the action-value functions and the value functions, which greatly accelerate the convergence for the subsequent regression phase. We apply the proposed approach successfully to two challenging control problems with Ginzburg-Landau potential in 1D and 2D with 64 variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving high-dimensional Hamilton-Jacobi-Bellman equation with functional hierarchical tensor Tang, Xun Sheng, Nan Ying, Lexing Numerical Analysis This work proposes a novel numerical scheme for solving the high-dimensional Hamilton-Jacobi-Bellman equation with a functional hierarchical tensor ansatz. We consider the setting of stochastic control, whereby one applies control to a particle under Brownian motion. In particular, the existence of diffusion presents a new challenge to conventional tensor network methods for deterministic optimal control. To overcome the difficulty, we use a general regression-based formulation where the loss term is the Bellman consistency error combined with a Sobolev-type penalization term. We propose two novel sketching-based subroutines for obtaining the tensor-network approximation to the action-value functions and the value functions, which greatly accelerate the convergence for the subsequent regression phase. We apply the proposed approach successfully to two challenging control problems with Ginzburg-Landau potential in 1D and 2D with 64 variables. |
| title | Solving high-dimensional Hamilton-Jacobi-Bellman equation with functional hierarchical tensor |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2408.04209 |