Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917049073139712 |
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| author | Chaudhary, Abhishek |
| author_facet | Chaudhary, Abhishek |
| contents | We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin's maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By a new technical approach we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus. For the practical computation we develop a gradient-descent algorithm based on artificial iterates that employs an exact computation for the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation Chaudhary, Abhishek Optimization and Control Numerical Analysis Analysis of PDEs Probability We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin's maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By a new technical approach we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus. For the practical computation we develop a gradient-descent algorithm based on artificial iterates that employs an exact computation for the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency. |
| title | Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2510.24876 |