A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866917462266609664 |
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| author | Bal, Guillaume Ma, Shengbo Zhang, Su Zhang, Zhiwen |
| author_facet | Bal, Guillaume Ma, Shengbo Zhang, Su Zhang, Zhiwen |
| contents | Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03853 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations Bal, Guillaume Ma, Shengbo Zhang, Su Zhang, Zhiwen Numerical Analysis 65C30, 60H35, 41A10, 65M70 Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost. |
| title | A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations |
| topic | Numerical Analysis 65C30, 60H35, 41A10, 65M70 |
| url | https://arxiv.org/abs/2605.03853 |