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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.12167 |
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| _version_ | 1866910060496551936 |
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| author | Bensoussan, Alain Nguyen, Thien P. B. Tran, Minh-Binh Tu, Son N. T. |
| author_facet | Bensoussan, Alain Nguyen, Thien P. B. Tran, Minh-Binh Tu, Son N. T. |
| contents | We propose a splitting approach to solve the second-order Hamilton--Jacobi equation, reducing it to a heat step and a purely first-order step. The latter is implemented using a gradient value policy iteration algorithm, enabling efficient characteristic-based machine learning methods. We establish convergence rates for the splitting method. In particular, with $h$ the splitting step, the $L^\infty$ error is bounded between $\mathcal{O}(h)$ and $\mathcal{O}(h^{1/5})$ for Lipschitz data, improving to $\mathcal{O}(h^{1/3})$ for semiconcave data. In the periodic setting, we also obtain an $L^1$ error of order $\mathcal{O}(h^{1/2})$. For the first-order step, we provide a weighted $L^2$ error analysis that shows exponential convergence. Each iteration solves linear characteristic equations and learns the value function by minimizing a weighted value gradient loss. The approach yields stable and accurate numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operator Splitting, Policy Iteration, and Machine Learning for Stochastic Optimal Control Bensoussan, Alain Nguyen, Thien P. B. Tran, Minh-Binh Tu, Son N. T. Optimization and Control Numerical Analysis Analysis of PDEs 35D40, 70H20, 49L25, 65M15, 49H25, 35L60 We propose a splitting approach to solve the second-order Hamilton--Jacobi equation, reducing it to a heat step and a purely first-order step. The latter is implemented using a gradient value policy iteration algorithm, enabling efficient characteristic-based machine learning methods. We establish convergence rates for the splitting method. In particular, with $h$ the splitting step, the $L^\infty$ error is bounded between $\mathcal{O}(h)$ and $\mathcal{O}(h^{1/5})$ for Lipschitz data, improving to $\mathcal{O}(h^{1/3})$ for semiconcave data. In the periodic setting, we also obtain an $L^1$ error of order $\mathcal{O}(h^{1/2})$. For the first-order step, we provide a weighted $L^2$ error analysis that shows exponential convergence. Each iteration solves linear characteristic equations and learns the value function by minimizing a weighted value gradient loss. The approach yields stable and accurate numerical results. |
| title | Operator Splitting, Policy Iteration, and Machine Learning for Stochastic Optimal Control |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs 35D40, 70H20, 49L25, 65M15, 49H25, 35L60 |
| url | https://arxiv.org/abs/2603.12167 |