Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914210627190784 |
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| author | Bréhier, Charles-Edouard Laurent, Adrien Busnot Debussche, Arnaud Vilmart, Gilles |
| author_facet | Bréhier, Charles-Edouard Laurent, Adrien Busnot Debussche, Arnaud Vilmart, Gilles |
| contents | For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs Bréhier, Charles-Edouard Laurent, Adrien Busnot Debussche, Arnaud Vilmart, Gilles Numerical Analysis Probability 60H35, 60H15, 37M25 For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach. |
| title | Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs |
| topic | Numerical Analysis Probability 60H35, 60H15, 37M25 |
| url | https://arxiv.org/abs/2512.17714 |