Numerical approximations for partially observed optimal control of stochastic partial differential equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912302327922688 |
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| author | Bao, Feng Cao, Yanzhao Qian, Hongjiang |
| author_facet | Bao, Feng Cao, Yanzhao Qian, Hongjiang |
| contents | In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in finite-dimensional space $\rr^d$. We begin by establishing stochastic maximum principles (SMPs) for such problems, where the system state is driven by a cylindrical Wiener process. The corresponding adjoint equations are characterized by backward stochastic partial differential equations. We then develop numerical algorithms to solve the partially observed optimal control. Our approach combines the stochastic gradient descent method, guided by the SMP, with a particle filtering algorithm to estimate the conditional distributions of the state of the system. Finally, we demonstrate the effectiveness of our proposed algorithm through numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_00381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical approximations for partially observed optimal control of stochastic partial differential equations Bao, Feng Cao, Yanzhao Qian, Hongjiang Optimization and Control Numerical Analysis 93E11, 60G35, 65K10, 60H15, 60H10 In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in finite-dimensional space $\rr^d$. We begin by establishing stochastic maximum principles (SMPs) for such problems, where the system state is driven by a cylindrical Wiener process. The corresponding adjoint equations are characterized by backward stochastic partial differential equations. We then develop numerical algorithms to solve the partially observed optimal control. Our approach combines the stochastic gradient descent method, guided by the SMP, with a particle filtering algorithm to estimate the conditional distributions of the state of the system. Finally, we demonstrate the effectiveness of our proposed algorithm through numerical experiments. |
| title | Numerical approximations for partially observed optimal control of stochastic partial differential equations |
| topic | Optimization and Control Numerical Analysis 93E11, 60G35, 65K10, 60H15, 60H10 |
| url | https://arxiv.org/abs/2504.00381 |