Higher order approximation of nonlinear SPDEs with additive space-time white noise
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917412170891264 |
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| author | Djurdjevac, Ana Gerencsér, Máté Kremp, Helena |
| author_facet | Djurdjevac, Ana Gerencsér, Máté Kremp, Helena |
| contents | We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order approximation of nonlinear SPDEs with additive space-time white noise Djurdjevac, Ana Gerencsér, Máté Kremp, Helena Probability 60H15, 60H35 We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature. |
| title | Higher order approximation of nonlinear SPDEs with additive space-time white noise |
| topic | Probability 60H15, 60H35 |
| url | https://arxiv.org/abs/2406.03058 |